www.tnschools.co.in
E R K HSS –ERUMIYAMPATTI
Page 1
+2 STUDY MATERIALS
www.tnschools.co.in
md;ghh;e;j khzt , khztpah;fNs ePq;fs; gbg;gjw;Fk; gbj;jit kwf;fhky; ,Ug;gjw;Fk; rpy mwpTiufs;… gbg;gpy; ve;j msT Mh;tk; mjpfk; cs;sNjh mNj mstpw;F epidthw;wYk; mjpfkhFk;. tFg;gpy; ghlj;jpy; tUk; re;Njfq;fis Nfs;tpfs; Nfl;L clDf;Fld; njspjy; Ntz;Lk;. Nfs;tpfs; Nfl;gjdhy; epidthw;wy; mjpfhpf;Fk;. njhlh;e;J gy kzp Neuk; gbf;Fk;nghOJ miu kzpNeuj;jpw;F xU Kiw 5 epkplq;fs; gbj;jij epidj;Jg; ghh;f;f Ntz;Lk;. ,t;thW gbg;gjhy; epidthw;wy; mjpfhpf;Fk;. Gj;Jzh;r;rpAld; ,Uf;Fk; Neuq;fspy; fbdkhd ghlq;fisAk;, Nrhh;thf ,Uf;Fk; Neuq;fspy; vspjhd ghlq;fisAk; gbj;jhy; kdjpy; ed;F gjpAk;. ,uz;L fbdkhd ghlq;fSf;F eLtpy; vspikahd ghlj;ijg; gbg;gjhy; kdr;Nrhh;T tuhJ. mjdhy; epidthw;wy; mjpfhpf;Fk;. fbdkhd ghlq;fisr; rpwpJ cuf;fg; gbj;J kdjpy; epidj;J mjid vOjpg; ghh;j;jhy; epidthw;wy; mjpfhpf;Fk;. ,uT J}q;fr; nry;tjw;F Kd;G kpfTk; fbdkhd ghlj;ij ,UKiw ed;F thrpj;Jtpl;Lr; nry;yTk;. ,jdhy; kWehs; ,g;ghlj;ijg; gbg;gjw;F vspjhf ,Uf;Fk;. Nfhgk;, gak;, gjl;lk; epidthw;wiyf; Fiwf;Fk; vd;gjdhy; ,itfisj; jtph;f;f Ntz;Lk;. cq;fSf;Ff; fz;lk; TV tbtpy; cs;sJ. mij Kiwahff; ifahSq;fs;. ,y;iynad;why; gbg;gpdpy; Mh;tj;ijf; Fiwj;J cq;fs; epidthw;wiyAk; Fiwj;J tpLk;. fhyj;jpd; mUik fUjp, gbg;gjw;fhd xU thuj;jpw;Fhpa fhy ml;ltiz Nghl;L mjidf; filg;gpbf;f Ntz;Lk;. rj;jhd czT (ntz;ilf;fha;, fPiu tiffs;, Kis fl;ba gaph;fs;, thiog; gok;, ghy;, Kl;il) kpjkhd clw;gapw;;rp, kpjkhd J}f;fk;, jpdrhp gpuhj;jid Mfpait epidthw;wiy mjpfhpf;Fk;. kQ;rs; fhkhiy, ilg;gha;L, jl;lk;ik Nghd;w Neha;fSf;Fj; jLg;G+rp NghLjy;, ntsp ,lq;fspy; czT mUe;Jtijj; jtph;j;jy;, < nkha;j;j gz;lq;fisr; rhg;gplhjpUj;jy;, ey;y nfhjpf;f itj;J, Fspu itj;j jz;zPiu mUe;Jjy; Mfpad cly; MNuhf;fpaj;ij Nkk;gLj;Jk;. ek; cly; MNuhf;fpak; Nkk;gl;lhy; epidthw;wYk; Nkk;gLk;. nja;tj;jhd; MfhnjdpDk; Kaw;rp jd; nka;tUj;jf; $ypjUk;.
E R K HSS –ERUMIYAMPATTI
Page 2
jpUf;Fws;
+2 STUDY MATERIALS
COME BOOK – TEN MARKS
www.tnschools.co.in
[16 Number Of Ten Mark Questions To Be Asked For Full Test] -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 1. MATRICES AND DETERMINANTS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 1.1. (3)
3 3 4 Find the adjoint of the matrix A 2 3 4 0 1 1
(6)
3 3 4 Find the inverse of the matrix A 2 3 4 and verify that A 3 A 1 . 0 1 1
(7)
Show that the adjoint of
(9)
If
(3) (4) (5)
(4)
A
2 2 1 2 1 3 1 2
1 2 2
1 2 2 A 2 1 2 2 2 1
and verify the result A ( adj A ) ( adj A ) A A .
is 3 A T .
, prove that A 1 A T .
EXERCISE 1.2. Solve by matrix inversion method each of the following system of linear equations: x y z 9, 2 x 5 y 7 z 52 , 2 x y z 0 . Solve by matrix inversion method each of the following system of linear equations: 2 x y z 7, 3 x y 5 z 13 , x y z 5 . Solve by matrix inversion method each of the following system of linear equations: x 3 y 8 z 10 0, 3 x y 4, 2 x 5 y 6 z 13 . EXERCISE 1.4 Solve the following non-homogeneous system of linear equations determinant method: x y z 4 ; x y z 2 ; 2x y z 1
(5)
Solve the following non-homogeneous system of linear equations determinant method: 2x y z 4 ;
(6)
x y 2z 0 ; 3 x 2y 3z 4
Solve the following non-homogeneous system of linear equations determinant method: 3 x y z 2 ; 2 x y 2z 6 ; 2 x y 2z 2
(7)
Solve the following non-homogeneous system of linear equations determinant method: x 2 y z 6 ; 3 x 3 y z 3 ; 2 x y 2z 3
(8)
Solve the following non-homogeneous system of linear equations determinant method: 2 x y z 2 ; 6 x 3 y 3z 6 ; 4 x 2y 2z 4
(9)
Solve the following non-homogeneous system of linear equations determinant method: 1 x
2 y
1 z
1 ;
2 x
4 y
1
5 ;
z
3 x
2 y
2
0
z
(10) A small seminar hall can hold 100 chairs. Three different colours ( red, blue and green) of chairs are available. The cost of a red chair is Rs. 240, cost of blue chair is Rs. 260 and the cost of a green chair is Rs. 300. The total cost of chair is Rs. 25,000. Find atleast 3 different solution of the number of chairs in each colour to be purchased. EXERCISE 1.5. (1) Examine the consistency of the following system of equations. If it is consistent then solve the same: (i) solve : 4 x 3 y 6 z 25 ; x 5 y 7 z 13 ; 2 x 9 y z 1 (ii) solve : x 3 y 8 z 10 ; 3 x y 4 z 0 ; 2 x 5 y 6 z 13 0
E R K HSS –ERUMIYAMPATTI
Page 3
+2 STUDY MATERIALS
www.tnschools.co.in x y z 1 ; 2 x 2 y 2z 2 ; 3 x 3 y 3 z 3
(v) solve :
Discuss the solutions of the system of equations x y z 2 , 2 x y 2 z values of (3) For what values of k, the system of equations kx y z 1, x ky z 1, have (i) unique solution (ii) more than one solution (iii) no solution Example 1.4 :
(2)
If
1 A 1 2
1 2 1
1 3 3
,verify
A adj A
adj
2,
x y 4 z 2 for all
x y kz 1
A A A I 3
Example 1.8 : Solve by matrix inversion method 2 x y 3 z 9 , x y z 6 , x y z 2 Example 1.18: Solve the following non-homogeneous equations of three unknowns. (1) x 2 y z 7 ; 2 x y 2 z 4 ; x y 2 z 1 x y 2z 6 ; 3 x y z 2 ; 4 x 2y z 8 (2) x y 2 z 4 ; 2 x 2 y 4 z 8 ; 3 x 3 y 6 z 12 (4) Example 1.19: A bag contains 3 types of coins namely Re.1, Rs. 2 and Rs. 5. There are 30 coins amounting to Rs. 100 in total. Find the number of coins in each category. Example 1.21: Solve: x y 2 z 0 ; 3 x 2 y z 0 ; 2 x y z 0 Example 1.22: Verify whether the given system of equations is consistent. If it is consistent, solve them. 2 x 5 y 7 z 52 , x y z 9 , 2 x y z 0
Example 1.23 : Examine the consistency of the equations. 2 x 3 y 7 z 5 , 3 x y 3 z 13 , 2 x 19 y 47 z 32 Example 1.24: Show that the equations x y z 6 , x 2 y 3 z 14 , x 4 y 7 z 30 are consistent and solve them. Example 1.25: Verify whether the given system of equations is consistent. If it is consistent, solve them: x y z 5 , x y z 5 , 2 x 2 y 2 z 10
Example 1.26: Investigate for what values of , the simultaneous equations x y z 6 , x 2 y 3 z 10 , x 2 y z have (i) no solution (ii) a unique solution and (iii) an infinite number of solutions. Example 1.27: Solve the following homogeneous linear equations. x 2 y 5 z 0 , 3 x 4 y 6 z 0 , x y z 0 Example 1.28: For what value of the equations x y 3 z 0 , 4 x 3 y z 0 , 2 x y 2 z 0 have a (i) trivial solution, (ii) non-trivial solution. -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 2. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST) ------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 2.2 (4) Prove that cos A B cos A cos B sin A sin B EXERCISE 2.4 (7) Prove that sin A B sin A cos B cos A sin B . EXERCISE 2.5 (5)
If a 2 i 3 j k ,
b 2i 5k ,
E R K HSS –ERUMIYAMPATTI
c j 3k
Verify that a b c a . c b a . b
Page 4
c
+2 STUDY MATERIALS
www.tnschools.co.in
(12) Verify ;
a b c d a
where a i j k ; b 2 i k ;
b d c a b c d
c 2 i j k
d i j 2k
EXERCISE 2.7 (3)
Show that the lines
x 1
y 1
1
1
z
x2
and
3
y 1
1
2
z 1 1
intersect and find their point of intersection.
EXERCISE 2.8. (7)
Find the vector and Cartesian equation of the plane containing the line
x2
2 x 1
and parallel to the line
y 1
3
(8)
z 1
2
y2
3
z 1 3
.
1
Find the vector and Cartesian equation of the plane through the point (1,3,2) and parallel to the lines x 1
2
y2 1
z3
x2
and
3
y 1
1
z2
2
2
(9)
Find the vector and Cartesian equation to the plane through the point (-1,3,2) and perpendicular to the planes x 2 y 2 z 5 and 3 x y 2 z 8 . (10) Find the vector and Cartesian equation of the plane passing through the points A ( 1,-2,3) and B (-1,2,-1) x2
and is parallel to the line
y 1
2
z 1
3
4
(11) Find the vector and Cartesian equation of the plane through the points (1,2,3) and (2,3,1) perpendicular to the plane 3 x 2 y 4 z 5 0 . (12) Find the vector and Cartesian equation of the plane containing the line
x2 2
y2 3
z 1 2
and
passing through the point (-1,1,-1). (13) Find the vector and Cartesian equation of the plane passing through points with position vectors 3i 4 j 2 k , 2 i 2 j k
and 7 i k . (14) Derive the equation of the plane in the intercept form.( both in vector and cartesion form ) Example 2.16: Altitudes of a triangle are concurrent – prove by vector method. Example 2.17: Prove that cos A B cos A cos B sin A sin B Example 2.29: Prove that sin A B sin A cos B cos A sin B Example 2.44: Show that the lines
x 1
y 1
3
1
z 1
and
0
x4 2
y 0
z 1 3
intersect and hence find the point of
intersection. Example 2.50: Find the vector and Cartesian equations of the plane through the point (2,-1,-3) and parallel to the lines. x2 3
y 1 2
z3 4
and
x 1 2
y 1 3
z2
.
2
Example 2.51: Find the vector and Cartesian equations of the plane passing through the points (-1,1,1) and (1,-1,1) and perpendicular to the plane x 2 y 2 z 5 Example 2.52: Find the vector and Cartesian equations of the plane passing through the points (2,2,-1) , (3,4,2) and (7,0,
E R K HSS –ERUMIYAMPATTI
Page 5
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 3.COMPLEX NUMBERS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 3.2 (8)
(i) If P represents the variable complex number z. Find the locus of P, if
2z 1 Im 2 iz 1
(iii) If P represents the variable complex number z. Find the locus of P, if
z 1 Re 1 zi
(v) If P represents the variable complex number z. Find the locus of P, if
z 1 arg z3 2
EXERCISE 3.4 (5)
If
and
are the roots of the equation
y n y n
q
(6)
If
and
n 1
sin n n
sin
are the roots of
x
2
x
2
2 px p
2
q
2
0
tan
and
q
Show that
y p
; n N
2x 4 0
Prove that n n i 2 n 1 sin
n
;n N
and deduct
3
(8)
9
If
(10) If (i)
x
9
1
2 cos
and
x
show that (i)
a cos 2 i sin 2 , 1
abc
b cos 2 i sin 2
2 cos
(ii)
a b
m
y
n
2
c
abc
abc
y
n
x
m
2 cos
c cos 2 i sin 2
and 2
x
2
m
n
(ii)
x
m
y
n
y
n
x
m
2 i sin
m
n
Prove that
2 cos 2
EXERCISE 3.5. (1)
Find all the values of the following:
(4)
(iii) Solve:
(5)
3 i x
4
x
2 3 3
x
2
x 1 0
Find all the values of
1 3 i 2 2
3
and hence prove that the product of the values is 1.
4
Example 3.11: (i) If P represents the variable complex number z, find the locus of P
z 1 Re 1 zi
(ii) If P represents the variable complex number z, find the locus of P
z 1 arg z 1 3
Example 3.22: If and are the roots of Show that
y
n
x
y
2
and
2x 2 0
n
sin n sin
n
cot y 1 ,
;n N
Example 3.23: Solve the equation x 9 x 5 x 4 1 0 Example 3.24: Solve the equation x 7 x 4 x 3 1 0
E R K HSS –ERUMIYAMPATTI
Page 6
+2 STUDY MATERIALS
www.tnschools.co.in
Example 3.25:
2
Find all the values of 3i 3 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 4. ANALYTICAL GEOMETRY (THREE QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 4.1 (2) Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the following parabolas and hence sketch their graphs. (v) x 2 6 x 12 y 3 0 (iv) y 2 8 x 6 y 1 0 (5) A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road way is 5 mts below the lowest point of the cable. If an extra support is provided across the cable 30 mts above the ground level find the length of the support if the height of the pillars are 55 mts. EXERCISE 4.2. Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram: (ii) x 2 4 y 2 8 x 16 y 68 0 (iv) 16 x 2 9 y 2 32 x 36 y 92 (7) A kho-kho player in a practice session while running realizes that the sum of the distances from the two kho-kho pples from him is always 8m. Find the equation of the path traced by him if the distance between the poles is 6m. (8) A satellite is traveling around the earth in an elliptical orbit having the earth at a focus and of eccentricity 1/2. The shortest distance that the satellite gets to the earth is 400 kms. Find the longest distance that the satellite gets from the earth. (9) The orbit of the planet mercury around the sun is in elliptical shape with sun at a focus. The semi-major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find (i) how close the mercury gets to sun? (ii) the greatest possible distance between mercury and sun. (10) The arch of a bridge is in the shape of a semi –ellipse having a horizontal span of 40ft and 16ft high at the centre. How high is the arch, 9ft from the right or left of the centre. EXERCISE 4.3 (5) Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams. (iii) x 2 4 y 2 6 x 16 y 11 0 (iv) x 2 3 y 2 6 x 6 y 18 0 EXERCISE 4.4 (5) Prove that the line 5 x 12 y 9 touches the hyperbola x 2 9 y 2 9 and find its point of contact. (6)
x y4 0
(6)
Show that the line of contact.
is a tangent to the ellipse
(2)
Find the equation of the hyperbola if (ii) its asymptotes are parallel to x 2 y 12 it passes through (2,0).
x
2
3 y
2
12 .
Find the co-ordinates of the point
EXERCISE 4.5. 0
and
x 2 y 8 0,
(2,4) is the centre of the hyperbola and
EXERCISE 4.6. (3) Find the equation of the rectangular hyperbola which has for one of its asymptotes the line x 2 y 5 0 and passes through the points (6,0) and (-3,0). Example 4.7: Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectumfor the following parabolas and hence draw their graphs. (iv) y 2 8 x 6 y 9 0 (v) x 2 2 x 8 y 17 0
E R K HSS –ERUMIYAMPATTI
Page 7
+2 STUDY MATERIALS
www.tnschools.co.in
Example 4.8: The girder of a railway bridge is in the parabolic form with span 100ft. and the highest point on the arch is 10ft, above the bridge. Find the height of the bridge at 10ft, to the left or right from the midpoint of the bridge. Example 4.10: On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4mts when it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the starting point. Find the angle of projection. Example 4.12: Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground? Example 4.13: A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of
3
radians with the axis of the orbit. Find (i) the equation of the comet‟s orbit (ii) how close does the comet nearer to the sun?( Take the orbit as open rightward ). Example 4.14: A cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers if 1500ft, the points of support of the cable on the towers are 200ft above the road way and the lowest point on the circle is 70ft above the roadway. Find the vertical distance to the cable from a pole whose height is 122 ft. Example 4.31: Find the eccentricity, centre, foci, vertices of the following ellipses: (iv) 36 x 2 4 y 2 72 x 32 y 44 0 Example 4.32: An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is 20 feet. How wide is the arch at a height of 10 feet above the base? Example 4.33: The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12ft. Example 4.35: A ladder of length 15m moves with its ends always touching the vertical wall and the horizontal floor. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. Example 4.56: Find the eccentricity, centre, foci and vertices of the hyperbola 9 x 2 16 y 2 18 x 64 y 199 0 and also trace the curve. Example 4.57: Find the eccentricity, centre, foci, and vertices of the following hyperbola and draw the diagram : 9 x 16 y 36 x 32 y 164 0 2
2
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 5.6 : A boy, who is standing on a pole of height 14.7m throws a stone vertically upwards. It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is x 9. 8 t 4. 9 t
2
(i) Find the time taken for upward and downward motions. (ii) Also find the maximum height reached by the stone from the ground.
E R K HSS –ERUMIYAMPATTI
Page 8
+2 STUDY MATERIALS
www.tnschools.co.in
Example 5.7 : A ladder 10m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 m/sec how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6m from the wall? Example 5.8 : A car A is travelling from west at 50 km/hr. and car B is traveling towards north at 60 km/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 kilometers and car B is 0.4 kilometers from the intersection? Example 5.9 : A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is being pumped into the tank at a rate of 2m3 / min , find the rate at which the water level is rising when the water is 3m deep. EXERCISE 5.1. (1)
A missile fired from ground level rises x metres vertically upwards in t seconds and
x 100 t
25 t
2
.
2
Find (i) the initial velocity of the missile, (ii) the time when the height of the missile is a maximum (iii) the maximum height reached and (iv) the velocity with which the missile strikes the ground. (3) The distance x metres traveled by a vehicle in time t seconds after the brakes are applied is given by : 2 x 20 t 5 / 3 t .Determine (i) the speed of the vehicle (in km/hr) at the instant the brakes are applied and (ii) the distance the car traveled before it stops. (5) The altitude of a triangle is increasing at a rate of 1 cm / min while the area of the triangle is increasing at a rate of 2 cm2 / min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2. (6) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km / hr and ship B is sailing north at 25 km / hr. How fast is the distance between the ships changing at 4.00 p.m. (8) Two sides of a triangle have length 12 m and 15 m. The angle between them is increasing at a rate of 2° / min. How fast is the length of third side increasing when the angle between the sides of fixed length is 60° ? (9) Gravel is being dumped from a conveyor belt at a rate of 30 ft3 / min and its coarsened such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? Example 5.13 : Find the equations of the tangent and normal at
to the curve
x a sin ,
y a
2
1 cos .
Example 5.14 : Find the equations of tangent and normal to the curve y 0 1
16 x
2
9y
2
144
at ( x 1 , y 1 ) where
x 2 1
and
.
Example 5.15 : Find the equations of the tangent and normal to the ellipse
x a cos , y b sin
at the point
4
Example 5.17 : Find the angle between the curves Example 5.18 :
y x
2
and
y x 2
2
.
at the point of intersection.
Find the condition for the curves ax 2 by 2 1 , a1 x 2 b1 y 2 1 to intersect orthogonally. Example 5.20 : Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the curve x a cos 4 , y a sin 4 , 0
(5)
2
is equal to a.
EXERCISE 5.2. Find the equations of those tangents to the circle x 2 y 2 52 , which are parallel to the straight line 2x 3y 6 .
E R K HSS –ERUMIYAMPATTI
Page 9
+2 STUDY MATERIALS
www.tnschools.co.in
(7)
Let P be a point on the curve y x 3 and suppose that the tangent line at P intersects the curve again at Q. Prove that the slope at Q is four times the slope at P. (10) Show that the equation of the normal to the curve 3 3 x a cos ; y a sin at „ ‟ is x cos y sin a cos 2 . (11) If the curve y 2 x and xy k are orthogonal then prove that 8 k 2 1
Example 5.34 :
Evaluate :
lim
cot x sin x
x 0
Example 5.35 :
Evaluate :
lim
x
sin x
x 0
EXERCISE 5.6. (11)
lim
x
tan x cos x
2
Example 5.48 (a) : Find the absolute maximum and absolute minimum values of f x x 2 sin Example 5.51 : Find the local minimum and maximum values of f x x 4 3 x 3 3 x 2 x . EXERCISE 5.9 (3) Find the local maximum and minimum values of the following functions:
x , 0 x 2
.
(iv) x 2 1 (v) sin 2 0, (vi) t cos t (iii) x 4 6 x 2 Example 5.52 : A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river. He needs no fence along the river. What ar the dimensions of the field that has the largest area? Example 5.53 : Find a point on the parabola y 2 2 x that is closest to the point (1,4) Example 5.54 : Find the area of the largest rectangle that can be inscribed in a semi circle of radius r. Example 5.55 : The top and bottom margins of a poster are each 6 cms and the side margins are each 4cms. If the area of the printed material on the poster is fixed at 384 cms2 , find the dimension of the poster with the smallest area. Example 5.56 : Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is 3
8 27
( volume of the sphere ).
Example 5.57 : A closed (cuboid) box with a square base is to have a volume of 2000 c.c. The material for the top and bottom of the box is to cost Rs. 3 per square cm. and the material for the sides is to cost Rs. 1.50 per square cm. If the cost of the materials is to be the least, find the dimensions of the box. Example 5.58 : A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km downstream on the opposite bank, as quickly as possible. He could row his boat directly across the river to point R and then run to Q, or he could row directly to Q, or he could row to some point to between Q and R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land to reach Q as soon as possible? EXERCISE 5.10. (3) Show that of all the rectangles with a given area the one with smallest perimeter is a square. (4) Show that of all the rectangle with a given perimeter the one with the greatest area is a square. (5) Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. Example 5.63 : Discuss the curve y x 4 4 x 3 with respect to concavity and points of inflection. Example 5.64 : Find the points of inflection and determine the intervals of convexity and concavity of the Gaussion curve y e
x
2
E R K HSS –ERUMIYAMPATTI
Page 10
+2 STUDY MATERIALS
www.tnschools.co.in
EXERCISE 5.11. Find the intervals of concavity and the points of inflection of the following functions : (5) f sin 2 in 0 , (6) y 12 x 2 2 x 3 x 4 (4) f x x 4 6 x 2 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 6.1. (3) Use differentials to find the an approximate value for the given number (iii) y 3 1 . 02 4 1 . 02 Example 6.9 :
Trace the curve
y x 1
Example 6.10 :
Trace the curve
y 2x
3
2
3
EXERCISE 6.2 Trace the following curve : 3 (1) yx Example 6.18 : If w u 2 e v where u
x
and v y log x , find
y
x
and
w y
1
f x, y
Example 6.20 : Verify Euler‟s theorem for
w
x
2
y
2
Example 6.22 : u
Using Euler‟s theorem, prove that x
x
y
u y
1
tan u if
2
u sin
1
x y x
y
EXERCISE 6.3. 2
(1)
u
Verify (ii)
x y
u
x y
(5)
2
2
u y x
for the following functions:
y x
(iii)
2
u sin 3 x cos 4 y
(iv)
u tan
1
x y
Using Euler‟s theorem prove the following : (i) If
u tan
1
x3 y3 x y
Prove that
x
u x
y
u y
sin 2 u .
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 7.25 : Find the area between the curves y x 2 x 2 , x-axis and the lines x 2 and x 4 Example 7.26 : Find the area between the line y x 1 and the curve y x 2 1 . Example 7.27 : Find the area bounded by the curve y x 3 and the line y x . Example 7.28 : Find the area of the region enclosed by y 2 x and y x 2 Example 7.29 : Find the area of the region common to the circle x 2 y 2 16 and the parabola y 2 6 x . Example 7.30 : Compute the area between the curve y sin x and y cos x and the lines x 0 and x
E R K HSS –ERUMIYAMPATTI
Page 11
+2 STUDY MATERIALS
Example 7.31 : Find the area of the region bounded by the ellipse
x
2
a
2
y
2
b
2
1
Example 7.32 : Find the area of the curve y 2 x 5 2 x 6 (i) between x = 5 and x = 6 (ii) between x = 6 and x = 7 Example 7.33 : Find the area of the loop of the curve 3 ay 2 x x a 2 Example 7.34 : Find the area bounded by x-axis and an arch of the cycloid x a 2 t sin 2 t , y a 1 cos 2 t . EXERCISE 7.4. (4) Find the area of the region bounded by the curve y 3 x 2 x and the x – axis between x = -1 and x = 1. (7)
Find the area of the region bounded by the ellipse
x
2
y
9
2
1
between the two latus rectums.
5
(8) Find the area of the region bounded by the parabola y 2 4 x and the line 2 x y 4 . (9) Find the common area enclosed by the parabolas 4 y 2 9 x and 3 x 2 16 y (15) Derive the formula for the volume of a right circular cone with radius „ r „ and height „ h „. Example 7. 37 : Find the length of the curve Example 7.38 : Find the length of the curve
4y
2
x
x a
2 3
3
between x 0 and
y a
x 1
2 3
1
Example 7.39 : Show that the surface area of the solid obtained by revolving the arc of the curve
y sin x
from x 0 to
about x-axis is 2 2 log 1 2 Example 7.40 : Find the surface area of the solid generated by revolving the cycloid x a t sin t , y a 1 cos t about its base ( x- axis ). EXERCISE 7.5. (1) Find the perimeter of the circle with radius a. (2) Find the length of the curve x a t sin t , y a 1 cos t between t 0 and . (3) Find the surface area of the solid generated by revolving the arc of the parabola y 2 4 ax , bounded by its latus rectum about x - axis. (4) Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a distance a and b from the centre of the sphere is 2 r b a and hence deduct the surface area of the x
sphere. b a . -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 8. DIFFERENTIAL EQUATIONS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 8.7 : Solve : x y 2
dy
a
2
dx
Example 8.10 : Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1 respectively. EXERCISE 8.2. (7)
Solve the following :
x y 2
E R K HSS –ERUMIYAMPATTI
dy
1
dx
Page 12
+2 STUDY MATERIALS
www.tnschools.co.in
xy x dy y dx 0
Example 8.13 : Solve : 2
x 3 xy dx y 3 x y dy 0 1 e dx e 1 x y dy 0 given that 3
Example 8.14 : Solve :
2
3
x y
Example 8.15 : Solve :
2
x y
y 1,
where x 0
EXERCISE 8.3. Solve the following : (5) x 2 y 2 dx 3 xy dy 0
1 x dy 2
Example 8.18 : Solve :
2 xy x
dx
1 y dx 2
Example 8.19 : Solve :
tan
1
1 x 2
y x dy
EXERCISE 8.4. Solve the following : (3)
dx
dy
(5)
dy
dx
(7) (9)
x 1 y
y x
2
tan
sin x
dx x dy e
1
y
1 y 2
y
2
sec
2
y dy
Show that the equation of the curve whose slope at any point is equal to through the origin is y 2 e x 1
y 2x
and which passes
x
EXERCISE 8.5. Solve the following differential equations : 2
(6) (10)
d y
dy
3x when x log 2 , y 0 and 2y2e dx dx D 2 6 D 9 y x e 2 x 2
3
x 0, y 0
(11) Solve the differential equation D 2 1 y cos 2 x 2 sin 2 x Example 8.34 : In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant. At the end of one hour. 60 grams remain and at the end of 4 hours 21 grams. How many grams of the first substance was there initially? Example 8.35 : A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year compounded continuously. Calculate the percentage increase in such an account over one year. [ Take e 08 1 . 0833 ]. Example 8.37 : For a postmortem report, a doctor requires to know approximately the time of death of the deceased. He records the first temperature at 10.00 a.m. to be 93.4 ° F. After 2 hours he finds the temperature to be 91.4° F. If the room temperature ( which is constant) is 72° F, estimate the time of death. (Assume normal temperature of a human body to be 98.6° F). 19.4 26.6 log e 21.4 0.0426 2.303 and log e 21.4 0.00945 2.303
Example 8.38 : A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient is administered 10 mg of drug at time t = 0 , which is excreted at a Rate of 3t 1 2 mg/h. (i) What is the general equation for the amount of drug in the patient at time t > 0 ? (ii) When will the patient be drug free? Example 8.39 : The number of bacteria in a yeast culture grows at a rate which is proportional to the number present. If the population of a colony of yeast bacteria triples in 1 hour. Show that the number of bacteria at the end of five hours will be 3 5 times of the population at initial time.
E R K HSS –ERUMIYAMPATTI
Page 13
+2 STUDY MATERIALS
www.tnschools.co.in
(1) (2) (3) (4)
EXERCISE 8.6. Radium disappears at a rate proportional to the amount present. If 5% of the original amount disappears in 50 years, how much will remain at the end of 100 years. [ Take A0 as the initial amount ]. The sum of Rs. 1000 is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be twice the original principal ? ( log e 2 0 .6931 ) A cup of coffee at temperature 100° C is placed in a room whose temperature is 15° C and it cools to 60°C in 5 minutes. Find its temperature after a further interval of 5 minutes. The rate at which the population of a city increases at any time is proportional to the population at that time. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990 what population may be anticipated in 2020?
16 . 42 1 . 52 . 2070 , e 13
[ log e
]
(5)
A radioactive substance disintegrates at a rate proportional to its mass. When its mass is 10 mgm, the rate of disintegration is 0.051 mgm per day. How long will it take for the mass to be reduced from 10 mgm to 5 mgm. ( log e 2 0 .6931 ) -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 9. DISCRETE MATHEMATICS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 9.18 : Show that Z , * is an infinite abelian group where * is defined as a * b a b 2 . Example 9.21 : x x
x x
Show the set G of all matrices of the form
where
x R 0 ,
is a group under matrix multiplication.
Example 9.22 : Show that the set G a b 2 / a , b Q is an infinite abelian group with respect to addition. Example 9.23 : Let G be the set of all rational numbers except 1 and * be defined on G by a * b a b ab for all a , b G . Show that ( G , * ) is an infinite abelian group. Example 9.24 : Prove that the set of four functions f 1 , f 2 , f 3 , f 4 on the set of non-zero complex numbers C 0
defined by
f1 z z, f 2 z z, f 3 z
1
and
z
f 4 z
1 z
z C 0
forms an abelian group with
respect to the composition of functions. Example 9.25 : Show that Z n , n forms group.
Example 9.26 :Show that Z 7 0 , 7 forms a group. Example 9.27 : Show that the nth roots of unity form an abelian group of finite order with usual multiplication.
(5)
EXERCISE 9.4. Show that the set G of all positive rational forms a group under the composition * defined by a a *b
ab
a , b G .
for all
3
(6)
1
Show that
0
0 , 1
0
0 , 2
2 0
0
0 , 1
1 , 0
0
2 0
, 2 0
0
Where 3 1 , 1 form a
group with respect to matrix multiplication. z 1 forms a group with respect to the
(7)
Show that the set M of complex numbers z with the condition
(8)
operation of multiplication of complex numbers. Show that the set G of all rational numbers except -1 forms an abelian group with respect to the
E R K HSS –ERUMIYAMPATTI
Page 14
+2 STUDY MATERIALS
www.tnschools.co.in
(9)
operation * given by a * b a b ab for all a , b G . Show that the set 1 , 3 , 4 , 5 , 9 forms an abelian group under multiplication modulo 11.
(11) Show that the set of all matrices of the form multiplication. (12) Show that the set
G 2
n
/ n Z
a 0
0 , a R 0 forms 0
an abelian group under matrix
an abelian group under multiplication.
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 10. PROBABILITY DISTRIBUTIONS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 10.2 : A random variable X has the following probability mass function X P(X= x) (i) Find k. (ii) Evaluate
0 k
1 3k
2 5k
3 7k
4 9k
5 11k
6 13k
P X 4 , P X 5 and P 3 X 6
(iii) What is the smallest value of x for which
P
X x
1 2
?
Example 10.3 : An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in three draws one by one from the urn. (i)With replacement (ii) without replacement Example 10.10 : The total life time (in year) of 5 year old dog of a certain breed is a Random Variable whose distribution function is given by , for x 5 0 F x 25 1 2 , for x 5 x
(i) beyond 10 years
(7)
Find the probability that such a five year old dog will live (ii) less than 8 years
(iii) anywhere between 12 to 15 years.
EXERCISE 10.1 The probability density function of a random variable X is k x f x 0
1
e
x
, x , , 0
Find (i) k
(ii) P ( X > 10 )
, elsewhere
Example 10.26 : If the number of incoming buses per minute at a bus terminus is a random variable having a Poisson distribution with 0 . 9 , find the probability that there will be (i) Exactly 9 incoming buses during a period of 5 minutes. (ii) Fewer than 10 incoming buses during a period of 8 minutes. (iii) At least 14 incoming buses during a period of 11 minutes. EXERCISE 10.4. The number of accidents in a year involving taxi drivers in a city follows a Poisson distribution with mean equal to 3. Out of 1000 taxi drivers find approximately the number of drivers with (i) no accident in a year (ii) more than 3 accidents in a year e 3 0 . 0498 . Example 10.29 : If X is normally distributed with mean 6 and standard deviation 5 find. (i) P 0 X 8 (ii) P X 6 10 (5)
E R K HSS –ERUMIYAMPATTI
Page 15
+2 STUDY MATERIALS
www.tnschools.co.in
[Area table: P(0
x
ke
2x
2
4x
X
.
Example 10.32 : The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31 psi and standard deviation 0.2 psi. (i) What is the probability that the pressure for a randomly selected tyre (a) between 30.5 and 31.5 psi (b) between 30 and 32 psi (ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi? [ Area table: P(0
(5)
(8)
EXERCISE 10.5 The mean weight of 500 male students in a certain college in 151 pounds and the standard deviation is 15 pounds. Assuming the weights are normally distributed, find how many students weigh (i) between 120 and 155 pounds (ii) more than 185 pounds. [ Area table: P(0
x ce x
2
3x
X
.
************
E R K HSS –ERUMIYAMPATTI
Page 16
+2 STUDY MATERIALS
www.tnschools.co.in
COME BOOK – SIX MARKS [16 Number Of Six Mark Questions To Be Asked For Full Test] -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT:1. MATRICES AND DETERMINANTS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 1.1 1 A 3
2 5
and verify the result A ( adj A ) ( adj A ) A A .
(2)
Find the adjoint of the matrix
(4)
Find the inverse of each of the following matrices : (i)
(iv)
1 2 1
3 1 1
0 1 1
8 5 10
1
5 A 7
2 3
(ii)
3 2 4
1 1
(v) 2 B 1
(5)
If
(8)
Show that the adjoint of
(10) For
1 A 4 4
and
2 3 4
2 4 5
1 1 A
1 4 1
3
2 1 1
2
2 2
3 2
7 3 1
show that
3 0 4
A A
3 2
(i) AB 1 B 1 A 1
3 1 3 1
2
2 0 1
1 1 2
verify that
4 1 4
(iii)
1 1 0
(ii) AB T
T
B A
T
is A itself.
.
PROPERTIES: (1) State and prove reversal law for inverses of matrices. [OR]
Prove that (AB)-1 = B-1A-1, where A and B are two non-singular matrices. EXERCISE 1.2. Solve by matrix inversion method each of the following system of linear equations: (1) (2)
2 x y 7 , 3 x 2 y 11 7 x 3 y 1, 2 x y 0
EXERCISE 1.3. Find the rank of the following matrices:
(1)
(4)
(1)
1 3 4
1 3 2
2
0 2 1
1
2
3
0
1
1
1 3
(2)
1 1 0
(5)
6 1 4 1 2 3
12 2 8
6 1 4
2
1
4
1
6
3
(3)
3 7 2
(6)
3 1 2 1 2 1
1
2
0
1
1
3
0 0 0
2
3
4
1
2
7
4 6
3
EXERCISE 1.4. Solve the following non-homogeneous system of linear equations by determinant method: 4 x 5 y 9 ; 8 x 10 y 18 (iii)
E R K HSS –ERUMIYAMPATTI
Page 17
+2 STUDY MATERIALS
www.tnschools.co.in
(1)
EXERCISE 1.5. Examine the consistency of the following system of equations. If it is consistent then solve the same. (iii)
x y z 7 ; x 2 y 3 z 18 ; y 2 z 6
(iv)
x 4 y 7 z 14 ; 3 x 8 y 2 z 13 ; 7 x 8 y 26 z 5 1 A 1 2
Example 1.2: Find the adjoint of the matrix
Example 1.3: If
1 A 1
2 , 4
1 2 1
1 3 3
A adj A adj A A A I 2
verify the result
Example 1.5: (iv) Find the inverse of the following matrix:
3 A 2 1
1 2 2
1 0 1
Example 1.6: If
1 A 1
2 1
and
0 B 1
1 2
verify that AB B 1 A 1 . 1
Example 1.7: Solve by matrix inversion method Example 1.12: Find the rank of the matrix
1 2 5
x y 3, 2x 3y 8
1
1
1
3
1
7
3 4 11
Example 1.13: Find the rank of the matrix
1 2 3
1 4 3
1 3 2
Example 1.14: Find the rank of the matrix
1 2 3
2
1 2 3
3
4
6
6
9
Example 1.15: Find the rank of the matrix
4 6 2
2
1
3
4
1
0
3 1 7
Example 1.16: Find the rank of the matrix
3 1 1
1
5
2
1
5
7
1 5 2
Example 1.17: Solve the following system of linear equations by determinant method. (2) 2 x 3 y 8 ; 4 x 6 y 16 Example 1.18: Solve the following non-homogeneous equations of three unknowns. (3) 2 x 2 y z 5 ; x y z 1 ; 3 x y 2 z 4 (5) x y 2 z 4 ; 2 x 2 y 4 z 8 ; 3 x 3 y 6 z 10
E R K HSS –ERUMIYAMPATTI
Page 18
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 2. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 2.2 Prove by vector method: (1) If the diagonals of a parallelogram are equal then it is a rectangle. (2) The mid point of the hypotenuse of a right angled triangle is equidistant from its vertices. (3) The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides. (8)
Forces of magnitudes 3 and 4 units acting in the directions 6 i 2 j 3 k and 3 i 2 j 6 k respectively act on a particle which is displaced from the point (2,2-1) to (4,3,1). Find the work done by the forces. EXERCISE 2.3
(9)
Let that
(5) (6) (8)
be unit vectors such that
a , b , c
a 2 b c
a .b a .c 0
and the angle between
and
b
c
is
. Prove
6
EXERCISE 2.4. Prove by vector method that the parallelogram on the same base and between the same parallels are equal in area. Prove that twice the area of a parallelogram is equal to the area of another parallelogram formed by taking as its adjacent sides the diagonals of the former parallelogram. 2 i 7 j , 2 i 5 j 6 k , i 2 j k act at a point P whose position vector is 4 i 3 j 2 k . Forces Find the moment of the resultant of three forces acting at P about the point Q whose position vector is
6 i j 3 k . (10) Find the magnitude and direction cosines of the moment about the point (1,-2,3) of a force 2 i 3 j 6 k whose line of action passes through the origin. EXERCISE 2.5. (4) Show that the points (1,3,1) , (1,1,-1) , (-1,1,1) (2,2,-1) are lying on the same plane. ( Hint : It is enough to prove any three vectors formed by these four points are coplanar). (7) If a 2 i 3 j 5 k , b i j 2 k and c 4 i 2 j 3 k , show that a b c a b c
(8) (10) (11)
c a b c iff a and c Prove that a b . c d b c . a d c Find a b . c d if a i j k , b 2 i k Prove that a
b
are collinear.(vector triple products is non-zero ) a
. b d 0
, c 2i j k , d i j 2k
EXERCISE 2.6. (4) (6)
A vector r has length 35 2 and direction ratios (3,4,5) , find the direction cosines and components of Find the vector and Cartesian equation of the line through the point (3, -4, -2) and parallel to the vector
r
.
9 i 6 j 2 k .
(7) (1)
Find the vector and Cartesian equation of the line joining the points (1,-2,1) and (0,-2,3) EXERCISE 2.7. Find the shortest distance between the parallel lines (i) (ii)
(2)
r x 1 1
2 i
y 3
j k z3 2
t i 2
and
x3 1
j 3 k y 1 3
and
r
i 2
s i 2
j 3 k
z 1 2
Show that the following two lines are skew lines: r 3 i 5 j 7 k t i 2 j k and r i
j k
(4)
Find the shortest distance between the skew lines
(5)
Show that (2,-1,3) , (1,-1,0) and (3,-1,6) are collinear.
E R K HSS –ERUMIYAMPATTI
j k
x6 3
Page 19
s 7 i 6
y7 1
z4 1
j 7 k
and
x 3
y9 2
z2 4
+2 STUDY MATERIALS
www.tnschools.co.in
(1)
EXERCISE 2.8. Find the vector and Cartesian equations of a plane which is at a distance of 18 units from the origin and
(4) (5)
which is normal to the vector 2 i 7 j 8 k The foot of the perpendicular drawn from the origin to a plane is (8,-4,3). Find the equation of the plane. Find the equation of the plane through the point whose p.v. is 2 i j k and perpendicular to the vector 4 i 2 j 3 k .
(6)
Find the vector and Cartesian equations of the plane through the point (2,-1,4) and parallel to the plane
r 4 i 12 j 3 k
7.
(15) Find the Cartesian form of the following planes: (i) r s 2 t i 3 t j 2 s t k (ii) EXERCISE 2.9. (1)
r 1 s t i 2 s t j 3 2 s 2 t k
x 1
Find the equation of the plane which contains the two lines
2
(2)
i 2
j 4 k
t 2 i 3
j 6 k
z3
x4
and
3
4
j k s 2 i
s 2 i 3
r 3 i 3 j 5 k
and Find the point of intersection of the line
and xz – plane. Find the meeting point of the line r 2 i j 3 k t 2 i r
(4)
3
Can you draw a plane through the given two lines? Justify your answer. r
(3)
y2
j k
j k
y 1
z8
2
j 8 k
and the plane
x 2 y 3z 7 0
EXERCISE 2.11 (1) (2)
(3)
Find the vector equation of a sphere with centre having position vector 2 i j 3 k and radius 4 units. Also find the equation in Cartesian form. Find the vector and Cartesian equation of the sphere on the join of the points A and B having position vectors 2 i 6 j 7 k and 2 i 4 j 3 k respectively as a diameter. Find also the centre and radius of the sphere. Obtain the vector and Cartesian equation of the sphere whose centre is (1, -1, 1) and radius is the same as that of the
r
sphere
i
j 2 k
5.
(6) Show that diameter of a sphere subtends a right angle at a point on the surface by vector method. Example 2.12: With usual notations in triangle ABC prove that cos A
b
2
c
2
a
2
2 bc
Example 2.13: With usual notations prove (i) a b cos C c cos B Example 2.14: Angle in a semi-circle is a right angle. Prove by vector method. Example 2.15: Diagonals of a rhombus are at right angles. Prove by vector methods. Example 2.24: If
p 3i 4 j 7 k
and
q 6i 2 j 3k
each other and also verify that Example 2.25:
q
and
p q
then find
p q.
Verify that
AC . Interpret
and
p q
are perpendicular to
are perpendicular to each other.
If the position vectors of three points A, B and C are respectively Find AB Example 2.26:
p
i 2 j 3 k , 4i j 5 k
and
7
i k .
the result geometrically.
Prove that the area of a quadrilateral
1
AC BD
where AC and BD are its diagonals.
2
E R K HSS –ERUMIYAMPATTI
Page 20
+2 STUDY MATERIALS
Example 2.27: If a
, b , c
are the position vectors of the vertices A, B, C of a triangle ABC, then prove that the area of
triangle ABC is
1
Deduce the condition for the points
a b b c c a
to be collinear.
a , b , c
2
Example 2.28: Prove that in triangle ABC with usual notations,
a
sin A
Example 2.33: For any three vectors
a , b , c
Example 2.36: If a 3 i 2 j 4 k , b 5 i 3 j 6 k show that they are not equal. Example 2.37: Let (i)
a , b , c
and
d
by vector method.
sin C
, c a 2 a b c
, c 5i j 2k
,find (i)
a b c
(ii)
a b c
and
be any four vectors then
a b c d a a b c d a
(ii) Example 2.38:
c
sin B
a b , b c
prove that
b
c a d b b
d d a
b d
b c
c
c
Prove that a b , b c , c a a , b , c Example 2.39: Find the vector and Cartesian equations of the straight line passing through the point A with position vector 3 i j 4 k and parallel to the vector 5 i 7 j 3 k Example 2.40: Find the vector and Cartesian equations of the straight line passing through the points (-5,2,3) and (4,-3,6) Example 2.42: Find the shortest distance between the parallel lines 2
r
i j t 2 i j k
and
r
2 i
j k s 2i jk
Example 2.43:
Show that the two lines r i j t 2 i k and distance between them. Example 2.45: Find the shortest distance between the skew lines r
i j 2 i j k
and
r
i
j k
r
2 i j s i j k are skew lines and find the
2 i
j k
Example 2.46: Show that the points (3,-1,-1) , (1,0,-1) and (5,-2,-1) are collinear. Example 2.48: Find the vector and Cartesian equation of a plane which is at a distance of 8 units from the origin and which is normal to the vector 3 i 2 j 2 k Example 2.49: The foot of perpendicular drawn from the origin to the plane is (4,-2,-5) , find the equation of the plane. Example 2.53: Find the equation of the plane passing through the line of intersection of the plane 2 x 3 y 4 z 1 and x y 4 and passing through the point (1,1,1) Example 2.54: Find the equation of the plane passing through the intersection of the planes 2 x 8 y 4 z 3 and 3 x 5 y 4 z 10 0 and perpendicular to the plane 3 x y 2 z 4 0 Example 2.55: Find the distance from the point (1,-1,2) to the plane
E R K HSS –ERUMIYAMPATTI
r
Page 21
i
j k
s i j t j k
+2 STUDY MATERIALS
www.tnschools.co.in
Example 2.56: Find the distance between the parallel planes Example 2.57:
3
r i j k
x 1
Find the equation of the plane which contains the two lines
2
and y2 3
r
i
z3
5
j k
x4
and
4
5
y 1 2
z 1
Example 2.58: Find the point of intersection of the line passing through the two points (1,1,-1) ; (-1,0,1) and the xy -plane. Example 2.59: r
Find the co-ordinates of the point where the line
r 2 i 4 j k
3
i 2
j 5 k
t 2 i 3
j 4 k
meets the plane
Example 2.62: Find the vector and Cartesian equations of the sphere whose centre is 2 i j 2 k and radius is 3. Example 2.63: Find the vector and Cartesian equation of the sphere whose centre is (1,2,3) and which passes through the point (5,5,3) Example 2.64: Find the equation of the sphere on the join of the points A and B having position vectors
2 i 6
j 7 k
and 2 i 4 j 3 k respectively as a diameter. Example 2.65: Find the coordinates of the centre and the radius of the sphere whose vector equation is 2
r
r 8 i 6 j 10 k
50
0
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 3.COMPLEX NUMBERS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 3.1. (1) Express the following in the standard form a ib (ii) (4)
1 i 1 2 i
(iv)
1 3i
i
4
i i
9
16
8
i
3 2i i
10
15
Find the real values of x and y for which the following equations are satisfied. 1 i x 2 i 2 3i y i (i) 1 i x 1 i y 1 3i (ii) (iii) x 2 3 x 8 i 3i
3i
and x 2 EXERCISE 3.2.
x 4 i y 2 i
are complex conjugate of each other?
(5)
For what values of x and y, the numbers
(2) (4)
Find the square root of 8 6 i Prove that the triangle formed by the points representing the complex numbers 10 8i , 2 4 i and 11 31 i on the Argand plane is right angled. Prove that the points representing the complex numbers 7 5i , 5 2 i , 4 7 i and 2 4 i form a parallelogram.( Plot the points and use midpoint formula).
(5) (7) (8)
(1) (2) (3)
If
arg z 1
6
and
arg z 1 2
3 ix
2
y
then prove that
y 4i
z 1
3
P represents the variable complex number z. Find the locus of P, if (ii) z 5 i z 5 i (iv) 2 z 3 2 . EXERCISE 3.3. Solve the equation x 4 8 x 3 24 x 2 32 x 20 0 if 3 i is a root. Solve the equation x 4 4 x 3 11 x 2 14 x 10 0 if one root is 1 2 i Solve : 6 x 4 25 x 3 32 x 2 3 x 10 0 given that one of the root is 2 i EXERCISE 3.4.
E R K HSS –ERUMIYAMPATTI
Page 22
+2 STUDY MATERIALS
www.tnschools.co.in
cos i sin 3 sin i cos 4
(2)
Simplify:
(3)
If cos cos cos 0 sin sin sin ,prove that (i) cos 3 cos 3 cos 3 3 cos (ii) sin 3 sin 3 sin 3 3 sin (iii) cos 2 cos 2 cos 2 0 (iv) sin 2 sin 2 sin 2 0 (v)
2
cos
cos
cos
2
sin
2
2
sin
sin
2
2
3
2
(4)
Prove that n2
(i) 1 i n (ii)
1 i 2 n
1 i 3 1 i 3
(iv) 1 i If
x
1
i sin
n
and 1 i
4n
2 cos
cos
n 4
n
(iii) 1 cos
(7)
2
n
2
n 1
cos
n 3
1 cos i sin 2 n
4n2
If
prove that
(i)
x
n
1
x
x cos i sin ;
cos
n
2
2
y cos i sin
n
2 cos n
(ii)
(2)
(3)
x
n
1
x
prove that
x
m
y
n
1
x
(1)
n
cos
are real and purely imaginary respectively
x
(9)
n 1
m
y
n
n
2 i sin n
2 cos m n
EXERCISE 3.5. Find all the values of the following: (ii) 8i 1 3 If x a b , y a b 2 , z a 2 b show that (i) xyz a 3 b 3 (ii) x 3 y 3 z 3 3 a 3 b 3 where is the complex cube root of unity. Prove that if 3 1 , then (i) a b c a b c 2 a b 2 c a 3 b 3 c 3 3 abc (iii)
1 1 2
1 1
1 2
0
(4) Solve: (i) x 4 4 0 Example 3.9: Find the modulus and argument of the following complex numbers: (i) 2 i 2 (ii) 1 i 3 (iii) 1 i 3 Example 3.10: If a 1 ib1 a 2 ib 2 ... a n ib n A iB , prove that (i) (ii)
a
2 1
tan
b1 1
2
a
b1 a 1
2 2
b2
tan
1
2
... a
b2 a 2
2 n
bn
2
.... tan
1
2
A B bn a n
2
k tan
1
B , A
Example 3.13: Prove that the complex numbers 3 3i , 3 3i , 3 3 3 complex plane. Example 3.14: Prove that the points representing the complex numbers vertices of a rectangle.
E R K HSS –ERUMIYAMPATTI
Page 23
3i
k Z
are the vertices of an equilateral triangle in the
2i , 1 i , 4 4i
and
3 5i
on the Argand plane are the
+2 STUDY MATERIALS
www.tnschools.co.in
Example 3.15: Show that the points representing the complex numbers 7 9 i , the Argand diagram. Example 3.16: Find the square root of 7 24 i Example 3.17: Solve the equation x 4 4 x 2 8 x 35 0 ,if one of its roots is 2 Example 3.19: cos i sin 4 Simplify: sin i cos 5 Example 3.20: If n is a positive integer, prove that
1 sin i cos 1 sin i cos
n
3 7 i , 3 3i
form a right angled triangle on
3i
cos n i sin n 2 2
Example 3.21: If n is a positive integer, prove that
3i
n
3 i
n
2
n 1
cos
n 6
PROPERTIES: (1) State and prove the triangle inequality of complex numbers. (2) For any two complex numbers Z 1 , Z 2 , show that (i) Z 1 Z 2 Z 1 Z 2 (ii) arg Z 1 Z 2 arg Z 1 arg Z 2 (3)
For any two complex numbers (i)
Z1 Z2
Z1
Z 1 , Z 2 , show
(ii)
Z2
that
Z1 arg Z 2
arg Z 1 arg Z 2
Show that for any polynomial equation P x 0 with real coefficients imaginary roots occur in conjugate pairs. -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 4. ANALYTICAL GEOMETRY (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 4.1. (2) Find the axis, vertex ,focus equation of directrix, latus rectum, length of the latus rectum fro the following parabolas and hence sketch their graphs. (iii) x 4 2 4 y 2 (3) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of the reflector. (4) The focus of a parabolic mirror is at a distance of 8cm from its centre (vertex). If the mirror is 25cm deep, find the diameter of the mirror. EXERCISE 4.2. (1) Find the equation of the ellipse if (ii) the foci are (2,-1) , (0,-1) and e = 1 / 2. (iii) the foci are 3 , 0 and the vertices are 5 , 0 (4)
(iv) the centre is (3,-4) , one of the foci is
3
3 ,4
and
e
3 2
(v) the centre at the origin , the major axis is along x axis , e = 2 / 3 and passes through the point 5 2, 3
(vi) the length of the semi major axis, and the latus rectum are 7 and 80 / 7 respectively, the centre is (2,5) and the major axis is parallel to y-axis. (vii) the centre is (3,-1) , one of the foci is (6,-1) and passing through the point (8,-1). (viii) the foci are 3 , 0 , and the length of the latus rectum is 32 / 5 .
E R K HSS –ERUMIYAMPATTI
Page 24
+2 STUDY MATERIALS
www.tnschools.co.in
(ix) the vertices are 4 , 0 and (3) (4) (5)
e
3 2
Find the locus of a point which moves so that the sum of its distances from (3,0) and (-3,0) is 9. Find the equations and length of major and minor axes of (iv) 16 x 2 9 y 2 32 x 36 y 92 0 (ii) 5 x 2 9 y 2 10 x 36 y 4 0 Find the equations of directrices , latus rectum and length of latus rectums of the following ellipses: (iv) 3 x 2 2 y 2 30 x 4 y 23 0 (iii) x 2 4 y 2 8 x 16 y 68 0
EXERCISE 4.3. Find the equatin of the hyperbola if focus : (2,3) ; corresponding directrix : x 2 y 5 , e 2 Centre : (0,0) ; length of the semi-transverse axis is 5 ; e = 7 / 5 the conjugate axis is along x-axis. centre : (0,0) length of the semi – transverse axis is 6 ; e = 3, transverse axis is parallel to y-axis. centre : (1,-2) ; length of the transverse axis is 8 ; e = 5 / 4 transverse axis is parallel to x-axis. centre ; (2,5) ; the distance between the directrices is 15, the distance between the foci is 20 and the transverse axis is parallel to y – axis. (vi) foci : 0 , 8 ; length of transverse axis is 12 (vii) foci : 3 , 5 ; e = 3 (viii) centre : (1,4) ; one of the foci (6,4) and the corresponding directrix is x = 9 / 4 . (ix) foci : (6,-1) and (-4,-1) and passing through the point (4,-1)
(1) (i) (ii) (iii) (iv) (v)
(2) (3) (4)
(1)
(2)
Find the equation and length of transverse and conjugate axes of the following hyperbolas: (iii) 16 x 2 9 y 2 96 x 36 y 36 0 Find the equations of directrices, latus rectums and length of latus rectum of the following hyperbolas. (ii) 9 x 2 4 y 2 36 x 32 y 8 0 Show that the locus of a point which moves so that the difference of its distances from the points (5,0) and (5,0) is 8 is 9 x 2 16 y 2 144 . EXERCISE 4.4. Find the equations of the tangent and normal to the parabolas. (i) y 2 12 x at 3 , 6 (ii) x 2 9 y at 3 , 1 (iii) x 2 2 x 4 y 4 0 at 0 , 1 (iv) to the ellipse 2 2 (v) to the hyperbola 9 x 5 y 31 at 2 , 1 Find the equations of the tangent and normal (i) to the parabola
y
2
at
8x
t
2x
2
3y
2
6
at
3 ,0
1 2
(ii) to the ellipse
x
2
2
4y
at
32
4
(iii) to the ellipse
16 x
2
25 y
2
at
400
t
1 3
x
(iv) to the hyperbola
2
9
(3)
1
12
at
6
Find the equations of the tangents (i) to the parabola y 2 6 x , parallel to 3 x 2 y 5 0 (ii) to the parabola y 2 16 x , perpendicular to the line (iii) to the ellipse
x
2
20
(4) (i) (ii) (iii)
y
2
y
3x y 8 0
2
1
, which are perpendicular to
x y2 0
5
(iv) to the hyperbola 4 x 2 y 2 64 , which are parallel to 10 x 3 y 9 0 Find the equations of the two tangents that can be drawn from the point (2,-3) to the parabola y 2 4 x from the point (1,3) to the ellipse 4 x 2 9 y 2 36 from the point (1,2) to the hyperbola 2 x 2 3 y 2 6 .
E R K HSS –ERUMIYAMPATTI
Page 25
+2 STUDY MATERIALS
www.tnschools.co.in
(1) (2) (3)
(1)
EXERCISE 4.5. Find the equation of the asymptotes to the hyperbola (ii) 8 x 2 10 xy 3 y 2 2 x 4 y 2 0 Find the equation of the hyperbola if (i) the asymptotes are 2 x 3 y 8 0 and 3 x 2 y 1 0 and (5,3) is a point on the hyperbola. Find the angle between the asymptotes of the hyperbola (iii) 4 x 2 5 y 2 16 x 10 y 31 0 EXERCISE 4.6. Find the equation of the rectangular hyperbola whose centre is point
(2)
(4) (5) (6) (7)
1 1 , 2 2
and which passes through the
1 1 , . 4
Find the equation of the tangent and normal (i) at (3,4) to the rectangular hyperbolas
xy 12
(ii) at
1 2, 4
to the rectangular hyperbola 2 xy 2 x 8 y 1 0 A standard rectangular hyperbola has its vertices at (5,7) and (-3,-1). Find its equation and asymptotes. Find the equation of the rectangular hyperbola which has its centre at (2,1) one of its asymptotes 3 x y 5 0 and which passes through the point (1,-1). Find the equations of the asymptotes of the following rectangular hyperbolas. (ii) 2 xy 3 x 4 y 1 0 (iii) 6 x 2 5 xy 6 y 2 12 x 5 y 3 0 Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.
Example 4.7 : Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabolas and hence draw their graphs. (iii) y 2 2 8 x 1 Example 4.9: The headlight of a motor vehicle is a parabolic reflector of diameter 12cm and depth 4cm. Find the position of bulb on the axis of the reflector for effective functioning of the headlight. Example 4.11: A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9mts. If the distance across (diameter) the top of the mirror is 160cm, how deep is the mirror at the middle? Example 4.15: Find the equation of the ellipse whose foci are (1,0) and (-1,0) and eccentricity is 1 / 2. Example 4.16: Find the equation of the ellipse whose one of the foci is (2,0) and the corresponding directrix is x=8 and eccentricity is 1 / 2. Example 4.17: Find the equation of the ellipse with focus (-1,-3), directrix x 2 y 0 and eccentricity 4 / 5. Example 4.18: Find the equation of the ellipse with foci 4 , 0 and vertices 5 , 0 Example 4.20: Find the equation of the ellipse whose centre is (1,2) , one of the foci is (1,3) and eccentricity is 1 / 2. Example 4.21: Find the equation of the ellipse whose major axis is along x-axis, centre at the origin,passes through the point (2,1) and eccentricity 1 / 2. Example 4.22: Find the equation of the ellipse if the major axis is parallel to y-axis, semi – major axis is 12, length of the latus rectum is 6 and the centre is (1,12). Example 4.23: Find the equation of the ellipse given that the centre is (4,-1) , focus is (1,-1) and passing through (8,0)
E R K HSS –ERUMIYAMPATTI
Page 26
+2 STUDY MATERIALS
www.tnschools.co.in
Example 4.24: Find the equation of the ellipse whose foci are (2,1) (-2,1) and length of the latus rectum is 6. Example 4.25: Find the equation of the ellipse whose vertices are (-1,4) and (-7,4) and eccentricity is 1 / 3. Example 4.26: Find the equation of the ellipse whose foci are (1,3) and (1,9) and eccentricity is 1 / 2. Example 4.27: Find the equation of a point which moves so that the sum of its distances from ( -4,0 ) and (4,0) is 10. Example 4.28: Find the equations and lengths of major and minor axes of x 1 2 y 1 2 (iii) 1 9
16
Example 4.29: Find the equations of axes and length of axes of the ellipse 6 x 2 9 y 2 12 x 36 y 12 0 Example 4.30: Find the equations of directrices, latus rectum and length of latus rectum of the following ellipses. (iii) 4 x 2 3 y 2 8 x 12 y 4 0 Example 4.31: Find the eccentricity, centre, foci, vertices of the following ellipse : x 32 y 5 2 (iii) 1 6
4
Example 4.34: The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatest possible distance between the earth and the sun. Example 4.36: Find the equation of hyperbola whose directrix is 2 x y 1 , focus (1,2) and eccentricity 3 . Example 4.37: Find the equation of the hyperbola whose transverse axis is along x-axis. The centre is (0,0) length of semitransverse axis is 6 and eccentricity is 3. Example 4.38: Find the equation of the hyperbola whose transverse axis is parallel to x-axis, centre is (1,2) , length of the conjugate axis is 4 and eccentricity e = 2. Example 4.39: Find the equation of the hyperbola whose centre is (1,2). The distance between the directrices is 20 / 3, the distance between the foci is 30 and the transverse axis is parallel to y – axis. Example 4.41: Find the equation of the hyperbola whose foci are 6 , 0 and length of the transverse axis is 8. Example 4.42: Find the equation of the hyperbola whose foci are 5 , 4 and eccentricity is 3 / 2. Example 4.43: Find the equation of the hyperbola whose centre is (2,1) , one of the foci are (8,1) and the corresponding directrix is x =4. Example 4.44: Find the equation of the hyperbola whose foci are 0 , 5 and the length of the transverse axis is 6. Example 4.45: Find the equation of the hyperbola whose foci are 0 , 10 and passing through (2,3). Example 4.48: Find the equations and length of transverse and conjugate axes of the hyperbola 9 x 2 36 x 4 y 2 16 y 56 0 . Example 4.51: Find the equations of directrices, latus rectum and length of latus rectum of the hyperbola 9x
2
36 x 4 y
2
16 y 56 0
E R K HSS –ERUMIYAMPATTI
Page 27
+2 STUDY MATERIALS
www.tnschools.co.in
Example 4.52: The foci of a hyperbola coincide with the foci of the ellipse
x
2
y
25
2
1.
Determine the equation of the
9
hyperbola if its eccentricity is 2. Example 4.53: Find the equation of the locus of all points such that the differences of their distances from (4,0) and (-4,0) is always equal to 2. Example 4.54: x
Find the eccentricity, centre, foci and vertices of the hyperbola
2
4
y
2
1
and also trace the curve.
5
Example 4.55: y
Find the eccentricity , centre , foci and vertices of the hyperbola
2
6
x
2
1
and also trace the curve
18
Example 4.58: Points A and B are 10km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it. Example 4.59: Find the equations of the tangents to the parabola y 2 5 x from the point (5,13). Also find the points of contact. Example 4.61: Find the equation of the tangent and normal to the parabola x 2 x 2 y 2 0 at (1,2). Example 4.62: Find the equations of the two tangents that can be drawn from the point (5,2) to the ellipse 2 x 2 7 y 2 14 Example 4.64: Find the separate equations of the asymptotes of the hyperbola 3 x 2 5 xy 2 y 2 17 x y 14 0 Example 4.65: Find the equation of the hyperbola which passes through the point (2,3) and has the asymptotes 4 x 3 y 7 0 and x 2 y 1 . Example 4.66: Find the angle between the asymptotes of the hyperbola Example 4.67: Find the angle between the asymptotes to the hyperbola Example 4.68:
3x
2
3x
y
2
2
12 x 6 y 9 0
5 xy 2 y
2
17 x y 14 0
Prove that the product of perpendiculars from any point on the hyperbola 2
a b
constant and the value is a
2
x
2
a
2
y
2
b
2
1
to its asymptotes is
2
b
2
Example 4.69: Find the equation of the standard rectangular hyperbola whose centre is through the point
3 2, 2
and which passes
2 1 , 3
Example 4.70: The tangent at any point of the rectangular hyperbola xy c 2 makes intercepts a, b and the normal at the point makes intercepts p,q on the axes. Prove that ap bq 0 Example 4.71: Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.
E R K HSS –ERUMIYAMPATTI
Page 28
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 5.1. (2) A Particle of unit mass moves so that displacement after t secs is given by x 3 cos 2 t 4 . Find the acceleration and kinetic energy at the end of 2 secs.
1 2 K .E . m v , m is mass 2
Newton‟s law of cooling is given by 0 e kt , where the excess of temperature at zero time is 0 C and at time t seconds is C . Determine the rate of change of temperature after 40 s, given that 0 16 C and k 0 .03 . e 1 .2 3 . 3201 (7) Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06 rad / sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is / 3 . Example 5.10 : Find the equations of the tangents and normal to the curve y x 3 at the point ( 1,1 ). Example 5.11 : Find the equations of the tangent and normal to the curve y x 2 x 2 at the point ( 1, -2 ). Example 5.12 : Find the equation of the tangent at the point (a,b) to the curve xy c 2 . Example 5.16 : Find the equation of the tangent to the parabola, y 2 20 x which forms an angle 45 with the x – axis. Example 5.19 : Show that x 2 y 2 a 2 and xy c 2 cut orthogonally. EXERCISE 5.2. (1) Find the equation of the tangent and normal to the curves (4)
(i)
2 y x 4 x 5 at x 2
(iii)
y 2 sin
y x sin x cos x ,
(ii)
at
x
2
2
3 x at x
(iv)
y
1 sin x
6
cos x
at
x
4
(2) Find the points on curve x 2 y 2 2 at which the slope of the tangent is 2. (3) Find at what points on the circle x 2 y 2 13 , the tangent is parallel to the line 2 x 3 y 7 (4) At what points on the curve x 2 y 2 2 x 4 y 1 0 the tangent is parallel to (i) x – axis (ii) y – axis. (6) Find the equation of a normal to y x 3 3 x that is parallel to 2 x 18 y 9 0 . (8) Prove that the curve 2 x 2 4 y 2 1 and 6 x 2 12 y 2 1 cut each other at right angles. (9) At what angle do the curves y a x and y b x intersect a b ? Example 5.22 : Verify Rolle‟s theorem for the following : (v) f x e x sin x , 0 x Example 5.23 : Apply Rolle‟s theorem to find points on curve y 1 cos x , where the tangent is parallel to x – axis in 0 , 2 . EXERCISE 5.3. (2) Using Rolle‟s theorem find the points on the curve y x 2 1 , 2 x 2 where the tangent is parallel to x – axis. Example 5.24 : Verify Lagrange‟s law of the mean for f x x 3 on 2 , 2 Example 5.25 : A cylindrical hole 4mm in diameter and 12 mm deep in a metal block is rebored to increase the diameter to 4.12 mm. Estimate the amount of metal removed.
E R K HSS –ERUMIYAMPATTI
Page 29
+2 STUDY MATERIALS
www.tnschools.co.in
Example 5.26 : Suppose that f 0 3 and f ' x 5 for all values of x, how large can f 2 possibly be? Example 5.27 : It took 14 sec for a thermometer to rise from -19° C to 100°C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at exactly 8.5 ° C / sec. EXERCISE 5.4 (1) Verify Lagrange‟s law of mean for the following functions: 2 (i) f x 1 x , 0 , 3 f x
(ii)
1 x
, 1, 2
3 2 (iii) f x 2 x x x 1 , 0,2 2 3 (iv) f x x , 2 , 2 3 2 (v) f x x 5 x 3 x , 1, 3 (2) If f 1 10 and f ' x 2 for 1 x 4 how small can f 4 possibly be ? (3) At 2.00 p.m. a car‟s speedometer reads 30 miles / hr., at 2.10 pm it reads 50 miles / hr. Show that sometime between 2.00 and 2.10 the acceleration is exactly 120 miles / hr2. Example 5.28 : Obtain the Maclaurin‟s Series for log e 1 x (2)
1
(3)
arc tan x
(1)
EXERCISE 5.5. Obtain the Maclaurin‟s Series expansion for : (ii) cos 2
or
tan
x
(iii)
x
1
(iv) tan
1 x
x,
2
x
2
Example 5.30 : sin
Find
1 x
lim
x
tan
1
1
if exists.
x
Example 5.31 :
log sin x
lim
2x
Example 5.33 : Evaluate :
lim
x
2
2
x 0
1 cos ec x x
Example 5.36 : The current at time t in a coil with resistance R, inductance L and subjected to a constant electromotive force E is given by
i
E 1 e R
RT L
Obtain a suitable formula to be used when R is very small. EXERCISE 5.6.
Evaluate the limit for the following if exists . (2)
tan x x lim
x 0
x sin x 1
(6)
lim
x
x
2
2 tan
1
1 x
1 x
E R K HSS –ERUMIYAMPATTI
Page 30
+2 STUDY MATERIALS
www.tnschools.co.in
(8)
cot x lim
cot 2 x
x 0
(9)
x
lim
2
log
e
x 0
x.
1
(10)
lim x
x 1
x 1
(12)
lim
x
x
x 0
(13)
lim
cos x
1 x
x 0
Example 5.37: f x sin x cos 2 x
Prove that the function
0, 4
is not monotonic on the interval
Example 5.38: Find the intervals in which f x 2 x 3 x 2 20 x is increasing and decreasing. Example 5.39 : Prove that the function f x x 2 x 1 is neither increasing nor decreasing in 0 ,1 Example 5.40 : Discuss monotonicity of the function f x sin x , x 0 , 2 Example 5.41: Determine for which values of x, the function
x2
y
x 1
,
x 1
is strictly increasing or strictly decreasing.
Example 5.42 : Determine for which values of x, the function f x 2 x 3 15 x 2 36 x 1 is increasing and for which it is decreasing. Also determine the points where the tangents to the graph of the function are parallel to the x axis. Example 5.43:Show that f x tan (3)
x cos x , x 0 is a strictly increasing function in the interval 0 ,
4
x x 1 x 1 on
2, 1
(v)
x sin x on 0 , 4
Prove that the following functions are not monotonic in the intervals given. 2 x x 1 x 1 on 0 , 2 (ii) (i) 2 x x 5 on 1, 0 (iii)
(5)
sin
EXERCISE 5.7 Which of the following functions increasing or decreasing on the interval given ? (iv)
(4)
1
x sin x on
0,
(iv)
Find the intervals on which f is increasing or decreasing. 2 (i) (ii) f x 20 x x 3 (iii) (iv) f x x x 1 (v)
f x x cos x
in 0 ,
Example 5.45 : Prove that the inequality 1 x n Example 5.46: Prove that
(vi) 1 nx
on 0 , 2
tan x cot x
f x x
3
3x 1
0 , 2
f x x 2 sin x , f x sin
is true whenever
4
x cos
x0
4
x
and
in
0, 2
n 1.
sin x x tan x , x 0 , 2
EXERCISE 5.8. (1)
Prove the following inequalities : (i) (iii)
cos x 1
x
2
, x0
(ii)
2 tan
1
x x
sin x x
x
3
, x0
6
for all
E R K HSS –ERUMIYAMPATTI
x0
(iv)
Page 31
log 1 x x
for all
x0.
+2 STUDY MATERIALS
www.tnschools.co.in
Example 5.48 : Find the absolute maximum and minimum values of the function.
f x x
3
2
3x 1 ,
1
x 4
2
Example 5.49 : Discuss the curve y x 4 4 x 3 with respect to local extrema. Example 5.50 : Locate the extreme point on the curve y 3 x 2 6 x and determine its nature by examine the sign of the gradient on either side. EXERCISE 5.9. (1) Find the critical numbers and stationary points of each of the following functions . (iii)
f x x
4 5
x 4 2
x 1
f x
(iv)
x
(vi) (2)
(3)
f sin
2
in
2
0,
2
x 1
f sin
(vii)
in
0 , 2
Find the absolute maximum and absolute minimum values of f on the given interval: 2 2 (ii) (i) 0 ,3 4 ,1 f x x 2 x 2 , f x 1 2 x x , (iii)
f x x
(v)
f x
(vii)
f x x 2 cos x ,
3
3 ,5
12 x 1 ,
x x 1
,
1, 2
(iv)
f x
(vi)
f x sin x cos x ,
9x
2
,
1, 2 0, 3
,
(1) (2)
Find the local maximum and minimum values of the following functions: (ii) 2 x 3 5 x 2 4 x (i) x 3 x EXERCISE 5.10. Find the numbers whose sum is 100 and whose product is a maximum. Find two positive numbers whose product is 100 and whose sum is minimum.
(6)
Resistance to motion F, of a moving vehicle is given by
F
5
100 x .
Determine the minimum value of
x
resistance. Example 5.62 : Determine where the curve y x 3 3 x 1 is can cave upward and where it is concave downward. Also find the inflection points. Example 5.65 : Determine the points of inflection if any, of the function y x 3 3 x 2 Example 5.66 : Test for points of inflection of the curve
y sin x , x 0 , 2
EXERCISE 5.11. Find the intervals of concavity and the points of inflection of the following functions: (1) f x x 11 3 (3) f x 2 x 3 5 x 2 4 x -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 6.2 : Compute the values of y and dy if y f x x 3 x 2 2 x 1 where x changes (i) from 2 to 2.05 and (ii) from 2 to 2.01. Example 6.3 : Use differentials to find an approximate value for 3 65 .
E R K HSS –ERUMIYAMPATTI
Page 32
+2 STUDY MATERIALS
www.tnschools.co.in
Example 6.5 : The time of swing T of a pendulum is given by T k where k is a constant. Determine the percentage error in the time of swing if the length of the pendulum / changes from 32.1 cm to 32.0 cm. Example 6.6 : A circular template has a radius of 10 cm 0 .02 . Determine the possible error in calculating the area of the templates. Find also the percentage error. Example 6.7. Show that the percentage error in the nth root of a number is approximately
1
times the percentage in the
n
number. (3)
EXERCISE 6.1. Use differentials to find an approximate value for the given number (i)
1
(ii)
36 . 1
(iv) 1 .97 6
10 . 1
(4)
The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error in computing (i) the volume of the cube and (ii) the surface area of cube. (5) The radius of a circular disc is given as 24cm with a maximum error in measurement of 0.02cm. (i) Use differentials to estimate the maximum error in the calculated area of the disc. (ii) Compute the relative error? Example 6.15: If
u log tan x tan y tan z ,
prove that
sin 2 x
Example 6.16: If U x y y z z x then show that
U
x
U
y
u x
2
U
z
0
Example 6.17: 2
x Suppose that z ye where
x 2t
and
y 1 t
then find
dz dt
Example 6.19: If
w x 2y z
and x cos t ; y sin t ; z t . Find
2
dw
.
dt
Example 6.21: 2
If u is a homogenous function of x and y of degree n , prove that
x
u x y
2
y
u y
2
n 1
u y
EXERCISE 6.3. 2
(1)
Verify (i)
(3)
2
u x y
u x
2
u
y x
3 xy y
for the following function:
2
Using chain rule find
dw
for each of the following:
dt
(i) (ii)
(4)
we
xy
where
w log x
2
x
(iii)
w
(iv)
w xy z
x
(i) Find (ii) Find (iii) Find
y
2
y
w r w u w u
2
2
x t
2
, yt
where
and and and
w w v w v
t
x e , y e
t
x cos t , y sin t
where
where
3
x cos t , y sin t , z t
if
w log x
if if
E R K HSS –ERUMIYAMPATTI
w x
2
w sin
2
y
y 1
2
xy
2
where
where
x r cos , y r sin
x u
where
2
v
2
, y 2 uv
x u v , y u v.
Page 33
+2 STUDY MATERIALS
www.tnschools.co.in
Using Euler‟s theorem prove the following:
(5)
(ii)
u xy
2
x sin y
show that
u
x
u
y
x
3u .
y
2
(iii) If u is a homogeneous function of x and y of degree n, prove that (iv) If x
ax by
V ze
V
V
y
x
x
u x
2
2
y
u x y
n 1
u x
and z is a homogenous function of degree n in x and y prove that ax by n V .
y
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS (ONE QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 7.1. Evaluate the following problems using second fundamental theorem: 1
(3) (7)
9 4x
2
2
4 2
(4)
dx
0
0
2
sin
1
1
dx
x
2
(8)
5x 6
(6)
2 sin x sin 2 x dx
0
1
3
x
1 x
0
sin x dx 9 cos
2
x
2
(9)
dx
1
(10)
sin 2 x cos x dx
2
2
x
x e dx 0
0
2
Example 7.7 :
Evaluate :
x sin x dx
2
2
Example 7.8 :
Evaluate
2
sin
x dx
2
f sin x
2
Example 7.9:
Evaluate
f sin x f cos x
0 1
Example 7.10:
x 1 x
Evaluate
dx
n
dx
0
2
Example 7.11:
log tan x dx
Evaluate
0 3
Example 7.12:
Evaluate
6
dx 1
cot x
EXERCISE 7.2. Evaluate the following Problems using properties of integration. 2
2
sin x cos x dx 3
(3) 1
(7)
2
3
(5)
cos x dx
3
(8)
1
x dx
x
0
Example 7.13:
Evaluate:
sin
Example 7.14:
Evaluate
sin
Example 7.15:
Evaluate
sin
2
x cos x dx
2
2
0
1 1 dx log x 0
(4)
5
6
3 x
3
x 1 x dx
(10)
6
0
dx 1
tan x
x dx
x dx 2
(iii)
(9)
10
sin
x
9
6
dx
4
0
(iv)
cos
7
3 x dx
0
2
Example 7.16:
Evaluate
4
sin x cos
2
x dx
0 1
Example 7.17:
Evaluate
(i)
3
x e
2x
dx
(ii)
xe
4x
dx
0
E R K HSS –ERUMIYAMPATTI
Page 34
+2 STUDY MATERIALS
www.tnschools.co.in
EXERCISE 7.3. (1)
Evaluate :
(i)
sin
4
x dx
(ii)
8
(ii)
4
(3)
Evaluate :
(i)
cos
cos
5
x dx
6
2 x dx
sin
7
3 x dx
0
0
1
(4)
Evaluate :
(i)
xe
2 x
dx
0
Example 7.18: Find the area of the region bounded by the line Example 7.19: Find the area of the region bounded by the line Example 7.20: Find the area of the region bounded by the line Example 7.21: Find the area of the region bounded by the line Example 7.22: Find the area of the region bounded by the line Example 7.23: (i) Evaluate the integral
3x 2 y 6 0 , 3 x 5 y 15 0 , y x
2
5x 4 ,
x 1, x 3
and x-axis.
x 1, x 4
and x-axis.
x 2, x 3
and the x-axis.
y 2x 1,
y 3, y 5
and y - axis.
y 2x 4 ,
y 1, y 3
and y - axis.
5
x 3 dx
1
(ii) Find the area of the region bounded by the line y 3 x , x = 1 and x = 5 Example 7.24: Find the area bounded by the curve y sin 2 x between the ordinates x = 0 , x = and x – axis. Example 7.35: Find the volume of the solid that results when the ellipse x
2
a
2
y
2
b
2
1
a
b 0
is revolved about the minor axis.
Example 7.36: Find the volume of the solid generated when the region enclosed by y x , y 2 and x 0 is revolved about the y – axis. EXERCISE 7.4. x y 1 and (1) Find the area of the region bounded by the line (i) x - axis , x = 2 and x = 4 (ii) x - axis , x = -2 and x = 0 x 2 y 12 0 and (2) Find the area of the region bounded by the line (i) y - axis , y = 2 and y = 5 (ii) y - axis , y = -1 and y = -3 (3) Find the area of the region bounded by the line y x 5 and the x-axis between the ordinates x = 3 and x = 7 (5) Find the area of the region bounded by x 2 36 y , y - axis , y = 2 and y = 4. (6) Find the area included between the parabola y 2 4 ax and its latus rectum. (10) Find the area of the circle whose radius is a. Find the volume of the solid that result when the region enclosed by the given curves: (11) y 1 x 2 , x 1 , x 2 , y 0 is revolved about the x – axis. (12) 2 ay 2 x x a (13) (14)
2
is revolved about x - axis, a>0.
y x , x 0 , y 1 is revolved about the y - axis. 3
x
2
a
2
y
2
b
2
1 is revolved about major axis a b 0 .
(16) The area of the region bounded by the curve xy 1 , x - axis, generated by revolving the area mentioned about x - axis.
E R K HSS –ERUMIYAMPATTI
Page 35
x 1
and x . Find the volume of the solid
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 8. DIFFERENTIAL EQUATIONS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 8.2: Form the differential equation from the following equations. (iii) Ax 2 By 2 1 EXERCISE 8.1. (2) Form the differential equations by eliminating arbitrary constants given in brackets against each (ix) y Ae 2 x cos 3 x B A , B (3)
Find the differential equation of the family of straight lines
y mx
a
when (i) m is the parameter ;
m
(ii) a is the parameter ; (iii) a , m both are parameters. Find the differential equation that will represent the family of all circles having centres on the x-axis and the radius is unity. Example 8.3:
(4)
dy
Solve :
1 x y xy
dx
Example 8.4: Solve: 3e x tan y dx 1 e x sec 2 y dy 0 Example 8.5: 1
1 y2 dx 1 x 2
dy
Solve:
2 0
Example 8.6: Solve:
e
x
1 y
2
dx
y
dy 0
x
Example 8.8:
4
Solve: x dy y 4 x 5 e x dx Example 8.9: Solve: x 2 y dx y 2 x dy 0 , if it passes through the origin. Example 8.11: The normal lines to a given curve at each point (x, y) on the curve pass through the point (2,0). The curve passes through the point (2, 3). Formulate the differential equation representing the problem and hence find the equation of the curve. EXERCISE 8.2. Solve the following: (1) sec 2 x dy sin 5 x sec 2 y dx 0 (2) cos 2 x dy ye tan x dx 0 (3)
x
2
yx
(5)
x
2
5 x 7 dy
2
dy y
2
xy
2
dx 0
(4) yx 2 dx e x dy 0
9 8 y y dx 0 2
(6)
dy dx
sin
x y
(8) y dx x dy e xy dx if it cuts the y-axis. Example 8.12: dy
Solve :
dx
y
tan
x
y x
Example 8.16: Solve: xdy ydx
x y dx 2
2
EXERCISE 8.3. Solve the following: (1)
dy dx
y x
y
2
x
2
E R K HSS –ERUMIYAMPATTI
Page 36
+2 STUDY MATERIALS
www.tnschools.co.in
(2)
dy
dx
(3)
x
x x 3 y
y
2
(4) x 2
y x 2 y
dy
2
dy xy
dx
y 2 xy given that y = 1, when x = 1. 2
dx
(6) Find the equation of the curve passing through (1,0) and which has slope
1
y x
Example 8.17:
dy
Solve:
at x , y .
y cot x 2 cos x
dx
Example 8.20: Example 8.21:
Solve: x 1 dy
Solve:
dy
ye
x
dx
x 1 2
2 y tan x sin x
dx
EXERCISE 8.4. Solve the following: (1)
dy
yx
dy
(2)
dx
(6)
dy
dx
(8) y x
xy x
dx
D D D
4x
y
x 1 2
dy
a
Solve:
Example 8.26:
Solve:
Example 8.27:
Solve:
Example 8.28: Example 8.29:
Solve: 2 D 2 5 D 2 y e Solve: D 2 4 y sin 2 x
Example 8.30:
Solve:
Example 8.31: Example 8.32: Example 8.33:
Solve: Solve: Solve:
6D 8 y e
6D 9 y e
2
D D D D
1
dy
2 xy cos x
dx
dx
Example 8.25:
2
x
2
2
13 D 12 y e
2
(4) 1 x 2
1 2
2 x
2 x 3x
1
x
2
4 D 13 y cos 3 x
2
9 y sin 3 x
2 2
2
3D 2 y x
4 D 1 y x
2
EXERCISE 8.5. Solve the following differential equations: (1) D 2 7 D 12 y e 2 x (3) (5) (7)
D D
D
(2)
2
14 D 49 y e
2
1 y 0 when x = 0 , y = 2 and when x
2
3D 4 y x
7 x
4
(4)
D D
2 2
4 D 13 y e
3 x
13 D 12 y e
2 x
5e
x
,y2
2
2
(8)
D
2
2 D 3 y sin x cos x
(12) D 2 5 y cos 2 x (9) D 2 y 9 sin 3 x (13) D 2 2 D 3 y sin 2 x (14) 3 D 2 4 D 1 y 3e x 3 Example 8.36 : The temperature T of a cooling object drops at a rate proportional to the difference T - S, where S is constant temperature of surrounding medium. If initially T 150 C , find the temperature of the cooling object at any time t.
E R K HSS –ERUMIYAMPATTI
Page 37
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 9. DISCRETE MATHEMATICS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 9.4 : Construct the truth table for the following statements : (iv) ~ ~ P ~ q (iii) p q ~ q Example 9.5: Construct the truth table for p q ~ r Example 9.6 : Construct the truth table for
pq
r
EXERCISE 9.2. Construct the truth tables for the following statements: p q ~ p q (7) pq r (9) (10) p q r Example 9.7 : Show that ~ p q ~ P ~ q Example 9.10 : (i) Show that ~ P ~ q p is a tautology. (ii) Show that ~ q p q is a contradiction. Example 9.11 : Use the truth table to determine whether the statement ~ P q p ~ q is a tautology. EXERCISE 9.3 (1) Use the truth table to establish which of the following statements are tautologies and which are contradictions . (i) ~ P q p (ii) p q ~ p q (iii) p ~ q ~ p q (iv) q p ~ q (v) p ~ p ~ q p (2) Show that p q ~ p q (3) Show that p q p q q p (4) Show that p q ~ p q ~ q p (5) Show that ~ p q ~ p ~ q (6) Show that p q and q p are not equivalent. (7) Show that p q p q is a tautology. Example 9.12 : Prove that Z , is an infinite abelian group. Example 9.13 : Show that R 0, . is an infinite abelian group. Here denotes usual multiplication. Example 9.14 : Show that the cube roots of unity forms a finite abelian group under multiplication. Example 9.15 : Prove that the set of all 4th roots of unity forms an abelian group under multiplication. Example 9.16 : Prove that C , is an infinite abelian group. Example 9.17 : Show that the set of all non-zero complex numbers is an abelian group under the usual multiplication of complex numbers.
E R K HSS –ERUMIYAMPATTI
Page 38
+2 STUDY MATERIALS
www.tnschools.co.in
Example 9.19: Show that the set of all 2 X 2 non-singular matrices forms a non-abelian infinite group under matrix multiplication, (where the entries belong to R). Example 9.20 : Show that
1 0
0 1 , 1 0
0 1 , 1 0
0 1 , 1 0
0 1
form an abelian group, under multiplication of matrices.
PROPERTIES: (1) State and prove cancellation laws on groups. (2) State and prove reversal law on inverse of a group. -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 10. PROBABILITY DISTRIBUTIONS (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------Example 10.1 : Find the probability mass function, and the cumulative distribution function for getting „ 3 „ s when two dice are thrown. Example 10.4 : A continuous random variable X follows the probability law, kx 1 x 10 , 0 x 1 f x 0 , elsewhere
Find k.
Example 10.5 : A continuous random variable X has p.d.f. f x 3 x 2 , (i) P X a P X a and Example 10.6 :
(ii)
0 x 1
, Find a and b such that.
P X b 0 . 05
If the probability density function of a random variable is given by Find (i) k Example 10.8 : If
k 1 x 2 , 0 x 1 f x 0 , elsewhere
(ii) the distribution function of the random variable .
A 3 , 1 x e f x x 0 , elsewhere
is a probability density function of a continuous random variable X ,
find p x e (1) (2) (3) (4)
EXERCISE 10.1 Find the probability distribution of the number of sixes in throwing three dice once. Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of queens. Two bad oranges are accidentally mixed with ten good ones. Three oranges are drawn at random without replacement from this lot. Obtain the probability distribution for the number of bad oranges. A discrete random variable X has the following probability distributions.
X
0
1
2
3
4
5
6
7
P(x)
a
3a
5a
7a
9a
11a
13a
15a
(i) Find the value of a
E R K HSS –ERUMIYAMPATTI
(ii) Find P x 3
8
17a
(iii) Find P 3 x 7
Page 39
+2 STUDY MATERIALS
www.tnschools.co.in
(6)
For the p.d.f
cx 1 x 3 , 0 x 1 f x 0 , elsewhere
1 P x 2
find (i) the constant C (ii)
(8)
For the distribution function given by
0 2 F x x 1
x0 0 x 1 x 1
Find the density function. Also evaluate (i) P 0 .5 X 0 .75 (10) A random variable X has a probability density function k f x 0
Find (i) k
(ii)
P X 0 .5
(iii)
P X 0 . 75
, 0 x 2 , elsewhere
(ii)
P 0 X 2
(iii)
3 X P 2 2
Example 10.11 : Two unbiased dice are thrown together at random. Find the expected value of the total number of points shown up. Example 10.12 : The probability of success of an event is p and that of failure is q. Find the expected number of trials to get a first success. Example10.13 : An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in three draws when a ball is drawn at random with replacement. Also find its mean and variance. Example 10.14 : A game is played with a single fair die. A player wins Rs. 20 if a 2 turns up, Rs. 40 if a turns up, loses Rs. 30 if a 6 turns up. While he neither wins nor loses if any other face turns up. Find the expected sum of money he can win. Example 10.15 : In a continuous distribution the p.d.f of X is 3 x 2 x ,0 x 2 f x 4 0 , otherwise
Find the mean and the variance of the distribution.
Example 10.16: Find the mean and variance of the distribution
(1) (3)
(4) (5)
3 e 3 x f x 0
, 0 x , elsewhere
EXERCISE 10.2 A die is tossed twice. A success is getting an odd number on a toss. Find the mean and the variance of the probability distribution of the number of successes. In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses half mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random Two cards are drawn with replacement from a well shuffled deck of 52 cards. Find the mean and variance for the number of aces. In a gambling game a man wins Rs. 10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.
E R K HSS –ERUMIYAMPATTI
Page 40
+2 STUDY MATERIALS
www.tnschools.co.in
(6)
The probability distribution of a random variable X is given below: X
0
1
2
3
P ( X =x )
0.1
0.3
0.5
0.1
If Y X 2 X find the mean and variance of Y. Find the Mean and Variance for the following probability density functions: 2
(7)
1 f x 24 0
(i)
e f x 0
(ii) (iii)
, 12 x 12 , otherwise x
x ex f x 0
, if x 0 , otherwise , if x 0 , otherwise
Example 10.17 : Let X be a binomially distributed variable with mean 2 and standard deviation
2
. Find
the corresponding
3
probability function. Example 10.18 : A pair of dice is thrown 10 times. If getting a doublet is considered a success find the probability of (i) 4 success (ii) No success. EXERCISE 10.3 (4) Four coins are tossed simultaneously. What is the probability of getting (a) exactly 2 heads (b) at least two heads (c) at most two heads. (5) The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the examination what is the probability that at least 5 pass the examination. Example 10.23 : If a publisher of non-technical books takes a great pain to ensure that his books are free of typological errors, so that the probability of any given page containing atleast one such error is 0.005 and errors are independent from page to page (i) what is the probability that one of Rs. 400 page novels will contain exactly one page with error. (ii) atmost three pages with errors. e 2 0 . 1363 ; e 0 .2 0 . 819 . Example 10.24 : Suppose that the probability of suffering a side effect from a certain vaccine is 0.005. If 1000 persons are inoculated. Find approximately the probability that (i) atmost 1 person suffer. (ii) 4,5 or 6 persons suffer. e 5 0 .0067 . Example 10.25 : In a Poisson distribution if P X 2 P X 3 find P X 5 given e 3 0 . 050 . EXERCISE 10.4 (3) 20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii) Poisson distribution.
e
(4)
(3)
2
0 . 1353 .
Alpha particles are emitted by a radioactive source at an average rate of 5 in a 20 minutes interval. Using Poisson distribution find the probability that there will be (i) 2 emission (ii) at least 2 emission in a particular 20 minutes interval. e 5 0 .0067 . EXERCISE 10.5 Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet across the United States is a random variable having a normal distribution with a mean of 4.35 m rem and a standard deviation of 0.59 m rem. What is the probability that a person will be exposed to more than 5.20 m rem of cosmic radiation of such a flight. [Area table : P(0
E R K HSS –ERUMIYAMPATTI
Page 41
+2 STUDY MATERIALS
www.tnschools.co.in
(4)
(6)
(7)
The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are issued, how many pairs would be expected to need replacement within 12 months. [ Area table : P(0
E R K HSS –ERUMIYAMPATTI
Page 42
+2 STUDY MATERIALS
www.tnschools.co.in
COME BOOK – THREE MARKS (For two subdivisions – Each 3 Mark ) -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT:1. MATRICES AND DETERMINANTS -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 1.1 (1) Find the adjoint of the following matrices: (i)
(1)
1 4
3 2
(ii)
1 0 2
3 0 3
2 5 4
(iii)
2 3 1
3 2 1
5 1 2
EXERCISE 1.4 Solve the following non-homogeneous system of linear equations by determinant method: (ii) 2 x 3 y 5 ; 4 x 6 y 12 (i) 3 x 2 y 5 ; x 3 y 4 a A c
Example 1.1: Find the adjoint of the matrix
b d
Example 1.5: Find the inverse of the following matrices: 1 1
(i)
2 4
Example 1.11:
(ii)
2 4
1 2
cos sin
(iii) 2
1 2 5
Find the rank of the matrix
sin cos
3 6 1
4 1
Example 1.17: Solve the following system of linear equations by determinant method. (1) x y 3 ; 2 x 3 y 7 (3) x y 2 ; 3 y 3 x 7 Example 1.20: Solve: x y 2 z 0 ; 2 x y z 0 ; 2 x 2 y z 0 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 2. VECTOR ALGEBRA -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 2.1 a i j 2k
b 3i 2 j k
and
(2)
If
(5)
Find the angles which the vector
(6)
Show that the vector
(7)
If (i)
a
and
cos
2
(8)
b
1
i j k
find
i j
a 3 b 2 a b
makes with the coordinate axes.
2 k
is equally inclined with the coordinate axes.
are unit vectors inclined at an angle ab
(ii)
tan
2
2
, then ab ab
prove that
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is
(12) Show that the vectors
3i 2 j k , i 3 j 5k
(13) Show that the points whose position vectors
and
2 i j 4 k
3.
form a right angled triangle.
4i 3j k , 2i 4 j 5k ,
i j
form a right angle.
EXERCISE 2.2 (6)
A force magnitude 5 units acting parallel to (5,3,7). Find the work done.
(7)
The constant forces position
(3)
2 i 2 j k
2 i 5 j 6 k , i 2 j k
4 i 3 j 2 k
to position
6 i
and
displaces the point of application from (1,2,3) to
2 i 7 j
act on a particle which is displaced from
j 3 k .
Find the work done. EXERCISE 2.3
Find the unit vectors perpendicular to the plane containing the vectors
E R K HSS –ERUMIYAMPATTI
Page 43
2 i
j k
and
i 2 j k
+2 STUDY MATERIALS
www.tnschools.co.in
(4)
Find the vectors whose length 5 and which are perpendicular to the vectors
a 3i j 4 k
and
b 6 i 5 j 2k
(7) (8)
If a i 3 j 2 k and separately. For any three vectors
b i 3k
then find
a b
. Verify that
show that a b c b c a
a , b , c
and
a
b
are perpendicular to
a b
c a b 0
(2)
show that a d and b c parallel. EXERCISE 2.4 Find the area of parallelogram ABCD whose vertices are A (-5,2,5) , B (-3,6,7) , C (4,-1,5) and D (2,-5,3) Find the area of the parallelogram whose diagonals are represented by 2 i 3 j 6 k and 3 i 6
(3) (4) (9)
Find the area of the parallelogram determined by the sides i 2 j 3 k and 3 i 2 j k Find the area of the triangle whose vertices are (3,-1,2) , (1,-1,-3) and (4,-3,1) Show that torque about the point A (3,-1,3) of a force 4 i 2 j k trough the point B (5,2,4) is
(10) If a (1)
b c d
and
a c b d ,
j 2 k
i 2 j 8 k .
EXERCISE 2.5 a b , b c ,c a
(1)
Show that vectors
(2)
The volume of a parallelepiped whose edges are represented by 546. Find the value of .
(3)
Prove that
(6)
Prove that a b c b c a
(9)
prove that i a i j a j k a k 2 a EXERCISE 2.6 (i) Can a vector have direction angles 30 , 45 , 60 . (ii) Can a vector have direction angles 45 , 60 , 120 ? What are the d.c.s of the vector equally inclined to the axes? Find the direction cosines of the line joining (2,-3,1) and (3,1,-2). Find the angle between the following lines.
(2) (3) (5) (8)
x 1 2
(9)
a
For any vector
y 1
abc
if and only if
x 1
y2
a , b , c
c a b 0
2
are coplanar.
12 i k , 3 j k , 2 i j 15 k
is
are mutually perpendicular.
z4 2
Find the angle between the lines.
i 4 j 2 k 3 i 4 k
r 2 i k
(1)
c
and
6
r 5i 7 j
(6)
b
a
z4
3
are coplanar if and only if
a , b , c
EXERCISE 2.7. If the points ( , 0,3 ) , (1,3,-1) and (-5,-3,7) are collinear then find EXERCISE 2.10 Find the angle between the following planes: (i) 2 x y z 9 and x 2 y z 7 (ii) 2 x 3 y 4 z 1 and x y 4 (iii)
r 3i
j k
7
r
and
i 4
j 2k
Show that the following planes are at right angles. r 2 i j k 15 and r i j 3 k 3
(3)
The planes
(4)
Find the angle between the line
E R K HSS –ERUMIYAMPATTI
10
and
x2 3
r i 3 j k y 1 1
.
10
(2)
r 2i j 3k
z3 2
5 are perpendicular. Find .
and the plane
Page 44
3x 4 y z 5 0
+2 STUDY MATERIALS
www.tnschools.co.in
(5) (4) (5)
2 i
r i j 3 k
Find the angle between the line
EXERCISE 2.11 If A (-1,4,-3) is one end of a diameter AB of the sphere x 2 y 2 coordinates of B. Find the centre and radius of each the following spheres.
(iii) x 2 Example 2.6:
y
2
z
2
For any vector Example 2.8:
4x 8y 2z 5
r
prove that
For any two vectors
r
and
a
2
(iv)
b
r i i r j prove that
r
j
r k k
2
and the plane
z
4 i 2
r
a b
j k
a b
2
i j 1.
3 x 2 y 2 z 15 0
j 6 k
2 a
2
r
2
, then find the
11 0
2
b
Example 2.9: a
If
and
Example 2.10: If a b c Example 2.11:
b
are unit vectors inclined at an angle , then prove that
0 ,
a 3 , b 5
Show that the vectors Example 2.20: If a
, b
and
2 i j k
c 7 , ,
sin
find the angle between
i 3 j 5 k
are any two vectors, then prove that
a b
3 i 4 j 4 k
, 2
a b
2
a
a b
1
2
2
and
b
.
form the sides of a right angled triangle.
2
a
2
b
Example 2.21: Find the vectors of magnitude 6 which are perpendicular to both the vectors
4 i j 3 k
and
2 i j 2 k
Example 2.22: If
a 13 , b 5 and
a b 60
then find
a b
Example 2.23: Find the angle between the vectors 2 i j k and Example 2.30:
i 2 j k
Show that the area of a parallelogram having diagonals Example 2.31: A force given by (2,-1,3). Example 2.32:
3 i 2 j 4 k
If the edges a 3 i 7 j of the parallelopiped. Example 2.34: If
5k ,
x .a 0 , x .b 0 , x .c 0
Example 2.35:
If a b c Example 2.41:
a b c
and
3 i
by using cross product. j 2 k
and
is
5 3
.
is applied at the point (1,-1,2) . Find the moment of the force about the point
b 5i 7 j 3k ,
x 0
c 7i 5j 3k
then show that
then prove that
c a b
a , b , c
meet at a vertex, find the volume
are coplanar.
0
Find the angle between the lines r 3 i 2 j k t i 2 j 2 k and Example 2.47: Find the value of if the points (3,2,-4) , (9,8,-10) and (,4,-6) are collinear. Example 2.60: Find the angle between 2 x y z 4 and x y 2 z 4
E R K HSS –ERUMIYAMPATTI
i 3 j 4 k
Page 45
r 5 j 2 k s 3 i 2 j 6 k
+2 STUDY MATERIALS
www.tnschools.co.in
Example 2.61: Find the angle between the line
r 3i 2 j 6 k
i 2
r
0
2 i
j k
j 2 k
and the plane
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 3.COMPLEX NUMBERS -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 3.1 (1) Express the following in the standard form a + ib 2 i 3 (i) 1 i 2 (2) Find the real and imaginary parts of the following complex numbers: (i) (3) (1) (3) (6)
(1) (1)
1
(ii)
1 i
2 5i 4 3i
Find the least positive integer n such that
1 i 1 i
n
1
EXERCISE 3.2 show that 2 . 5 . 10 . . . 1 n 2 x 2
If 1 i 1 2 i 1 3i . . . 1 ni x iy If z 2 0 ,1 find z. Express the following complex numbers in polar form. (i) 2 2 3 i (ii) 1 i 3 (iii) 1 i EXERCISE 3.4 cos 2 i sin 2 7 cos 3 i sin 3 5 Simplify : cos 4 i sin 4 12 cos 5 i sin 5 6 EXERCISE 3.5 Find all the values of the following:
y
2
(iv)
1 i
1
(i) i 3 5
(3)
Prove that if
3
1,
then
(ii)
5
1 i 3 1 i 3 1 2 2
Example 3.4: Express the following in the standard form of a + ib (iv)
5 5i 3 4i
Example 3.5: Find the real and imaginary parts of the complex number z
3i
20
i
19
2i 1
Example 3.6: If
z 1 2 i , z 2 3 2i
and
z3
1
2
3
find the conjugate of (i)
i
2
Example 3.8: Find the modulus or the absolute value of Example 3.12: Graphically prove that
z1 z 2 z 3
z1 z 2
(ii) z 3 4
1 3 i 1 2 i 3 4 i
z1 z 2 z 3
Example 3.18: Simplify :
cos 2 i sin 2 3 cos 3 i sin 3 3 cos 4 i sin 4 6 cos i sin 8
E R K HSS –ERUMIYAMPATTI
Page 46
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 4. ANALYTICAL GEOMETRY -------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 4.1 (1) Find the equation of the parabola if (i) Focus : (2,-3) ; directrix : 2 y 3 0 (ii) Focus : (-1,3) ; directrix : 2 x 3 y 3 (iii) Vertex : (0,0) ; focus : (0,-4) (iv) Vertex : (1,4) ; focus : ( -2,4) (v) Vertex : (1,2) ; latus rectum : y = 5 (vi) Vertex : (1,4) ; open leftward and passing through the point (-2,10) (vii) Vertex : (3,-2) ; open downward and the length of the latus rectum is 8. (viii) Vertex : (3,-1) ; open rightward ; the distance between the latus rectum and the directrix is 4. (ix) Vertex : (2,3) ; open upeard ; and passing through the point (6,4). EXERCISE 4.2 (1) Find the equation of the ellipse if (i) one of the foci is (0,-1) , the corresponding directrix is
3 x 16 0
and
e
3 5
Example 4.1: Find the equation of the following parabola with indicated focus and directrix. (ii) 1, 2 ; x 2 y 3 0 (iii) 2 , 3 ; y 2 0 (i) a ,0 ; x a a 0 Example 4.2: Find the equation of the parabola if (i) the vertex is (0,0) and the focus is (-a,0) , a > 0 (ii) the vertex is (4,1) and the focus is (4,-3) Example 4.3: Find the equation of the parabola whose vertex is (1,2) and the equation of the latus rectum is x = 3. Example 4.4: Find the equation of the parabola if the curve is open rightward, vertex is (2,1) and passing through point (6,5) Example 4.5: Find the equation of the parabola if the curve is open upward, vertex is (-1,-2) and the length of the latus rectum is 4. Example 4.6: Find the equation of the parabola if the curve is open leftward, vertex is (2,0) and the distance between the latus rectum and directrix is 2 Example 4.40: Find the equation of the hyperbola whose transverse axis is parallel to y - axis , centre (0,0) , length of semiconjugate axis is 4 and eccentricity is 2. Example 4.60: Find the equation of the tangent at t = 1 to the parabola y 2 12 x . Example 4.63: Find the equation of chord of contact of tangents from the point (2,4) to the ellipse 2 x 2 5 y 2 20 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I -------------------------------------------------------------------------------------------------------------------------------------------------------Example 5.2: The luminous intensity I candelas of a lamp at varying voltage V is given by : I 4 10 4 V 2 . Determine the voltage at which the light is increasing at a rate of 0.6 candelas per volt. Example 5.21: Using Rolle‟s theorem find the value (s) of c. (i) (ii)
f x
1 x
2
,
f x x a b x ,
1 x 1
a x b , a b.
E R K HSS –ERUMIYAMPATTI
Page 47
+2 STUDY MATERIALS
www.tnschools.co.in f x 2 x 5 x 3
(iii)
2
4x 3,
1
x3
2
Example 5.22: Verify Rolle‟s theorem for the following: 3 f x x 3x 3 0 x 1 (i) (ii) f x tan x , 0 x 1 x 1 (iii) f x x , (iv) (vi)
(1)
x x
f
f
sin
2
x,
0 x
x x 1 x 2 ,
0 x2
EXERCISE 5.3. Verify Rolle‟s theorem for the following functions: (i) f x sin x , (ii) f x x 2 , 0 x (iii)
x
f
x 1 ,
0 x2
(iv)
0 x 1
f x 4 x 9 x, 3
3
x
2
3 2
Example 5.28: Obtain the Maclaurin‟s Series for x e 1) EXERCISE 5.5 (1) Obtain the Maclaurin‟s Series expansion for: (i) e 2 x Example 5.32: Evaluate:
x
2
e
x
lim
x
EXERCISE 5.6 Evaluate the limit for the following if exists. (1)
sin x lim
x 2
2x
lim
x
lim
x 0
1
x
x
(4)
x lim
x 2
n
2
n
x2
2
sin
(5)
sin
(3)
x
log
(7)
1
lim
x
e
x
x
x
(1) (2)
EXERCISE 5.7 Prove that e is strictly increasing function on R. Prove that log x is strictly increasing function on 0 ,
(3)
Which of the following functions are increasing or decreasing on the interval given?
x
(i)
x
2
0 , 2
1 on
Example 5.44: Prove that Example 5.47:
e
x
1 x
(ii) for all
Find the critical numbers of
2x
2
3x
on
1 1 , 2 2
(iii)
e
x
on 0 , 1
x 0. 3
x
5
4 x
EXERCISE 5.9 Find the critical numbers and stationary points of each of the following functions. (i) f x 2 x 3 x 2 (ii) f x x 3 3 x 1 Example 5.59: Determine the domain of concavity (convexity) of the curve y 2 x 2 . Example 5.60: Determine the domain of convexity of the function y e x (1)
E R K HSS –ERUMIYAMPATTI
Page 48
+2 STUDY MATERIALS
www.tnschools.co.in
Example 5.61: Test the curve
y x
4
for points of inflection. EXERCISE 5.11 Find the intervals of concavity and the points of inflection of the following functions: (2) f x x 2 x -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II -------------------------------------------------------------------------------------------------------------------------------------------------------Example 6.4: The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of atmost 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? Example 6.8: Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 1% EXERCISE 6.1 (2) Find the differential dy and evaluate dy for the given values of x and dx. (i) y 1 x 2 ,
x 5 , dx
1 2
(ii) y x 3 x x 1 , x 2 , dx 0 . 1 4
3
(iii) y x 2 5 , x 1 , dx 0 . 05 3
(iv) y 1 x , x 0 , dx 0 . 02
(v) y cos x , x
dx 0 . 05
6
EXERCISE 6.3 (2)
(i) If u
x y 2
x
2
x
(ii) If u e sin y
, show that x y
e cos x
y
y
u
y
x
u y
u.
, show that x
x
u x
y
u y
0.
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS -------------------------------------------------------------------------------------------------------------------------------------------------------Example 7.1: 2
Evaluate
sin x
1 cos
0
2
dx x
Example 7.2: 1
Evaluate
x
x e dx 0
Example 7.3: a
Evaluate
a
2
x
2
dx
0
Example 7.4: 2
Evaluate:
e
2x
cos x dx
0
EXERCISE 7.1 Evaluate the following problems using second fundamental theorem: 2
(1)
2
sin
2
x dx
(2)
1
0
3
x dx
0
0
(5)
cos
4x
2
2
dx
(11) 2
E R K HSS –ERUMIYAMPATTI
e
3x
cos x dx
(12)
e
x
sin x dx
0
0
Page 49
+2 STUDY MATERIALS
www.tnschools.co.in
Example 7.5: 4
Evaluate
x
3
sin
2
x dx
4
Example 7.6: 3 x dx log 3 x 1 1
Evaluate
EXERCISE 7.2 Evaluate the following problems using properties of integration. 1
(1)
sin x cos
4
4
(2)
x dx
1
x
3
cos
3
x dx
4
4
(6)
x sin
2
x dx
4
Example 7.15: 2
Evaluate (i)
2
sin
7
(ii)
x dx
0
cos
8
x dx
0
EXERCISE 7.3 2
(2) Evaluate : (i)
2
sin
6
(ii)
x dx
cos
9
x dx
0
0
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 8. DIFFERENTIAL EQUATIONS -------------------------------------------------------------------------------------------------------------------------------------------------------Example 8.2: Form the differential equation from the equations. (i) y e 2 x A Bx (ii) y e x A cos 3 x B sin 3 x (iv) y 2 4 a x a EXERCISE 8.1 (2) Form the differential equations by eliminating arbitrary constants given in brackets against each a (i) y 2 4 a x (ii) (iv) (v) (vi)
y ax
x
2
a
2
2
a, b
bx c
y
2
b
2
y Ae
2x
a ,b
1 Be
y A Bx e
5 x
3x
A ,B A,B
C, D (vii) y e C cos 2 x D sin 2 x Example 8.20: Solve : D 2 5 D 6 y 0 Example 8.21: Solve: D 2 6 D 9 y 0 Example 8.22: Solve: D 2 D 1 y 0 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 9. DISCRETE MATHEMATICS -------------------------------------------------------------------------------------------------------------------------------------------------------Example 9.4: Construct the truth table for the following statements: (i) ~ p ~ q (ii) ~ ~ p q EXERCISE 9.2 Construct the truth tables for the following statements: (2) ~ p ~ q (3) ~ p q (1) p ~ q (4) p q ~ p (5) p q ~ q (6) ~ p ~ q (8) p q ~ q 3x
E R K HSS –ERUMIYAMPATTI
Page 50
+2 STUDY MATERIALS
www.tnschools.co.in
PROPERTIES: (1) Prove that identity element of a group is unique. (2) Prove that inverse element of an element of a group is unique.
(3) Show that a 1 a a G , a group. -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 10. PROBABILITY DISTRIBUTIONS -------------------------------------------------------------------------------------------------------------------------------------------------------Example 10.7: 1
If
F x
1 tan
2
1
x
x
is a distribution function of a continuous variable X , find
P 0 x 1
Example 10.9: For the probability density function
(5)
2x
, x 0 , x0
, find F(2)
EXERCISE 10.1 Verify that following are probability density functions. (a)
(9)
2 e f x 0
2x f x 9 0
, 0 x3 , elsewhere
(b)
f x
1
1
1 x 2
,
x
A continuous random variable x has the p.d.f defined by ce f x 0
ax
, 0 x
Find the value of c if a > 0.
, elsewhere
EXERCISE 10.2 (2) Find the expected value of the number on a die when thrown. Example 10.19: In a Binomial distribution if n = 5 and P X 3 2 P X 2 find p. Example 10.20: If the sum of mean and variance of a Binomial Distribution is 4.8 for 5 trials find the distribution. Example 10.21: The difference between the mean and the variance of a Binomial distribution is 1 and the difference between their squares is 11. Find n. EXERCISE 10.3 (1) The mean of a binomial distribution is 6 and its standard deviation is 3. Is this statement true or false? Comment. (2) A die is thrown 120 times ad getting 1 or 5 is considered a success. Find the mean and variance of the number of successes. (3) If on an average 1 ship out of 10 do not arrive safely to ports. Find the mean and the standard deviation of ships returning safely out of a total of 500 ships. (6) In a hurdle race a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5 / 6. What is the probability that he will knock down less than 2 hurdles. Example 10.22: Prove that the total probability is one. EXERCISE 10.4 (ii) P 2 X 5 e 4 0 . 0183 . (1) Let X have a Poisson distribution with mean 4. Find (i) P X 3 (2) If the probability of a defective fuse from a manufacturing unit is 2% in a box of 200 fuses find the probability that (i) exactly 4 fuses are defective (ii) more than 3 fuses are defective e 4 0 . 0183 . Example 10.27: Let Z be a standard normal variate. Calculate the following probabilities. (i) P 0 Z 1 .2 (ii) P 1 .2 Z 0 (iii) Area to the right of Z = 1.3 (iv) Area to the left of Z = 1.5 (v) P 1 .2 Z 2 .5 (vi) P 1 .2 Z 0 .5 (vii) P 1 .5 Z 2 .5
E R K HSS –ERUMIYAMPATTI
Page 51
+2 STUDY MATERIALS
www.tnschools.co.in
Example 10.28: Let Z be a standard normal variate. Find the value of c in the following problems. (ii) P c Z c 0 .94 (i) P Z c 0 .05 (iii) P Z c 0 . 05 (iv) P c Z 0 0 .31 EXERCISE 10.5 (1) If X is a normal variate with mean 80 and standard deviation 10, compute the following probabilities by standardizing. (ii) P X 80 (i) P X 100 (iii) P 65 X 100 (iv) P 70 X (v) P 85 X 95 (2) If Z is a standard normal variate, find the value of c for the following (ii) P c Z c 0 .40 (iii) P Z c 0 . 85 (i) P 0 Z c 0 .25 *********
E R K HSS –ERUMIYAMPATTI
Page 52
+2 STUDY MATERIALS
www.tnschools.co.in
COME BOOK – ONE MARK [10 Number Of One Mark Questions to be Asked For Full Test] -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 1. MATRICES AND DETERMINANTS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------1.
2
The rank of the matrix
1
1.1 2.
4 2
2.2 7
The rank of the matrix
2
is 3.0
4. 8
1 is 1
10.
1. 9 2. 2 3. 1 4. 5 If A and B are matrices conformable to multiplication then (AB)T is 1. ATBT 2. BTAT 3. AB 4. BA T -1 (A ) is equal to 1. A-1 2. AT 3. A 4. (A-1)T If (A) = r , then which of the following is correct? 1. all the minors of order r which do not vanish. 2. A has atleast one minor of r which does not vanish and all higher order minors vanish. 3. A has atleast one (r+1) order minor which vanishes. 4. all (r+1) and higher order minors should not vanish. Which of the following is not elementary transformation? 2. Ri 2Ri+Rj 3. Ci Cj+Ci 4. Ri Ri+Cj 1. Ri Rj Equivalent matrices are obtained by 1. taking inverses 2. taking transposes 3. taking adjoints 4. taking finite number of elementary transformations In echelon form, which of the following is incorrect? 1. Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. 2. The first non-zero entry in each non-zero row is 1. 3. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row. 4. Two rows can have same number of zeros before the first non-zero entry If 0 then the system is 1. consistent and has unique solution 2. consistent and has infinitely many solutions 3. inconsistent 4. either consistent or inconsistent In the system of 3 linear equations with three unknowns, if 0 and one of x , y , or z is non-zero then
11.
the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. In the system of 3 linear equations with three unknowns, if 0 , x 0 , y 0 , z 0 and atleast
12.
one 2x2 minor of 0 then the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. In the system of 3 linear equations with three unknowns, if 0 and all 2x2 minors of 0 and atleast
3. 4. 5.
6. 7.
8.
9.
one 2x2 minor of x or y or z is non-zero then the system is
E R K HSS –ERUMIYAMPATTI
Page 53
+2 STUDY MATERIALS
www.tnschools.co.in
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. In the system of 3 linear equations with three unknowns, if 0 and all 2x2 minors of , x , y , z are zeros and atleast one non-zero element is in then the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. Every homogeneous system(linear) 1. is always consistent 2. has only trivial solution 3. has infinitely many solution 4. need not be consistent If ( A ) A , B then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent If ( A ) A , B = the number of unknowns then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent ( A ) A , B then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent In the system of 3 linear equations with three unknowns, ( A ) A , B = 1, then the system 1.has unique solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent In the homogeneous system with three unknowns, ( A ) = number of unknowns then the system has 1. only trivial solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent In the system of three linear equations with three unknowns, in the non-homogeneous system ( A ) A , B = 2, then the system 1. has unique solution 2. reduces to two equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent In the homogeneous system ( A ) < the number of unknowns then the system has 1. only trivial solution 2. trivial solution and infinitely many non-trivial solutions 3. only non-trivial solutions 4. no solution Cramer‟s rule is applicable only (with three unknowns) when 1. 0 2. 0 3. 0 , x 0 4. x y z 0
E R K HSS –ERUMIYAMPATTI
Page 54
+2 STUDY MATERIALS
www.tnschools.co.in
23.
Which of the following statement is correct regarding homogeneous system? 1. always inconsistent 2. has only trivial solution 3. has only non-trivial solutions 4. has only trivial solution only if rank of the coefficient matrix is equal to the number of unknowns
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 2. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------1. 2. 3. 4. 5.
is
The value of a b when a j 2 k and b 2 i k is
2. -2
3. 3
4. 4
1. 7
2. -7
3. 5
4.6
1.-4
2. 8
3. 4
4. 12
The value of a b when a j 2 k and b 2 i 3 j 2 k is
If m i 2 j k and 4 i 9 j 2 k are perpendicular, then m is
If
5i 9 j 2k
and m i 2 j k are perpendicular, then m is
5
5
2. -
3.
16
4. -
2. -
3.
4.
7.
3 The angle between the vectors 3 i 2 j 6 k and 4 i j 8 k is
8.
34 3. sin 63 The angle between the vectors i j and j k is
6
1. cos
1.
1
34 63
1
2. sin
2.
2
3. -
1
34 63
3
4. cos
4.
3
7
8
2.
3.
8
2. -4
66
4.
3. 3
34 63
3
8
8 7 66 a b when a 2 i 2 j k and b 6 i 3 j 2 k is
1. 4
1
2
3 The projection of the vector 7 i j 4 k on 2 i 6 j 3 k is
1.
11.
6
3
10.
16
16 16 5 5 If a and b are two vectors such that a 4 , b 3 and a b = 6, then the angle between a and b is
1.
9.
4. 14
1. 2
1. 6.
The value of a b when a i 2 j k and b 4 i 4 j 7 k 1. 19 2. 3 3. -19
4. 5
If the vectors 2 i j k and i 2 j k are perpendicular to each other, then is
1.
2
2.
2
3.
3
4.
3
3
12.
3 2 2 If the vectors a = 3 i 2 j 9 k and b = i m j 3 k are perpendicular, then „m‟ is
1. -15
13.
2. 15
3. 30
4. -30
If the vectors a = 3 i 2 j 9 k and b = i m j 3 k are parallel, then „m‟ is 1.
3
2.
2
E R K HSS –ERUMIYAMPATTI
2 3
3.
3 2
Page 55
4.
2 3
+2 STUDY MATERIALS
www.tnschools.co.in
14.
1. 3 15. 16. 17. 18.
2. 9 3. 3 3 If a b 60, a b 40 and b 46, then a is
4.
1. 22
3. 18
4. 11
3. 5
4.
3. -1
4. 2
2. 21
1. 25 2. -25 The projection of i j on z-axis is 1. 0 2. 1
The projection of i 2 j 2 k on 2 i j 5 k is 10
10
2.
3.
1
9
9
2. -
21
22.
23.
81
21
21
position vector 3 i j 5 k by a force F i 3 j k is
2. 26
3. 27
2i 6 j 7 k
to the point B, with
4. 28
The work done by the force F a i j k in moving the point of application from (1, 1, 1) to (2, 2, 2)
along a straight line is given to be 5 units. The value of a is 1. -3 2. 3 3. 8
4. -8
1.3 7
4. 69
If a =3, b 4 and a b =9, then a b is
2.63
3.69
if a b = a b is
The angle between the two vectors a and
2.
4 If a
1.
3.
4.
3 6 2 =2, b =7 and a b 3 i 2 j 6 k then the angle between a and b is
2.
4
25.
4. -
The work done in moving a particle from the point A with position vector
1. 24.
81
3.
21
1. 25
21.
10 30
3 30 The projection of 3 i j k on 4 i j 2 k is
1.
5
4.
30
20.
3
Let u , v and w be vector such that u v w 0 . If u 3 , v 4 and w 5 , then u v v w w u is
1. 19.
If a , b , c are three mutually perpendicular unit vectors, then a b c
3.
3
4.
6
2
The direction cosines of a vector whose direction ratios are 2, 3, -6 are
2 3 6 2 3 6 4. , , , , 7 7 49 49 7 7 7 7 7 7 26. The unit normal vectors to the plane 2 x y 2 z 5 are 1 1 1 1. 2 i j 2 k 2. 3. 4. 2i j 2k 2 i j 2k 2 i j 2k 3 3 3 27. The length of the perpendicular from the origin to the plane r . 3 i 4 j 12 k 26 is 2
1.
,
3
,
6 7
2
2.
,
3
,
6 49
3.
1. 26
2.
26 169
28.
The distance from the origin to the plane r .
1.
7
2.
30
E R K HSS –ERUMIYAMPATTI
30 7
3. 2
2 i j 5k
3.
30 7
Page 56
4.
1 2
7 is 4.
7 30
+2 STUDY MATERIALS
www.tnschools.co.in
29.
30. 31.
33. 34.
35.
18 with coordinate of A as
3. (-1, 0, 10)
4. (1, 0, -10)
1. (2, -1, 4) and 5
3. (-2, 1, 4) and 6
4. (2, 1, -4) and 5
The centre and radius of the sphere r ( 2 i j 4 k ) 5 are
2. (2, 1, 4) and 5
The centre and radius of the sphere 2 r ( 3 i j 4 k ) 4 are
3
2
,
, 2 , 4 2
1
3
2
2.
,
, 2 and 2 2
1
3
2
3.
, 2 , 6 2
1
,
4.
3
, 2 and 5 2
1
,
2 a and perpendicular to a The vector equation of a plane passing through a point whose position vector is vector n is 2. r n a n 3. r n a n 4. r n a n 1. r n a n
The vector equation of a plane whose distance from the origin is p and perpendicular to a unit vector nˆ is 1. r . n p 2. r . nˆ q 3. r n p 4. r . nˆ p The non-parametric vector equation of a plane passing through a point whose position vector is a and parallel to u and v is 2. r u v 0 3. r a u v 0 4. a u v 0 1. r a , u , v 0 The non-parametric vector equation of a plane passing through the points whose position vectors are a . b and parallel to v is b a v 0 2. 1. r a
36.
(3, 2, -2). Then the coordinates of B is 1. (1, 0, 10) 2. (-1, 0, -10)
1.
32.
Chord AB is a diameter of the sphere r 2 i j 6 k
b a
r
v 0
3. a b v 0 4. r a b 0 The non-parametric vector equation of a plane passing through three non-collinear points whose position
vectors are a , b , c is
39.
r a
b a
b 0
b
b c 0 37. The vector equation of a plane passing through the line of intersection of the planes r . n 1 q 1 and r . n 2 q 2 is 1. r . n 1 q 1 r . n 2 q 2 0 2. r . n 1 r . n 2 q1 q 2 3. r n 1 r n 2 q1 q 2 4. r n 1 r n 2 q 1 q 2 38. The angle between the line r a t b and the plane r n q is connected by the relation a b b n b n a n 1. cos 2. cos 3. sin 4. sin n q b n b n
1.
ca 0
2.
r
a
3.
r
c
0
The vector equation of a sphere whose centre is origin and radius „a‟ is 1. r a 2. r c a 3. r a
4.
a
4. r a
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 3.COMPLEX NUMBERS (ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------1. The complex number form of 35 is 1. i 35
2.
The complex number form of 31. -3+i 7
3.
2. - i 35
1. 4 ,
3
2. 4 , 3
E R K HSS –ERUMIYAMPATTI
4. 35i
3. 3- i7
4. 3+ i7
3.
4.
7 is
2. 3- i 7
Real and imaginary parts of 4 i
3. i 35
3 are 3 ,4
Page 57
3,4
+2 STUDY MATERIALS
www.tnschools.co.in
4.
3
Real and imaginary parts of
i are
2
1. 0 ,
3
2.
2
3
, 0
5.
The complex conjugate of 2 i 7 is
6.
1. 2 i 7 2. 2 i 7 The complex conjugate of 4 i 9 is 2. 4 i 9 1. 4 i 9
7.
3. 2 , 3
4. 3, 2
3. 2 i 7
4. 2 i 7
3. 4 – i9
4 .– 4 – i9
2
The complex conjugate of
5 is
13.
2. 5 3. i 5 1. 5 The standard form ( a ib ) of 3 2 i ( 7 i ) is 1. 4 i 2. - 4 i 3. 4 i If a ib ( 8 6 i ) ( 2 i 7 ) then the values of „a‟ and „b‟ are 1. 8 , 15 2. 8 , 15 3. 15 , 9 If p+iq = (2-3i)(4+2i), then q is 1. 14 2. -14 3. -8 The conjugate of (2+i)(3-2i) is 1. 8-i 2. -8-i 3. -8+i The real and imaginary parts of (2+i)(3-2i) are 1. -1, 8 2. -8, 1 3. 8, -1 The modulus values of -2+2i and 2-3i are
4. - 4, 1
14.
2. 2 5 , 13 3. 2 2 , 13 1. 5 , 5 The modulus values of -3-2i and 4+3i are 1. 5,5 2. 5 , 7 The cube roots of unity are 1. in G.P. with common ratio
4. 13 ,5
8. 9. 10. 11. 12.
15.
16.
19. 20.
21.
4. 8 4. 8+i 4. -8, -1
3. In A.P. with common difference 4. in A.P. with common difference th The arguments of n roots of a complex number differ by
2
2
2.
3.
n
3
4.
n
Which of the following statements is correct? 1. negative complex numbers exist 3. order relation exist in complex numbers Which of the following are correct?
4 n
2. order relation does not exist in real numbers 4. (1 i ) > (3 2 i ) is meaningless
a. Re ( z ) z
b. I m ( z ) z
c. z z
n n d. z z
1. (a), (b)
2. (b), (c)
3. (b), (c) and (d)
4. (a), (c) and (d)
The values of z z is 2. Re ( z ) 1. 2 Re ( z )
3. Im ( z )
4. 2 Im ( z )
The value of z z 1. 2 Im ( z )
is 2. 2 i Im ( z )
3. I m ( z )
4. i Im ( z )
The value of
is 3. 2 | z |
4. 2 | z | 2
If | z z
zz
2. | z | 2
1. | z | 22.
4. 15 , 8
2
n
18.
6 ,1
4. 4 4 i
2. in G.P. with common difference
1. 17.
3.
4. i 5
1
| | z z 2 | then the locus of z is
1. a circle with centre at the origin 2. a circle with centre at z 1
E R K HSS –ERUMIYAMPATTI
Page 58
+2 STUDY MATERIALS
www.tnschools.co.in
23.
3. a straight line passing through the origin 4. is a perpendicular bisector of the line joining z 1 and z 2 If is a cube roots of unity, then
24.
1. 2 1 2. 1 0 3. 1 The principal value of arg z lies in the interval
1. 0 ,
25.
26.
27. 28.
2. ( , ]
2
2
4. 1
0
0 ,
3.
2
0
4. ( , 0 ]
If z 1 and z 2 are any two complex numbers then which one of the following is false? 1. Re ( z 1 z 2 ) Re ( z 1 ) Re ( z 2 )
2. Im ( z 1 z 2 ) I m ( z 1 ) I m ( z 2 )
3. arg ( z 1 z 2 ) arg z 1 arg z 2 The fourth roots of unity are 1. 1 i , 1 i 2. i , 1 i
4. | z 1 z 2 | | z 1 | | z 2 |
The fourth roots of unity form the vertices of 1. an equilateral triangle 2. a square Cube roots of unity are 1. 1,
1 i
3
2.
i, 1
i
2
3
3. 1, i
4. 1, 1
3. a hexagon
4. a rectangle
3.
1,
1 i
3
4. i ,
2
2
1 i
3
2
p
The number of values of cos i sin where p and q are non-zero integers prime to each other, is 1. p 2. q 3. p+q 4.( p-q) i i is The value of e e 1. 2 cos 2. cos 3. 2 sin 4. sin q
29. 30. 31. 32.
33. 34.
The value of e i e i is 1. sin 2. 2 sin
4. 2 i sin
3. i sin
Geometrical interpretation of z is 1. reflection of z on real axis 2. reflection of z on imaginary axis 3. rotation of z about origin 4. rotation of z about origin through / 2 in clockwise direction If z 1 a ib , z 2 a ib then z 1 z 2 lies on 1. real axis 2. imaginary axis 3. the line y= x Which one of the following is incorrect? 1. (cos i sin ) n cos n i sin n
4. the line y = -x
2. (cos i sin ) n cos n i sin n 3. (sin i cos ) n sin n i cos n 4. 35. 36.
37.
1 cos i sin
cos i sin
Polynomial equation P(x)=0 admits conjugate pairs of imaginary roots only if the coefficients are 1. imaginary 2. complex 3. real 4. either real or complex Identify the correct statement 1. Sum of the moduli of two complex numbers is equal to their modulus of the sum 2. Modulus of the product of the complex numbers is equal to the sum of their moduli 3. Arguments of the product of two complex numbers is the product of their arguments. 4. Arguments of the product of two complex numbers is equal to sum of their arguments. Which of the following is not true? 1. z 1 z 2
z1 z 2
E R K HSS –ERUMIYAMPATTI
2. z 1 z 2
z1 z 2
Page 59
3. Re ( z )
z z 2
4.
Im ( z )
z z 2i
+2 STUDY MATERIALS
www.tnschools.co.in
38.
If z 1 and z 2 are complex numbers then which of the following is meaningful? 1. z 1 z 2
39.
Which of the following is incorrect?
40.
2. Im ( z ) | z | 1. Re ( z ) | z | Which of the following is incorrect? 1. | z 1 z 2 | | z 1 | | z 2 |
41.
3. z 1 z 2
2. z 1 z 2
4. z 1 z 2
3. z z | z |
4. Re ( z ) | z |
2
2. | z 1 z 2 | | z 1 | | z 2 |
3. | z 1 z 2 | | z 1 | | z 2 | Which of the following is incorrect?
4. | z
1
z
2
| | z1 | | z 2 |
1. z is the mirror image of z on the real axis
42.
43.
2. The polar form of z is r , 3. –z is the point symmetrical to z about the origin 4. The polar form of -z is r , Which of the following is incorrect? 1. Multiplying a complex number by i is equivalent to rotating the number counter clockwise about the origin through an angle 90 . 2. Multiplying a complex number by - i is equivalent to rotating the number clockwise about the origin through an angle 90 . 3. Dividing a complex number by i is equivalent to rotating the number counter clockwise about the origin through an angle 90 . 4. Dividing a complex number by i is equivalent to rotating the number clockwise about the origin . through an angle 90 Which of the following is incorrect regarding nth roots of unity? 1. The number of distinct roots is n 2. the roots are in G.P. with common ratio cis
2 n
3. the arguments are in A.P. with common difference
2
.
n 4. product of the roots is 0 and the sum of the roots is 1 .
44.
Which of the following are true? 1. If n is a positive integer then cos i sin cos n i sin n n
2. If n is a negative integer then cos i sin cos n i sin n n
3. If n is a fraction then cos n i sin n is one of the values of ( cos i sin ) n . 4. If n is a negative integer then cos i sin cos n i sin n 1. (i), (ii), (iii), (iv) 2. (i), (iii), (iv) 3. (i), (iv) 4. (i) only If O ( 0 , 0 ), A ( Z 1 ), B ( Z 2 ), B ' ( Z 2 ) are the complex numbers in a argand plane then which of the following are correct? (i) In the parallelogram OACB, represents Z 1 Z 2 n
45.
(ii) In the argand plane E represents Z 1 Z 2 where OE = OA.OB and OE makes an angle arg( z 1 )+arg( z 2 ) with positive real axis. (iii) In the argand parallelogram OB ' DA , D represents Z 1 Z 2 (iv) In the argand plane F represents
Z1
where OF
Z2
with positive real axis. 1. (i), (ii), (iii), (iv)
E R K HSS –ERUMIYAMPATTI
2. (i), (iii), (iv)
OA OB
and OF makes an angle arg( z 1 )-arg( z 2 )
3. (i), (iv)
Page 60
4. (i) only
+2 STUDY MATERIALS
www.tnschools.co.in
46.
If Z = 0, then the arg(Z) is 2.
1. 0
3.
4. Indeterminate
2
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 4. ANALYTICAL GEOMETRY (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. The axis of the parabola y 2 4 x is 1. x 0 2. y 0 3. x 1 4. y 1 2.
The vertex of the parabola y 2 4 x is 2. ( 0 , 1) 1. ( 1, 0 )
3. ( 0 , 0 )
4. ( 0 , 1)
3.
The focus of the parabola y 4 x is 1. ( 0 , 1) 2. (1, 1)
3. ( 0 , 0 )
4. ( 1, 0 )
4.
The directrix of the parabola y 4 x is 1. y 1 2. x 1
3. y 1
4. x 1
2
2
5.
The equation of the latus rectum of y 2 4 x is 2. y 1 3. x 4 1. x 1
4. y 1
6.
The length of the latus rectum of y 4 x is 1. 2 2. 3 3. 1
4. 4
7.
The axis of the parabola x 2 4 y is 1. y 1 2. x 0
3. y 0
4. x 1
8.
The vertex of the parabola x 4 y is 1. ( 0 , 1) 2. ( 0 , 1)
3. ( 1, 0 )
4. ( 0 , 0 )
9.
The focus of the parabola x 4 y is 1. ( 0 , 0 ) 2. ( 0 , 1)
3. ( 0 , 1)
4. (1, 0 )
10.
The directrix of the parabola x 2 4 y is 1. x 1 2. x 0 3. y 1
4. y 0
11.
The equation of the latus rectum of x 4 y is 1. x 1 2. y 1 3. x 1
4. y 1
12.
The length of the latus rectum of x 4 y is 1. 1 2. 2 3. 3 2 The axis of the parabola y 8 x is 1. x 0 2. x 2 3. y 2
13.
2
2
2
2
2
4. 4 4. y 0
14.
The vertex of the parabola y 8 x is 1. ( 0 , 0 ) 2. ( 2 , 0 )
3. ( 0 , 2 )
4. ( 2 , 2 )
15.
The focus of the parabola y 8 x is 1. ( 0 , 2 ) 2. ( 0 , 2 )
3. ( 2 , 0 )
4. ( 2 , 0 )
2
2
16.
The equation of the directrix of the parabola y 8 x is 1. y 2 0 2. x 2 0 3. y 2 0
4. x 2 0
17.
The equation of the latus rectum of y 8 x is 1. y 2 0 2. y 2 0 3. x 2 0
4. x 2 0
18.
The length of the latus rectum of y 8 x is 1. 8 2. 6 3. 4
4. - 8
2
2
E R K HSS –ERUMIYAMPATTI
2
Page 61
+2 STUDY MATERIALS
www.tnschools.co.in
19.
The axis of the parabola x 2 20 y is 2. x 5 1. y 5
3. x 0
4. y 0
20.
The vertex of the parabola x 20 y is 1. ( 0 , 5 ) 2. ( 0 , 0 )
3. ( 5 , 0 )
4. ( 0 , 5 )
21.
The focus of the parabola x 20 y is 2. ( 5 , 0 ) 1. ( 0 , 0 )
3. ( 0 , 5 )
4. ( 5 , 0 )
2
2
22.
The equation of the directrix of the parabola x 20 y is 1. y 5 0 2. x 5 0 3. x 5 0
4. y 5 0
23.
The equation of the latus rectum of the parabola x 20 y is 1. x 5 0 2. y 5 0 3. y 5 0
4. x 5 0
24.
The length of the latus rectum of the parabola x 20 y is 1. 20 2. 10 3. 5 4. 4 If the centre of the ellipse is (2, 3) one of the foci is (3, 3) then the other focus is 1. (1, 3 ) 2. ( 1, 3 ) 3. (1, 3 ) 4. ( 1, 3 )
25.
26.
2
2
2
The equation of the major and minor axes of 1. x 3 , y 2
27.
2
2
y
1 are 9 4 3. x 0 , y 0
4. y 0 , x 0
The equation of the major and minor axes of 4 x 3 y 12 are 2
1. x 28.
2. x 3, y 2
x
3, y 2
2
3. x 3 , y 2
2. x 0 , y 0
x
The length of the minor and major axes of
2
y
4. y 0 , x 0
2
1 are
9
1. 6 , 4 29.
2. 3 , 2 2. 2 ,
3. 2 3 , 4
3 x
The equation of the directrices 4
1. y
2. x
y
3. x
4. y
16
7
7
The equation of the directrices 25x +9y = 225 are 4
2. x
25
3. y
4
7
2. x
4
x
2
y
1 are
3. x 7
The equation of the latus rectum of 25 x 9 y 225 are 1. y 5 2. x 5 3. y 4 x
16
1.
9
2.
2
4
9
7
2
25
2
2
The length of the latus rectum of
4. y
25
The equation of the latus rectum of 1. y
35.
16
2
16
34.
3, 2
1 are
7
25
33.
4.
2
9
2
1. x 32.
2
16 16
7
31.
4. 2 , 3
The length of the major and minor axes of 4 x 2 3 y 2 12 are 1. 4 , 2 3
30.
4 3. 4 , 6
y
4. y 7
2
4. x 4
2
1 is
9
2
3.
9
9
4.
16
16 9
The length of the latus rectum of 25 x 9 y 225 is 2
1.
9
2.
5
E R K HSS –ERUMIYAMPATTI
18 5
2
3.
25 9
Page 62
4.
5 18
+2 STUDY MATERIALS
www.tnschools.co.in
36.
2
x
The eccentricity of the ellipse
y
25
1.
1
2.
5
37.
3.
5
5
x
y
4
2.
1 is
3.
3
4.
5
The foci of the ellipse
3.
2 3
x
x
4.
y
2
y
2
2 5
2
1 is
3.
( 3, 5 )
4. ( 0 , 5 )
2
1
4 9 2. ( 2 , 3 ) x
3 4
2
25 9 2. ( 5 , 0 )
The centre of the ellipse 1. ( 0 , 3 )
3 5
The centre of the ellipse 1. ( 0 , 0 )
41.
5
9
5
5
40.
4
The eccentricity of the ellipse 16x2+25y2= 400 is 1.
39.
4.
2
3
2.
3
2 5
2
4
38.
1 is
9
3
The eccentricity of the ellipse 1.
2
is 3.
(0, 0)
4. ( 3 , 0 )
3.
( 5 , 0 )
4. ( 4 , 0 )
2
y
1 are 9 2. ( 0 , 4 ) 25
1. ( 0 , 5 ) 42.
The foci of the ellipse
x
2
4
43.
44.
2
1 are
9
1. 5 , 0 2. 0 , 5 The foci of the ellipse 16x2+25y2= 400 are 1. 3 , 0 2. 0 , 3 The vertices of the ellipse 1. ( 0 , 5 )
45.
y
x
2
y
4.
3. 0 , 5
4. 5 , 0
5, 0
1 are
25 9 2. ( 0 , 3 )
( 5, 0 )
4. ( 3 , 0 )
1. ( 0 , 3 ) 3. ( 3, 0 ) 2 2 The vertices of the ellipse 16x +25y = 400 are 1. ( 0 , 4 ) 2. ( 5 , 0 ) 3. ( 4 , 0 )
4. ( 0 , 2 )
The vertices of the ellipse
x
2
y
2
3.
2
1 are
4 9 2. ( 2 , 0 )
46.
3. 0 , 5
4. ( 0 , 5 )
47.
If the centre of the ellipse is (4, -2) and one of the foci is (4, 2), then the other focus is 1. ( 4 , 6 ) 2. ( 6 , 4 ) 3. ( 4 , 6 ) 4. ( 6 , 4 )
48.
The equations of transverse and conjugate axes of the hyperbola
x
2
y
2
1 are 4 4. x 0 , y 0 9
1. x 2 ; y 3 49.
3. x 3 ; y 2
The equations of transverse and conjugate axes of the hyperbola 16 y 2 9 x 2 144 are 1. y 0 ; x 0
50.
2. y 0 ; x 0 2. x 3 ; y 4
3. x 0 ; y 0
4. y 3 ; x 4
The equations of transverse and conjugate axes of the hyperbola 144 x 2 25 y 2 3600 are 1. y 0 ; x 0
2. x 12 ; y 5
E R K HSS –ERUMIYAMPATTI
3. x 0 ; y 0
Page 63
4. x 5 ; y 12
+2 STUDY MATERIALS
www.tnschools.co.in
51.
The equations of transverse and conjugate axes of the hyperbola 8y2-2x2 = 16 are 2. x 2 ; y 2 2 3. x = 0 ; y = 0 4. y = 0 ; x = 0 1. x 2 2 ; y 2
52.
The equation of the directrices of the hyperbola
x
2
y
9
1. y =
9
2. x =
13
53.
13
3. y =
4. x =
9
The equation of the directrices of the hyperbola 16 y 9 x 144 are 5
2. y
9
9
3. x
5
2
9
4. y
5
x
The equation of the latus rectum of the hyperbola 1. y 13
y
13
The length of the latus rectum of the hyperbola 4
8
2.
3.
3
34
y
2
x
y
9 3
2
1 is
4
4.
4
1 is
3.
34
5
The centre of the hyperbola 25 x 16 y 400 are 2. ( 0 , 5 ) 3. ( 4 , 5 ) 1. ( 0 , 4 ) y
2
9
1. ( 0 ,
34 )
34
4.
3 2
The foci of the hyperbola
9
2
3
3
2
5
25
5
2.
x
4. x
5
2
The eccentricity of the hyperbola 1.
2
3. y
9
59.
4. x
3. x 13
13
x 5
2.
3
58.
1 are
The equation of the latus rectum of the hyperbola 16 y 9 x 144 are
1. 57.
2
4 2
1. y 5 56.
2. y
5 9
2
9
55.
9 13
2
1. x 54.
1 are
4
9
13
2
2
x
4. ( 0 , 0 )
2
1 are
25
2. ( 34 , 0 )
3. ( 0 , 34 )
4. (
34 , 0 )
60.
The vertices of the hyperbola 25 x 2 16 y 2 400 are 1. ( 0 , 4 ) 2. ( 4 , 0 ) 3. ( 0 , 5 )
4. ( 5 , 0 )
61.
The equation of the tangent at (3, -6) to the parabola y 12 x is 1. x y 3 0 2. x y 3 0 3. x y 3 0
4. x y 3 0
62.
The equation of the tangent at (-3, 1) to the parabola x 9 y is 1. 3 x 2 y 3 0 2. 2 x 3 y 3 0 3. 2 x 3 y 3 0
2
2
4. 3 x 2 y 3 0
63.
The equation of chord of contact of tangents from the point (-3, 1) to the parabola y 2 8 x is 1. 4 x y 12 0 2. 4 x y 12 0 3. 4 y x 12 0 4. 4 y x 12 0
64.
The equation of chord of contact of tangents from the point (2, 4) to the ellipse 2 x 2 5 y 2 20 is 1. x 5 y 5 0 2. 5 x y 5 0 3. x 5 y 5 0 4. 5 y y 5 0
65.
The equation of chord of contact of tangents from the point (5, 3) to the hyperbola 4 x 2 6 y 2 24 is 1. 9 x 10 y 12 0 2. 10 x 9 y 12 0 3. 9 x 10 y 12 0 4. 10 x 9 y 12 0
66.
The combined equation of the asymptotes to the hyperbola 36 x 2 25 y 2 900 is 1. 25 x 2 36 x 2 0
E R K HSS –ERUMIYAMPATTI
2. 36 x 2 25 y 2 0
3. 36 x 2 25 y 2 0
Page 64
4. 25 x 2 36 y 2 0
+2 STUDY MATERIALS
www.tnschools.co.in
67.
The angle between the asymptotes of the hyperbola 24 x 2 8 y 2 27 is 1.
2.
a
2a m
,
2
m
b2
3
3
m
2
,
a m
a
3.
,
m
a 2m
2.
c
,
2 b c
a 2m
3.
am2
c
,
2 b c
a 2m
2.
c
,
2 b c
a
4.
m
x
2
a
2
a 2m
y
2
b
2
2 b c
,
c
3.
4 ax is
2
2a 2 m
,
c
2a m
,
2
1 is a 2m
4.
The point of contact of the tangent y m x c and the hyperbola 1.
72.
2a
2.
2 a m c
,
c
71.
3
2
4.
The point of contact of the tangent y m x c and the ellipse 1.
70.
2
3.
The point of contact of the tangent y m x c and the parabola y 1.
69.
2
3
3
68.
or
x
2
a
2
y
2
b
2
c
1 is am
2 b c
2 b c
,
4.
2 b c
2
,
c
The true statements of the following are a. Two tangents and 3 normals can be drawn to a parabola from a point. b. Two tangents and 4 normals can be drawn to an ellipse from a point. c. Two tangents and 4 normals can be drawn to an hyperbola from a point. d. Two tangents and 4 normals can be drawn to an rectangular hyperbola from a point. 1. a, b, c and d 2. a, b only 3. c, d only 4. a, b and c 2 If ' t 1 ' , ' t 2 ' are the extremities of any focal chord of a parabola y 4 ax then ; t 1 t 2 is 1. 1
3. 1
2. 0
1
4.
2
73.
1. t 2 74.
1
4.
t2
The condition that the line lx my n 0 may be a normal to the ellipse
3.
a
2
l
2
b
2
m
2
2
m n 0 2
(a
b )
2
2
n
2. 2
4.
2
a
2
l
2
a
2
l
2
b
2
m
2
b
2
m
2
(a
2
b ) 2
n
(a
2
x
2
a
2
3.
a
2
l
2
b
2
m
2
2
m n 0 2
(a
b )
2
2
n
2. 2
4.
2
a
2
l
2
a
2
l
2
b
2
m
2
b
2
m
2
(a
2
b ) 2
(a
2
2
b
2
1
is
2
2
b ) 2
n
y
2
n
2
The condition that the line lx my n 0 may be a normal to the hyperbola 1. al 3 2 alm
76.
3. t 1 t 2
2. t 2
1. al 3 2 alm
75.
2
is The normal at ' t 1 ' on the parabola y 2 4 ax meets the parabola at ' t 2 ' then t 1 t 1
x
2
a
2
y
2
b
2
1 is
2
2
b ) 2
n
2
2
The condition that the line lx my n 0 may be a normal to the parabola y 2 4 ax is 1. al 3 2 alm 3.
a
2
l
2
b
2
m
2
2
m n 0 2
(a
2
b ) 2
n
2
E R K HSS –ERUMIYAMPATTI
2. 2
4.
a
2
l
2
a
2
l
2
b
2
m
2
b
2
m
2
Page 65
(a
2
b ) 2
n
(a
2
2
2
b ) 2
n
2
2
+2 STUDY MATERIALS
www.tnschools.co.in
77.
78.
The chord of contact of tangents from any point on the directrix of the parabola y 2 4 ax passes through its 1. vertex 2. focus 3. directrix 4. latus rectum
80.
2. focus
82.
83.
4. latus rectum
2. ( a t1 t 2 , a ( t1 t 2 ))
x
2
a
2
y
2
b
2
1 passes
4. ( a t1 t 2 , a ( t1 t 2 ))
3. ( at 2 , 2 at )
3
2. x 2 y 2 a 2
3. x 2 y 2 a 2 b 2
x
2
y
2
a
2
b 4. x 0
2
x
2. x 2 y 2 a 2
1 is
2 2
a 4. x 0
3. x 2 y 2 a 2 b 2
2. x 2 y 2 a 2
y
2
b
2
1 is
4. x 0
3. x 2 y 2 a 2 b 2
The locus of the point of intersection of perpendicular tangents to the ellipse 2. x 2 y 2 a 2
x
2
a
2
2. x 2 y 2 a 2
y
2
b
2
1 is
4. x 0
3. x 2 y 2 a 2 b 2
The locus of the point of intersection of perpendicular tangents to the hyperbola
x
2
a
2
y
2 2
1 is
b 4. x 0
3. x 2 y 2 a 2 b 2
The condition that the line lx my n 0 may be a tangent to the parabola y 2 4 ax is 2. am 2 ln
3. a 2 l 2 b 2 m 2 n 2
4. 4 c 2 lm n 2
The condition that the line lx my n 0 may be a tangent to the ellipse 1. a 2 l 2 b 2 m 2 n 2
90.
1 passes through
The locus of the foot of perpendicular from the focus on any tangent to the parabola y 2 4 ax is
1. a 2 l 2 b 2 m 2 n 2 89.
2
The locus of the foot of perpendicular from the focus on any tangent to the hyperbola
1. x 2 y 2 a 2 b 2 88.
b
The locus of the foot of perpendicular from the focus on any tangent to the ellipse
1. x 2 y 2 a 2 b 2 87.
2
If the normal to the R.H. xy c at ' t 1 ' meets the curve again at ' t 2 ' then t 1 t 2 1. 1 2. 0 3. -1 4. -2 The locus of the point of intersection of perpendicular tangents to the parabola y 2 4 ax is 1. latus rectum 2. Directrix 3. tangent at the vertex 4. axis of the parabola
1. x 2 y 2 a 2 b 2 86.
3. directrix
2
1. x 2 y 2 a 2 b 2 85.
2
y
through its 1. vertex 2. focus 3. directrix 4. latus rectum ' t ' ' t ' The point of intersection of tangents at 1 and 2 to the parabola y 2 4 ax is
1. x 2 y 2 a 2 b 2 84.
a
The chord of contact of tangents from any point on the directrix of the hyperbola
1. ( a ( t 1 t 2 ), at 1 t 2 ) 81.
2
The chord of contact of tangents from any point on the directrix of the ellipse its 1. vertex
79.
x
2. am 2 ln
3. a 2 l 2 b 2 m 2 n 2
x
2
2. am 2 ln
3. a 2 l 2 b 2 m 2 n 2
2
1 is 2 2 a b 4. 4 c 2 lm n 2
The condition that the line lx my n 0 may be a tangent to the hyperbola 1. a 2 l 2 b 2 m 2 n 2
y
x
2
y
2
2 1 is 2 a b 2 4. 4 c lm n 2
91.
The condition that the line lx my n 0 may be a tangent to the rectangular hyperbola xy c 2 is
92.
1. a 2 l 2 b 2 m 2 n 2 2. am 2 l n 3. a 2 l 2 b 2 m 2 n 2 4. 4 c 2 lm n 2 The foot of a perpendicular from a focus of the hyperbola on an asymptote lies on the 1. centre 2. corresponding directrix 3. vertex 4. L.R.
E R K HSS –ERUMIYAMPATTI
Page 66
+2 STUDY MATERIALS
www.tnschools.co.in
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. Let „h' be the height of the tank. Then the rate of change of pressure „p‟ of the tank with respect to height is 1.
dh
2.
dp
dt
2.
3.
3.
dt
dh
4.
dp dh
dp
If the temperature C of the certain metal rod of „ l ‟ metres is given by l 1 0 . 00005 0 . 0000004 2
then the rate of change of „ l ‟in m/ C when the temperature is 100 C is 1. 0.00013 m/ C 2. 0.00023 m/ C 3. 0.00026 m/ C 4. 0.00033 m/ C The following graph gives the functional relationship between distance and time of a moving car in m/sec. The speed of the car is
1.
2.
t
m/s
x
3.
dx
4.
m/s
dt
dt
m/s
dx
4.
The distance-time relationship of a moving body is given by y = F(t), then the acceleration of the body is the 1. gradient of the velocity/time graph 2. gradient of the distance/time graph 3. gradient of the acceleration/time graph 4. gradient of the velocity/distance graph
5.
The distance travelled by a car in „t‟ seconds is given by x = 3t3-2t2+4t-1. Then the initial velocity and initial acceleration respectively are 2. (4m/s, -4m/ s 2 ) 3. (0, 0) 4. (18.25m/s, 23m/ s 2 ) 1. (-4m/s, 4m/ s 2 ) The angular displacement of a fly wheel in radians is given by 9 t 2 2 t 3 . The time when the angular acceleration zero is 1. 2.5s 2. 3.5s 3. 1.5s 4. 4.5s Food pockets were dropped from an helicopter during the flood and distance fallen in „t‟ seconds is given by
6.
7.
y
1
gt (g = 9.8 m/ s ).Then the speed of the food pocket after it has fallen for „2‟ seconds is 2
2
2
1. 19.6 m/sec 8.
2. 9.8 m/sec
3. -19.6 m/sec
An object dropped from the sky follows the law of motion x
4. -9.8 m/sec 1
2
2
gt (g = 9.8 m/ s ). The acceleration of the
2
9.
10.
object when t = 2 is 1. -9.8 m/sec2 2. 9.8 m/sec2 3. 19.6 m/sec2 4. -19.6 m/sec2 A missile fired from ground level rises x metres vertically upwards in „t‟ seconds and x = t(100-12.5t). Then the maximum height reached by the missile is 1. 100m 2. 150 m 3. 250 m 4. 200 m A continuous graph y = f (x) is such that f ' ( x ) as x x 1 at x 1 , y 1 . Then y f ( x ) has a 1. vertical tangent y = x 1
11.
2. horizontal tangent x = x 1
3. vertical tangent x = x 1 4. horizontal tangent y = y 1 The curve y f ( x ) and y g ( x ) cut orthogonally if at the point of intersection 1. slope of f(x) = slope of g(x) 2. slope of f(x) + slope of g(x) = 0 3.. slope of f(x) / slope of g(x) = -1 4. [slope of f(x)][slope of g(x)] = -1
E R K HSS –ERUMIYAMPATTI
Page 67
+2 STUDY MATERIALS
www.tnschools.co.in
12.
The law of the mean can also be put in the form 1. f ( a h ) f ( a ) hf ' ( a h ), 0 1 3. f ( a h ) f ( a ) hf ' ( a h ), 0 1 x 1
0 1 0 1
as x 0 because f(x) = x+1 and g(x) = x+3are
13.
l ' H oˆ pital ' s rule cannot be applied to
14.
1. not continuous 3. not in the indeterminate form as x 0 If lim g ( x ) b and f is continuous at x = b then
x3
2. f ( a h ) f ( a ) hf ' ( a h ), 4. f ( a h ) f ( a ) hf ' ( a h ),
2. not differentiable 4. in the indeterminate form as x 0
x a
1.
lim
g ( f ( x )) f lim g ( x ) x a
2.
lim
f ( g ( x )) g lim f ( x ) x a
4.
x a
3.
x a
15.
lim x 0
16.
17.
x
20. 21.
22.
23.
24.
25. 26.
lim
f ( g ( x )) f lim g ( x ) x a
x a
x a
is
tan x
3. f ( x 1 ) f ( x 2 ) whenever x 1 x 2 , x 1 , x 2 I 4. f ( x 1 ) f ( x 2 ) whenever x 1 x 2 , x 1 , x 2 I If a real valued differentiable function y = f (x) defined on an open interval I is increasing then dy
0
2.
dx
19.
f ( g ( x )) f lim g ( x ) x a
1. 1 2. -1 3. 0 4. f is a real valued function defined on an interval I R (R being the set of real numbers) increases on I. Then 1. f ( x 1 ) f ( x 2 ) whenever x 1 x 2 , x 1 , x 2 I 2. f ( x 1 ) f ( x 2 ) whenever x 1 x 2 , x 1 , x 2 I
1. 18.
lim
dy dx
0
3.
dy
0
dx
4.
dy
0
dx
f is differentiable function defined on an interval I with positive derivative. Then f is 1. increasing on I 2. decreasing on I 3. strictly increasing on I 4. strictly decreasing on I 3 The function f (x) = x is 1. increasing 2. decreasing 3. strictly decreasing 4. strictly increasing If the gradient of a curve changes from positive just before P to negative just after then “P” is a 1. minimum point 2. maximum point 3. inflection point 4. discontinuous point The function f (x) = x2 has 1. a maximum value at x=0 2. minimum value at x=0 3. finite no. of maximum values 4. infinite no. of maximum values The function f(x) = x3 has 1. absolute maximum
2. absolute minimum
If f has a local extremum at a and if f ' ( a ) exists then 1. f ' ( a ) <0 2. f ' ( a ) >0
3. local maximum
4. no extrema
3. f ' ( a ) =0
4. f " ( a ) =0
In the following figure, the curve y = f(x) is Y 1. concave upward 2. convex upward 3. changes from concavity to convexity X 4. changes from convexity and concavity The point that separates the convex part of a continuous curve from the concave part is 1. the maximum point 2. the minimum point 3. the inflection point 4. critical point f is a twice differentiable function on an interval I and if f ' ' ( x ) 0 for all x in the domain I of f , then f is 1. concave upward 2.convex upward 3. increasing 4. decreasing
E R K HSS –ERUMIYAMPATTI
Page 68
+2 STUDY MATERIALS
www.tnschools.co.in
27.
x x 0 is a root of even order for the equation f ' ( x ) = 0, then x x 0 is a
28.
1. maximum point 2. minimum point 3. inflection point 4. critical point If x 0 is the x-coordinate of the point of inflection of a curve y = f(x), then (second derivative exists) 1. f ( x 0 ) 0
29.
30.
31.
32.
33.
34.
35.
2. f ' ( x 0 ) 0
3. f ' ' ( x 0 ) 0
4. f ' ' ( x 0 ) 0
The statement “If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c ) and an absolute minimum value f ( d ) at some number c and d in [a, b] ” is 1. The extreme value theorem 2. Fermat‟s theorem 3. Law of mean 4. Rolle‟s theorem The statement: “ If f has a local extremum (minimum or maximum) at c and if f ' ( c ) exists then f ' ( c ) 0 ” is 1. the extreme value theorem 2. Fermat‟s theorem 3. Law of mean 4. Rolle‟s theorem Identify the false statement: 1. all the stationary numbers are critical numbers 2. at the stationary point the first derivative is zero 3. at critical numbers the first derivative need not exist 4. all the critical numbers are stationary numbers Identify the correct statement: a. a continuous function has local maximum then it has absolute maximum b. a continuous function has local minimum then it has absolute minimum c. a continuous function has absolute maximum then it has local maximum d. a continuous function has absolute minimum then it has local minimum 1. a and b 2. a and c 3. c and d 4. a, c and d Identify the correct statements: a. Every constant function is an increasing function b. Every constant function is a decreasing function c. Every identity function is an increasing function d. Every identity function is a decreasing function 1. a, b and c 2. a and c 3. c and d 4. a, c and d Which of the following statement is incorrect? 1. Initial velocity means velocity at t = 0 2. Initial acceleration means acceleration at t = 0 3. If the motion is upward, at the maximum height, the velocity is not zero 4. If the motion is horizontal, v = 0 when the particle comes to rest Which of the following statements are correct ( m 1 and m 2 are slopes of two lines) a. If the two lines are perpendicular then m 1 m 2 = -1 b. If m 1 m 2 = -1, then two lines are perpendicular c. If m 1 = m 2 , then the two lines are parallel d. If m 1
1
1. b, c and d 36.
37.
38.
then the two lines are perpendicular
m2
2. a, b and d
3. c and b
4. a and b
One of the conditions of Rolle‟s theorem is 1. f is defined and continuous on (a, b) 2. f is differentiable on [a, b] 3. f ( a ) f ( b ) 4. f is differentiable on ( a , b ] If „a‟ and „b‟ are two roots of a polynomial f ( x ) 0 , then Rolle‟s theorem says that there exists atleast 1.one root between a and b for f ' ( x ) 0 2. two roots between a and b for f ' ( x ) 0 3. one root between a and b for f ' ' ( x ) 0 4. two roots between a and b for f ' ' ( x ) 0 A real valued function which is continuous on [a, b] and differentiable on (a, b) then there exists atleast one c in 1. [a, b] such that f ' ( c ) 0 2. (a, b) such that f ' ( c ) 0
E R K HSS –ERUMIYAMPATTI
Page 69
+2 STUDY MATERIALS
www.tnschools.co.in f (b ) f ( a )
3. (a, b) such that
ba
0
4. (a, b) such that
f (b ) f ( a ) ba
f ' (c )
In the law of mean, the value ' ' satisfies the condition 1. 0 2. 0 3. 1 4. 0 1 40. Which of the statements are correct? a. Rolle‟s theorem is a particular case of Lagranges law of mean b. Lagranges law of mean is a particular case of generalized law of mean(Cauchy) c. Lagranges law of mean is a particular case of Rolle‟s theorem d. Generalised law of mean is a particular case of Lagrange‟s law of mean (Cauchy) 1. b, c 2. c, d 3. a, b 4. a, d -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (*** Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. For the function y=x3+2x2 the value of dy when x=2 and dx = 0.1 is 1. 1 2. 2 3. 3 4. 4
39.
2.
If U x 4 y 3 3 x 2 y 2 3 x 2 y then 1. 4 x 3 6 xy 2 6 xy
3.
is
2. 3 x 4 6 x 2 y 3 xy 2
4. 4 x 3 6 x 2 y 2 3 xy
3. 4 x 3 6 x 2 y 6 xy 2
2. u x is continuous
3. u y is continuous
4. u , u x , u y are continuous
If u f ( x , y ) is a differentiable function of x and y ; x and y are differentiable functions of „t‟ then 1.
du
dt
3.
du
dt
5.
x
If u f ( x , y ) then with usual notations, u xy u yx if 1. u is continuous
4.
u
f
.
x f
.
x
x t dx
f
y
dt
f y
. .
y
du
2.
t
dt
dy
u
4.
t
dt
f x f x
.
dx dt
.
x
t
If f ( x , y ) is a homogeneous functions of degree n then x 1. f
f x
3. n ( n 1) f
2. nf
u
t y f y
. .
y
y t y t f y
=
4. n ( n 1) f
2
6.
If u ( x , y ) x y 3 x y 3 x y 4
3
1. 12 xy 6 x
2
2
2
then
2. 12 xy 6 x
is
xy
3. 12 x y 6 x 2
u
4. 12 xy 6 x 2
2
7.
If u ( x , y ) x y 3 x y 3 x y 4
3
1. 12 xy 6 x
2
2
2
then
2. 12 xy 6 x
yx
3. 12 x y 6 x 2
u
4. 12 xy 6 x 2
2
8.
If u ( x , y ) x y 3 x y 3 x y 4
1/ 3 y 6 x y 3 x 2
2
3
2
2
2
then
2. 6 y 6 x
2
x
2
3. 12 x y 6 x 2
2
u
4. 12 x
2
6y
2
6y
2
9.
If u ( x , y ) x y 3 x y 3 x y 4
1. 6 y 6 x 10.
2
3
2
2
1
x
3 / 4
then
y
2. 12 xy 6 x
The differential on y of the function y 1.
2.
4
11.
2
1
x
3 / 4
dx
2
3. 12 x y 6 x 2
4
4. 3 y 6 x y 3 x 2
2
2
x is
3. x 3 / 4 dx
4. 0
4
The differential of y if y x 5 is
E R K HSS –ERUMIYAMPATTI
Page 70
+2 STUDY MATERIALS
www.tnschools.co.in
1. 5 x 4 12.
2. 5 x 4 dx
The differential of y if y 1.
1
(4 x 2 x) 3
x x 4
1 is
2
1 2
1
2.
dx
1
(4 x 2 x) 3
1
1
4.
2
2
13.
14. 15.
16.
2
1
( 4 x 2 x ) dx 3
2
( x x 1) 4
2
1 2
(4 x 2 x) 3
2
The differential of y if y 1.
( x x 1) 4
2
2
3.
4. 5 x 5
3. 5 x 5 dx
7 ( 2 x 3)
2
x2 2x 3
2.
dx
is 1
( 2 x 3)
2
7
3.
dx
( 2 x 3)
2
The differential of y if y = sin2x is 2. 2 cos 2 x. dx 3. 2 cos 2 x.dx 1. 2 cos 2 x . The differential of x tan x is 1. ( x sec 2 x tan 2 x ) 2. ( x sec 2 x tan x ) dx 3. x sec 2 xdx If u ( x , y ) x y 3 x y 3 x y 4
3
2
2
1. 3 y 2 6 xy 3 x 2
2
then
u y
4.
dx
7 ( 2 x 3)
2
4. cos 2 x.dx 4. ( x sec
2
x tan x ) dx
is
2. 3 y 2 6 xy 2 3 x 2
3. 3 y 2 6 x 2 y 3 x 2
4. 3 y 2 6 x 2 y 2 3 x 2
17.
The curve y 2 x 2 (1 x 2 ) is defined only for 1. x 2 and x 2 2. x 1 and x 1
18.
The curve y x (1 x ) is symmetrical about 1. x-axis only 2. y-axis only 3. x and y axes only 4. x,y axes and the origin 2 2 2 The curve y x (1 x ) has 1. only one loop between x=0 and x=1 2. two loops between x=-1 and x=0 3. two loops between x=-1 and 0; 0 and 1 4. no loop The curve y 2 x 2 (1 x 2 ) has 1. an asymptote x = -1 2. an asymptote x= 1 3. two asymptotes x =1 and x = -1 4. no asymptote The curve y 2 ( 2 x ) x 2 ( 6 x ) exists for 1. 2 x 6 2. 2 x 6 3. 2 x 6 4. 2 x 6
19.
20.
21.
2
2
3. x 1 and x 1
22.
The x-intercept of the curve y 2 ( 2 x ) x 2 ( 6 x ) is 1. 0 2. 6, 0 3. 2
4. 2
23.
An asymptote to the curve y ( 2 x ) x ( 6 x ) is 1. x 2 2. x 2 3. x 6
4. x 6
24.
The curve y ( 2 x ) x ( 6 x ) has 1. only one loop between x = 0 and x = 6 3. only one loop between x = -2 and x = 6 The curve y 2 x 2 (1 x ) is defined only for 1. x 1 2. x 1
25. 26. 27. 28.
2
2
2
2
2. two loops between x = 0 and x = 6 4. two loops between x= -2 and x=6 3. x 1
The curve y x (1 x ) is symmetrical about 1. y-axis only 2. x-axis only 3. both the axes 2 2 The curve y x (1 x ) has 1. an asymptote y =0 2. an asymptote x = 1 3. an asymptote y = 1 2 2 The curve y x (1 x ) has 1. only one loop between x = -1 and x = 0 2. only one loop between x= 0 and x=1 3. two loops between x = -1 and x = 1 2
4. x 1 and x 1
2
4.
x 1
2
E R K HSS –ERUMIYAMPATTI
Page 71
4. origin only 4. no asymptote
+2 STUDY MATERIALS
www.tnschools.co.in
29. 30. 31. 32.
33. 34. 35. 36.
37.
38. 39. 40.
41.
4. no loop The curve y 2 ( x a ) ( x b ) 2 , a , b 0 and a b does not exist for 2. x b 3. b x a 1. x a
4.
x a
The curve y ( x a ) ( x b ) is symmetrical about 1. origin only 2. y-axis only 3. x-axis only 4. both x and y axis 2 2 The curve y ( x a ) ( x b ) , has a , b 0 and a b 1. an asymptote x = a 2. an asymptote x = b 3. an asymptote y = a 4. no asymptotes The curve y 2 ( x a ) ( x b ) 2 , a , b 0 and a b has 1. a loop between x = a and x = b 2. two loops between x = a and x = b 3. two loops between x=0 and x= a 4. no loop 2 2 The curve y (1 x ) x (1 x ) is defined for 1. 1 x 1 2. 1 x 1 3. 1 x 1 4. 1 x 1 2
2
The curve y 2 (1 x ) x 2 (1 x ) is symmetrical about 1. both the axes 2. origin only 3. y-axis only 2 2 The asymptote to the curve y (1 x ) x (1 x ) is 1. x 1 2. y 1 3. y 1 The curve y (1 x ) x (1 x ) has 1. a loop between x = -1 and x = 1 3. a loop between x = 0 and x = 1 The curve a 2 y 2 x 2 ( a 2 x 2 ) is defined for 1. x a and x a 3. x a and x a 2
4. x-axis only 4. x 1
2
2. a loop between x = -1 and x = 0 4. no loop 2. x a and x a 4. x a and x a
The curve a 2 y 2 x 2 ( a 2 x 2 ) is symmetrical about 1. x-axis only 2. y-axis only 3. both the axes 4. both the axes and origin The curve a 2 y 2 x 2 ( a 2 x 2 ) has 1. an asymptote x = a 2. an asymptote x = -a 3. an asymptote x = 0 4. no asymptote 2 2 2 2 2 The curve a y x ( a x ) has 1. a loop between x = a and x = -a 2. two loops between x = -a and x = 0; x = 0 and x = a 3. two loops between x = 0 and x = a 4. no loop The curve y 2 ( x 1) ( x 2 ) 2 is not defined for 1. x 1 2. x 2 3. x 2 4. x 1
The curve y 2 ( x 1) ( x 2 ) 2 is symmetrical about 1. both x and y axes 2. x-axis only 3. y-axis only 4. both the axes and origin 2 2 43. The curve y ( x 1) ( x 2 ) has 1. an asymptote x=1 2. an asymptote x=2 3. two asymptotes x=1 and x=2 4. no asymptote 44. The curve y 2 ( x 1) ( x 2 ) 2 has 1. two loops between x= 0 and x=2 2. one loop between x= 0 and x=1 3. one loop between x=1 and x= 2 4. no loop -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. If I n sin n xdx , then I n
42.
1.
1
sin
n 1
x cos x
n 1
n
E R K HSS –ERUMIYAMPATTI
n
I n2
2.
1
sin
n 1
n
Page 72
x cos x
n 1 n
I n2
+2 STUDY MATERIALS
www.tnschools.co.in 1
3.
sin
n 1
x cos x
n 1
n
n
2a
2.
1
4.
I n2
n 1
sin
x cos x
n 1
n
In
n
a
f ( x ) dx
= 2 f ( x ) dx , if 0
0
1. f(2a-x) = f(x)
2. f(a-x) = f(x)
3. f(x) = -f(x)
4. f(-x) = f(x)
2. f(2a-x) = -f(x)
3. f(x) = -f(x)
4. f(-x) = f(x)
2a
3.
f ( x ) dx
= 0 , if
0
1. f(2a-x) = f(x)
a
4.
If f(x) is an odd function then
is
f ( x ) dx
a
a
1. 2 f ( x ) dx
2.
0
4.
f ( a x ) dx
0
a
f ( x ) dx
+
f ( 2 a x ) dx =
0
0 a
1.
3.0
f ( x ) dx
0
a
5.
a
a
a
2a
2. 2 f ( x ) dx
f ( x ) dx
0
3.
0
2a
4.
f ( x ) dx
0
f ( a x ) dx
0
a
6.
If f(x) is even then
is
f ( x ) dx
a a
a
2. 2 f ( x ) dx
1. 0
3.
0
a
4. -2 f ( x ) dx
f ( x ) dx
0
0
a
7.
f ( x ) dx
is
0
a
1.
a
a
f ( x a ) dx
2.
0
f ( a x ) dx
3.
a
f ( 2 a x ) dx
4.
0
0
f ( x 2 a ) dx
0
b
8.
f ( x ) dx
is
a b
a
1. 2 f ( x ) dx
2.
b
f ( a x ) dx
3.
a
0
b
f ( b x ) dx
4.
a
f ( a b x ) dx
a
9.
If n is a positive integer , then
n
x e
ax
dx
=
0
1.
n a
2.
n
n 1 a
n
3.
n 1 a
n 1
4.
n a
n 1
2
10.
If n is odd, then
cos
n
xdx
is
0
1. 3.
n
n2 n4
n 1 n 3 n 5 n n2 n4 n 1 n 3 n 5
2 3 2
E R K HSS –ERUMIYAMPATTI
2. 1
4.
n 1 n 3 n 5 n n2 n4 n 1 n 3 n 5 n
1
2 2 2 1 n2 n4 3
Page 73
+2 STUDY MATERIALS
www.tnschools.co.in 2
11.
If n is even, then
n
sin
is
xdx
0
1. 3.
n2 n4
n
n 1 n 3 n 5 n n2 n4
2 3
n 1 n 3 n 5
n 1 n 3 n 5
2. 1
n n2 n4 n 1 n 3 n 5
4.
2
1
2 2 2 1 3 n2 n4
n
2
12.
If n is even, then
cos
n
is
xdx
0
3.
n2 n4
n
1.
n 1 n 3 n 5 n n2 n4 n 1 n 3 n 5
2 3
n 1 n 3 n 5
2. 1
n n2 n4 n 1 n 3 n 5
4.
2
1
2 2 2 1 3 n2 n4
n
/2
13.
If n is odd, then
sin
x dx
n
0
1. 3.
n2 n4
n
n 1 n 3 n 5 n n2 n4 n 1 n 3 n 5
n 1 n 3 n 5
2.
2 3
1
n n2 n4 n 1 n 3 n 5
4.
2
1
2 2 2 1 3 n2 n4
n
b
14.
f ( x ) dx
is
a b
a
1. - f ( x ) dx
2.
a
15.
b
1.
2.
xdx
3.
xdy
c
d
4.
ydy
c
xdy
c
The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then the volume of the solid is d
1. x dy
d
2. x dx
2
3.
2
c
c
d
y
2
4. y 2 dy
dx
c
c
The area bounded by the curve x = f(y) to the left of y-axis and between the lines y=c and y=d is d
1.
d
d
2. - xdy
xdy
c
3.
c
d
4. - ydx
ydx
c
c
The arc length of the curve y = f(x) from x=a to x=b is b
1.
a
19.
0
d
d
18.
0
a
c
17.
b
4. 2 f ( x ) dx
The area bounded by the curve x = g(y) to the right of y-axis and the two lines y=c and y=d is given by d
16.
a
3. - f ( x ) dx
f ( x ) dx
dy 1 dx
2
dx 1 dy
d
dx
2.
c
2
b
dx
3. 2
a
dy y 1 dx
2
b
4) 2
dx
y
a
dx 1 dy
2
dx
The surface area obtained by revolving the area bounded by the curve y= f(x), the two ordinates x=a, x=b and x-axis, about x-axis is b
1.
a
dy 1 dx
2
d
dx
E R K HSS –ERUMIYAMPATTI
2.
c
dx 1 dy
2
b
dx
3. 2
a
Page 74
dy y 1 dx
2
b
dx
4) 2
a
y
dx 1 dy
2
dx
+2 STUDY MATERIALS
www.tnschools.co.in
20.
5
x e
4 x
is
dx
0
1.
6 4
2.
6
6 4
3.
5
5 4
4.
6
5 4
5
21.
e
mx
7
is
x dx
0
1.
m 7
22.
2.
m
m
6
x e
7
m
3.
7
7
4.
m 1
7 m
8
x 2
is
dx
0
1.
6 2
23.
2.
7
6 2
6
3. 2 6 6
4. 2 7 6
If I n cos xdx , then I n n
1.
1
cos
n 1
x sin x
n
3.
1
cos
n
n 1
x sin x
n 1
n n 1 n
I n2
I n2
2. cos
n 1
x sin x
n 1 n
4.
1
n 1
cos
I n2
x sin x
n 1 n
n
I n2
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 8. DIFFERENTIAL EQUATIONS (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------3
1.
The order and degree of the differential equation
d y dx
1. 3, 1 2.
2. 1, 3
3
d2y 2 dx
4. 2, 3 dy
3x
dx
2. 1, 2
5.
are
dy
4. 2,2
The order and degree of the differential equation
d y dx
4.
dx
3. 1, 1 2
3.
5
3. 3, 5
The order and degree of the differential equation y 4 1. 2, 1
3
dy y 7 are dx
2
dy 4 dx 2
3 4
1. 2, 1 2. 1, 2 3. 2, 4 The order and degree of the differential equation (1 y ' ) 2 y ' 2 are 1. 2, 1 2. 1, 2 3. 2, 2 The order and degree of the differential equation
dy
yx
2
are 4. 4,2 4. 1, 1
are
dx
6. 7.
1. 1, 1 2. The order and degree of 1. 2, 1 2. The order and degree of 1. 2, 2 2.
1, 2 3. 2, 1 4. 0, 1 2 the differential equation y ' y x are 1, 1 3. 1, 0 4. 0, 1 the differential equation y " 3 y ' 2 y 3 0 are 2, 1 3. 1, 2 4. 3, 1 2
8.
The order and degree of the differential equation 1. 2, 1
2. 1, 2
3. 2,
d y dx 1
2
x
y
dy dx
are 4. 2,2
2
E R K HSS –ERUMIYAMPATTI
Page 75
+2 STUDY MATERIALS
www.tnschools.co.in 3 2
9.
The order and degree of the differential equation
d y dx
1. 2, 3
2. 3, 3
2
dy d 3 y y 3 dx dx
3. 3, 2
2 0 are
4. 2,2 2
10. 11. 12.
The order and degree of the differential equation 1. 2, 3 2. 3, 3 3. 3, The order and degree of the differential equation 1. 1, 1 2. 1, 2 3. 2, The order and degree of the differential equation 1. 2, 2 2. 2, 1 3. 1,
y " ( y y ' ) 3 are 3
2
4. 2,2 are 1 4. 2,2 2 2 y ' ( y " ) x ( x y " ) are 2 4. 1, 1
y ' ( y" ) ( x y" ) 2
2
2
13. 14. 15.
16.
dx dy 2 The order and degree of the differential equation x are x dy dx
1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 3 The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1. 1, 1 2. 0, 0 3. 1, 2 4. 2, 1 The differential equation corresponding to xy = c2 where c is an arbitrary constant, is 2. y " 0 3. xy ' y 0 4. xy " x 0 1. xy " x 0 In finding the differential equation corresponding to y = emx where m is the arbitrary constant, then m is 1.
y
2.
y'
y'
3. y '
4. y
y
17. The solution of a linear differential equation
dx
Px Q where P and Q are functions of
y is
dy
1. y I . F I . F Qdx c
2. x I . F I . F Qdy c
3. y I . F I . F Qdy c
4. x I . F I . F Qdx c
18. The solution of a linear differential equation
dy
Py Q where P and Q are functions of x is
dx
19.
1. y I . F I . F Qdx c
2. x I . F I . F Qdy c
3. y I . F I . F Qdy c
4. x I . F I . F Qdx c
Identify the incorrect statement. 1. The order of a differential equation is the order of the highest order derivative occurring in it. 2. The degree of the differential equation is the degree of the highest order derivative which occurs in it. (The derivatives are free from radicals and fractions). 3.
dy dx
4.
dy
f1 ( x, y )
is the first order first degree homogeneous differential equation.
f 2 ( x, y ) xy e
x
is a linear differential equation in x.
dx
-------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 9. DISCRETE MATHEMATICS (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. Which of the following are statements? (i) Chennai is the capital of TamilNadu. (ii) The earth is a planet. (iii) Rose is a flower. (iv) Every triangle is an isosceles triangle. 1. all 2. (i) and (ii) 3. (ii) and (iii) 4. (iv) only
E R K HSS –ERUMIYAMPATTI
Page 76
+2 STUDY MATERIALS
www.tnschools.co.in
2.
3.
4.
Which of the following are not statements? (i) Three plus four is eight. (ii) The sun is a planet. (iii) Switch on the light. (iv) Where are you going? 1. (i) and (ii) 2. (ii) and (iii) 3. (iii) and (iv) The truth values of the following statements are (i) Ooty is in Tamilnadu and 3+4 = 8 (ii) Ooty is in Tamilnadu and 3+4 = 7 (iii) Ooty is in Kerala and 3+4 = 7 (iv) Ooty is in Kerala and 3+4 = 8 1. FTFF 2. FFFT 3. TTFF The truth values of the following statements are (i) Chennai is in India or
2 is an integer.
(ii) Chennai is in India or
2 is an irrational number.
(iii) Chennai is in China or
5.
6.
7.
8.
9.
10.
11. 12. 13.
4. (iv) only
4.TFTF
2 is an integer.
(iv) Chennai is in China or 2 is an irrational number. 1. TFTF 2. TFFT 3. FTFT 4. TTFT Which of the following are not statements? (i) All natural numbers are integers. (ii) A square has five sides. (iii) The sky is blue. (iv) How are you? 1. (iv) only 2. (i) and (iv) 3. (i), (ii), (iii) 4. (iii) and (iv) Which of the following are statements? (i) 7+2<10. (ii) The set of rational numbers is finite. (iii) How beautiful you are! (iv) Wish you all success. 1. (iii), (iv) 2. (i), (ii) 3. (i), (iii) 4. (ii), (iv) The truth values of the following statements are (i) All the sides of a rhombus are equal in length. (ii) 1+ 19 is an irrational number. (iii) Milk is white. (iv) The number 30 has four prime factors. 1. TTTF 2. TTTT 3. TFTF 4. FTTT The truth values of the following statements are (i) Paris is in France. (ii) sinx is an even function. (iii) Every square matrix is non-singular. (iv) Jupiter is a planet. 1. TFFT 2. FTFT 3. FTTF 4.FFTT Let p be “Kamala is going to school” and q be “There are twenty students in the class”.“Kamala is not going to school or there are twenty students in the class” stands for 2. p q 3. p 4. p q 1. p q If p stands for the statement “Sita likes reading” and q for the statement “Sita likes playing”.“Sita likes neither reading nor playing” stands for 1. p q 2. p q 3. p q 4. p q If p is true and q is unknown then 1. p is true 2. p ( p ) is false 3. p ( p ) is true 4. p q is true If p is true and q is false then which of the following is not true? 1. p q is false 2. p q is true 3. p q is false 4. p q is true Which of the following is not true? 1. Negation of a negation of a statement is the statement itself. 2. If the last column of its truth table contains only T then it is tautology. 3. If the last column of its truth table contains only F then it is contradiction. 4. If p and q are any two statements then p q is a tautology.
E R K HSS –ERUMIYAMPATTI
Page 77
+2 STUDY MATERIALS
www.tnschools.co.in
14.
15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26.
27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
Which of the following are binary operations on R? a. a*b = min{a, b} b. a*b = max{a, b} c. a*b = a d. a*b = b 1. all 2. a, b and c 3. b,c and d 4. c, d „+‟ is not a binary operation on 1. N 2. Z 3. C 4. Q-{0} „-‟ is a binary operation on 1. N 2. Q-{0} 3. R-{0} 4. Z „ ‟ is a binary operation on 1. N 2. R 3. Z 4. C-{0} In congruence modulo5, { x z / x 5 k 2 , k z } represents 1. 0
2. 5
3. 7
4. 2 .
5 12 11 is 1. 55
2. 12
3. 7
4. 11
3 8 7 is 1. 10
2. 8 3. 5 4. 2 In the group (G, .), G = {1, -1, i, -i} the order of -1 is 1. -1 2. 1 3. 2 4. 0 In the group (G, .), G = {1, -1, i, -i} the order of -i is 1. 2 2. 0 3. 4 4. 3 2 In the group (G, .), G = {1, , } , is cube root of unity, then O( 2 ) is 1. 2 2. 1 3. 4 4. 3 ( Z , ) In the group , order of [0] is 4 4 1.1 2. 3. can‟t be determined 4. 0 In the group ( Z 4 , 4 ) , O ( [ 3 ] ) is 1.4 2. 3 3. 2 4. 1 S , In , x y x , x , y S then „ ‟ is 1. only associative 2. only commutative 3. associative and commutative 4. neither associative nor commutative In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1. only closed 2. only semi group 3. only monoid 4. a group The set of positive even integers, with usual multiplication forms 1. a finite group 2. only a semi group 3. only a monoid 4. an infinite group The set of positive even integers, with usual addition forms 1. a finite group 2. only a semi group 3. only a monoid 4. an infinite group In the group ( Z 5 0 , 5 ) , O [ 3 ] is 1. 5 2. 3 3. 4 4. 2 In the group (G, .), G = {1, -1, i, -i} the order of 1 is 1. 2 2. 0 3. 4 4. 1 In the group (G, .), G = {1, -1, i, -i} the order of i is 1. 2 2. 0 3. 4 4. 3 2 In the group (G, .), G = {1, , } , is cube root of unity, then O( ) is 1. 2 2. 1 3. 4 4. 3 2 In the group (G, .), G = {1, , } , is cube root of unity, then O(1) is 1. 2 2. 1 3. 4 4. 3 ( Z , ) In the group , order of O( [1]) is 4 4 1.1 2. 3. cannot be determined 4. 4 In the group ( Z 4 , 4 ) ,order of O([2]) is 1.1 2. 2 3.cannot be determined 4. 0 O [ 2 ] In the group ( Z 5 0 , 5 ) , is 1. 5 2. 3 3. 4 4. 2
E R K HSS –ERUMIYAMPATTI
Page 78
+2 STUDY MATERIALS
www.tnschools.co.in
In the group ( Z 5 0 , 5 ) , O [ 4 ] is 1. 5 2. 3 3. 4 4. 2 39. In the group ( Z 5 0 , 5 ) , O [1] is 1. 1 2. 2 3. 3 4. 4 -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 10. PROBABILITY DISTRIBUTIONS (One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------1. A discrete random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of value 2. A continuous random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values 3. If X is a discrete random variable then P(X a) = 1. P(X
a) 3. P(X>a) 4. 1-P(X a-1) 5 If X is a continuous random variable then P(a
1. 12.
13.
14.
1
e
1 2 z 2
2.
1
z
2
e
3.
1
e
1 2 z 2
4.
1
e
1 2 z 2
2 2 2 2 If X is a discrete random variable then which of the following is correct ? 1. 0 F ( x ) 1 2. F ( ) 0 and F ( ) 1 3. P [ X x n ] F x n F ( x n 1 ) 4. F(x) is a constant function If X is a continuous random variable then which of the following is incorrect ? 1. F ' ( x ) f ( x ) 2. F ( ) 1 and F ( ) 0 3. P [ a x b ] F ( b ) F ( a ) 4. P [ a x b ] F ( b ) F ( a ) Which of the following are correct? (i) E(aX+b) = aE(X)+b (ii) 2 2 ' ( 1 ' ) 2
(iii) 2 var iance 1. all
2. (i), (ii), (iii)
E R K HSS –ERUMIYAMPATTI
(iv) var(aX+b) = a2var(X) 3. (ii), (iii)
Page 79
4. (i), (iv)
+2 STUDY MATERIALS
www.tnschools.co.in
15.
Which of the following is not true regarding the normal distribution? 1. skewness is zero. 2. mean = median = mode 3. the points of inflection are at X = 4. maximum height of the curve is
1 2
*****************
E R K HSS –ERUMIYAMPATTI
Page 80
+2 STUDY MATERIALS
COME BOOK ONE MARK KEY www.tnschools.co.in
Q.NO ANS Q.NO ANS Q.NO ANS
1 1 11 3 21 2
2 2 12 2 22 1
3 2 13 4 23 4
4 4 14 1
5 2 15 3
6 4 16 2
7 4 17 4
8 4 18 3
9 1 19 1
10 2 20 2
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 1 11 3 21 2 31 2
2 1 12 1 22 1 32 1
3 1 13 2 23 1 33 4
4 3 14 4 24 3 34 1
5 3 15 1 25 1 35 1
6 4 16 2 26 4 36 1
7 4 17 1 27 3 37 1
8 4 18 1 28 1 38 4
9 3 19 1 29 4 39 3
10 1 20 4 30 1
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 1 11 4 21 2 31 4 41 4
2 2 12 3 22 4 32 1 42 3
3 2 13 3 23 3 33 1 43 4
4 1 14 4 24 2 34 3 44 2
5 3 15 1 25 3 35 3 45 1
6 1 16 1 26 3 36 4 46 4
7 1 17 4 27 2 37 4
8 2 18 4 28 1 38 4
9 4 19 1 29 2 39 4
10 3 20 2 30 1 40 4
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 2 11 2 21 3 31 4 41 4 51 3 61 4 71 4 81 3 91 4
2 3 12 4 22 4 32 2 42 2 52 4 62 3 72 1 82 2 92 2
3 4 13 4 23 2 33 3 43 1 53 2 63 1 73 1 83 2
4 2 14 1 24 1 34 1 44 3 54 4 64 3 74 3 84 2
5 1 15 3 25 1 35 2 45 1 55 1 65 4 75 4 85 4
6 4 16 2 26 4 36 4 46 2 56 2 66 2 76 1 86 3
7 2 17 4 27 2 37 1 47 3 57 3 67 3 77 2 87 1
8 4 18 1 28 3 38 2 48 2 58 4 68 1 78 2 88 2
9 2 19 3 29 1 39 1 49 3 59 1 69 2 79 2 89 1
10 3 20 2 30 2 40 3 50 1 60 2 70 3 80 2 90 3
E R K HSS –ERUMIYAMPATTI
Page 81
+2 STUDY MATERIALS
www.tnschools.co.in
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 4 11 4 21 2 31 4 41 1
2 1 12 2 22 4 32 1
3 1 13 3 23 3 33 1
4 1 14 2 24 1 34 3
5 2 15 1 25 3 35 1
6 3 16 1 26 1 36 3
7 1 17 2 27 3 37 1
8 2 18 3 28 3 38 4
9 4 19 4 29 1 39 4
10 3 20 2 30 2 40 3
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 2 11 2 21 1 31 4 41 4
2 1 12 2 22 2 32 4 42 2
3 4 13 3 23 2 33 2 43 4
4 3 14 2 24 1 34 4 44 3
5 2 15 4 25 1 35 4
6 1 16 3 26 2 36 3
7 1 17 2 27 4 37 1
8 4 18 4 28 2 38 4
9 1 19 3 29 3 39 4
10 2 20 4 30 3 40 2
Q.NO ANS Q.NO ANS Q.NO ANS
1 1 11 2 21 4
2 1 12 2 22 4
3 2 13 4 23 4
4 3 14 2
5 3 15 4
6 2 16 1
7 2 17 2
8 4 18 1
9 4 19 3
10 4 20 3
Q.NO ANS Q.NO ANS
1 1 11 3
2 2 12 1
3 3 13 4
4 4 14 1
5 1 15 3
6 2 16 2
7 2 17 2
8 4 18 1
9 2 19 4
10 1
Q.NO ANS Q.NO ANS Q.NO ANS Q.NO ANS
1 1 11 4 21 3 31 4
2 3 12 4 22 3 32 3
3 1 13 4 23 4 33 4
4 4 14 1 24 1 34 2
5 1 15 4 25 1 35 4
6 2 16 4 26 1 36 2
7 1 17 4 27 3 37 3
8 1 18 4 28 2 38 4
9 4 19 3 29 2 39 1
10 1 20 4 30 3
Q.NO ANS Q.NO ANS
1 4 11 4
2 2 12 3
3 3 13 4
4 3 14 1
5 4 15 4
6 2
7 1
8 4
9 4
10 3
E R K HSS –ERUMIYAMPATTI
Page 82
+2 STUDY MATERIALS
www.tnschools.co.in
BOOK BACK – ONE MARK [30 Number Of One Mark Questions to be Asked For Full Test] -------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 1. MATRICES AND DETERMINANTS (THREE QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------1 1
1.
4 4
1)1
2.
3.
2 4 is 8
The rank of the matrix 2 2 2) 2
3)3
1 2 The rank of the diagonal matrix
4)4 is 0
0
1) 0 2)2 If A = [2 0 1], then rank of AAT is 1)1 2)2
4
3)3
4)5
3)3
4)0
3)1
4)2
1
4.
If A = 2 then the rank of AAT is 3
1)3
2)0
5.
6.
If the rank of the matrix
0 1
1 k
2
2)
I
1 k
1 1 1
If the matrix
3
4) kI
I
k
2 3 has an inverse then the values of k 5
3 k 4
2) k = -4
3) k≠ -4
0 1
3)
4) k≠4
0 1 5
1 2) 0
5 0
0 5
5
4)
0
0 5
If A is a square matrix of order n then adj A is 1) A
10.
1
3)
4) any real number
2 1 If A= ,then (adj A)A = 3 4 1 1) 5 0
9.
0
I
1) k is any real number 8.
0 1 is 2, then is
1)1 2)2 3)3 If A is a scalar matrix with scalar k≠0, of order 3, then A-1 is 1)
7.
1
2
2) A
The inverse of the matrix
E R K HSS –ERUMIYAMPATTI
n
0 0 1
3) A 0 1 0
n 1
4) A
1 0 is 0
Page 83
+2 STUDY MATERIALS
www.tnschools.co.in
0 0 1 0 0 1
1 1) 0 0
11. 12. 13. 14.
0
1 0 4) 0
1 0 0
0 1 0
0 0 1
0 1 0
4) det (A) 4) kn-1(adj I) then 4) B = A
0
0 12 5
2)
0
0
1 0 0 1
0 0
3)
0
4)
3 1 is Inverse of 5 1 3
2
1)
5
2
2
2)
1
5 3
1 3
3
3)
5
3
4)
5 2
1
In a system of 3 linear non-homogeneous equation with three unknowns, if andx y 0 , z = 0 then the system has 1) unique solution 2) two solutions 3) infinitely many solutions 4) no solution The system of equations ax + y + z = 0 ; x +by + z = 0 ; x + y + cz = 0 has a non-trivial solution then 1 1 a
1 1 b
1)1 18.
0
0 60
0
17.
1
0 3) 0 1
0 12 , then A is 5
0
If A = 0
16.
1 0 0
0
If A is a matrix of order 3, then det(kA) 1) k3 det (A) 2) k2 det (A) 3) k det (A) If I is unit matrix of order n, where k≠0 is a constant, then adj(kI)= 1) kn(adj I) 2) k(adj I) 3) k2(adj I)) If A and B are any two matrices such that AB=O and A is non-singular, 1) B = О 2) B is singular 3) B is non-singular
1) 15.
0 2) 0 1
1 1 c
2)2
3)-1
If aex + bey = c ; pex +qey = d and 1 = 2
1)
1
,
3 1
2
1
2) log
,
log
a
b
p
q
4)0
; 2 =
3 1
c
b
d
q
; 3 =
1
3
3) log
,
log
a
c
p
d
then the value of (x ,y) is
1 2
1
2
4) log
,
log
1 3
19.
If the equation -2x + y + z = l ; x - 2y +z = m ; x + y -2z = n such that l+m+n = 0, then the system has 1) a non-zero unique solution 2) trivial solution 3) infinitely many solution 4) No solution -------------------------------------------------------------------------------------------------------------------------------------------------------2. VECTOR ALGEBRA (FOUR QUESTIONS FOR FULL TEST) ------------------------------------------------------------------------------------------------------------------------------------------------------- 20. If a is a non-zero vector and m is a non-zero scalar then m a is a unit vector if 1) m = ±1
21.
3) a =
1 m
4) a =1
If a and b are two unit vectors and is the angle between them, then ( a b ) is a unit vector if 1) =
22.
2)a = m
2) =
3) =
4 2 3 0 If a and b include an angle 120 and their magnitude are 2 and
1)
3
2) -
3
3)2
4) =
2 3
3 then a . b is equal to
4) -
3 2
E R K HSS –ERUMIYAMPATTI
Page 84
+2 STUDY MATERIALS
www.tnschools.co.in
23.
1) u is a unit vector 24.
3)
3 The vectors 2 i 3 j 4 k and
3) u 0
4) u 0
5
4)
2
3 a i bj ck
are perpendicular when 1) a = 2 , b = 3 , c = -4 2) a = 4 , b = 4 , c = 5 3) a = 4 , b = 4 , c = -5 4) a = -2 , b = 3 , c = 4 The area of the parallelogram having a diagonal 3 i j k and a side i 3 j 4 k is
2) 6 30
3)
3
4) 3 30
30
2
If a b a b then
1) a is parallel to b 28.
2
2)
1) 10 3 27.
6
26.
2) u a b c
If a b c 0 , a 3, b 4 , c 5 , then the angle between a and b is 1)
25.
If u a ( b c ) b ( c a ) c ( a b ) , then
2) a is perependicular to b
3) a
b
pq
If p , q and p q are vectors of magnitude then the magnitude of
4) a and b are unit vectors is
2) 3 3) 2 4) 1 If a ( b c ) b ( c a ) c ( a b ) x y then 1) x 0 2) y 0 3) x and y are parallel 4) x 0 or y 0 or x and y are parallel If PR 2 i j k , QS i 3 j 2 k then the area of the quadrilateral PQRS is 1) 2
29.
30.
2) 10 3
1) 5 3
3) 5
3
4)
2
31.
35. 36. 37.
3) sin-1
2) cos-1
1
4) sin-1
3 If the projection of a on b and projection of b on a are equal then the angle between a + b and a - b is
1)
34.
3 10
3 10
1 3
33.
2
The projection of OP on a unit vector OQ equals thrice the area of parallelogram OPRQ. Then POQ is 1) tan-1
32.
3
2)
2 If a ( b c ) 1) a parallel to
3)
3 4 ( a b ) c for non-coplanar vectors a , b , c then b 2) b parallel to c 3) c parallel to a
4.
2 3
4) a b c 0 with positive direction of axes x and y then the angle it makes with the
If a line makes 450, 600 z-axis is 1) 300 2) 900 If a b , b c , c a = 64, then a , b , c is 1) 32 2) 8 If a b , b c , c a = 8 then a , b , c is 1) 4 2) 16 The value of i j , j k , k i is equal to 1)0 2)1
E R K HSS –ERUMIYAMPATTI
3) 450
4) 600
3) 128
4) 0
3) 32
4) -4
3) 2
4) 4
Page 85
+2 STUDY MATERIALS
www.tnschools.co.in
38.
1) 2 26 39.
The shortest distance of the point (2 , 10 , 1) from the plane ⃗⃗ [ 3 i j 4 k ]= 2 26 is 2)
3)2
26
26
The vector ( a b ) ( c d ) is 1) perpendicular to a , b , c and d 2) parallel to the vectors a b and
1
4)
c d
3) parallel to the line of intersection of the plane containing a and b and the plane containing c and d
4) perpendicular to the line of intersection of the plane containing a and b and the plane containing
40.
c and d If a , b , c are a right handed triad of mutually perpendicular vectors of magnitude a , b , c then the value of [ a , b , c ] is
1) a2b2c2
2) 0
1
3)
4) abc abc 2 b c c a ] [a b b c
c a ] then [ a , b , c ]
42.
are non-coplanar and [ a b 2) 3 r s i t j is the equation of
43.
1) a straight line joining the points i and j 2) xoy plane 3) yoz plane 4) zox plane If the magnitude of moment about the point j k of a force i a j k acting through the point i j
41.
If a , b , c 1) 2
3) 1
is 8 then the value of a is 1) 1 2) 2 44.
3) 3
The equation of the line parallel to
x3
is
4) 0
1
y3
4) 4
2z 5
5
and passing through the point (1 , 3, 5) in
3
vector form is 1) r ( i 5 j 3 k ) t ( i 3 j 5 k )
2) r ( i 3 j 5 k ) t ( i 5 j 3 k )
3 3 4) r ( i 3 j 5 k ) t ( i 5 j k ) k ) t (i 3 j 5k ) 2 2 The point of intersection of the line r ( i k ) t ( 3 i 2 j 7 k ) and the plane r .( i j k ) 8 is
3) r ( i 5 j 45.
1) (8 , 6 , 22) 46.
47.
48.
2) (-8 , -6 , -22)
4) (-4 , -3 , -11)
The equation of the plane passing through the point (2 , 1 , -1) and the line of intersection of the planes r .( i 3 j k ) 0 ; and r .( j 2 k ) 0 is 1) x + 4y -z = 0 2) x + 9y +11z = 0 3) 2x + y - z +5 =0 4) 2x -y +z = 0 The work done by the force F i j k acting on a particle, if the particle is displaced from A (3,3,3) to the point B(4,4,4) is 1) 2 units 2) 3 units 3) 4 units 4) 7 units
If a i 2 j 3 k 1)
i j k
2)
i j k
3)
i j 2k
3
The point of intersection of the lines 1) ( 0 , 0 , -4)
and b 3 i j 2 k then a unit vector perpendicular to a and
3
49.
3) (4 , 3 , 11)
2) (1 , 0 , 0)
E R K HSS –ERUMIYAMPATTI
4)
3 x6 6
y4 4
z4 8
3)(0 , 2 , 0)
Page 86
b
i j k
is
3
and
x 1 2
y2 4
z3 2
is
4) (1 , 2 ,0)
+2 STUDY MATERIALS
www.tnschools.co.in
50.
2) (1 , 2 , 1)
The shortest distance between the lines 2
1)
x3
4
y 1
3
and
y 1
1
2
z
and
z5
is
5
1 2 6 x 1
y2
2
z3 3
is
4) 0 x 2
1
y 1
5
3
2) intersecting
4
4
3) 1 x 1
y4
3
z5
2
2) 2
x2
and 4)
4
1) parallel
3
The shortest distance between the lines
The following two lines are
z3
2
3)
6
1) 3
4) (1 , 1 ,1)
y2 3
1
2)
3
53.
3) (1 , 1 ,2) x 1 2
52.
r ( 2 i 3 j 5 k ) s ( i 2 j 3 k ) is
1) (2 , 1 , 1) 51.
The point of intersection of the lines r ( i 2 j 3 k ) t ( 2 i j k ) and
3) skew
z1 2
4) perpendicular
The centre and radius of the sphere given by x2+y 2 +z 2 - 6x + 8y - 10z +1 = 0 is 1) (-3, 4, -5), 49 2) (-6, 8, -10), 1 3)(3, -4, 5),7 4) ( 6, -8, 10 ), 7 -------------------------------------------------------------------------------------------------------------------------------------------------------3. COMPLEX NUMBERS (THREE QUESTIONS FOR FULL TEST) --------------------------------------------------------------------------------------------------------------------------------------------------------
54.
55.
56.
3
1) 2
2) 0
100
is
3) -1
4) 1
3i
2) e 9,
3) e 6,
3
, 8 2
1
2)
, 8 2
2) 2x
, 8 2 1
1
If x 2 + y 2 = 1, then the value of 1) x –iy
3) 1 x iy 1 x iy
3) -2iy
The modulus of the complex number 2 + i 3 is
60.
1) 3 2) 13 3) 7 If A + iB = (a1 + ib1) (a2 + ib2) (a3 + ib3 ), then A2 + B2 is
62.
4
3
are respectively 3
4) e 9,
1
4)
2
,
8
is
59.
61.
4 4 2 2 If ( m 5 ) i ( n 4 ) is the complex conjugate of (2m +3) + i (3n - 2), then ( n, m ) are
1) 58.
+
1 i 3 2
The modulus and amplitude of the complex number e 1) e 9,
57.
100
1 i The value of 2
4) x + iy
4) 7
1) a12 + b12 + a22 + b22 + a32 + b32 2) (a1 + a2 + a3) 2 + (b1 + b2 + b3) 2 2 2 2 2 2 2 3) (a1 + b1 ) (a2 + b2 ) (a3 + b3 ) 4) (a12 +a22+a32) (b12 + b22+b32) If a = 3+i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and-az are 1) Vertices of a right angled triangle 2) Vertices of an equilateral triangle 3) Vertices of an isosceles triangle 4) Collinear The points z 1 , z 2 , z 3 , z 4 in the complex plane are the vertices of a parallelogram taken in order if and only if
E R K HSS –ERUMIYAMPATTI
Page 87
+2 STUDY MATERIALS
www.tnschools.co.in
1) z 1 z 4 z 2 z 3 63.
2) z 1 z 3 z 2 z 4
4) z 1 z 2 z 3 z 4 3) z 1 z 2 z 3 z 4 If z represents a complex number then arg (z) + arg ( z ) is 1)
2)
4
64.
3) 0
4)
2
4
If the amplitude of a complex number is
, then the number is
2
1) purely imaginary 3) 0 65.
2) purely real 4) neither real nor imaginary
If the point represented by the complex number iz is rotated about the origin through the angle
in the
2
66.
counter clockwise direction then the complex number representing the new position is 1) iz 2) –iz 3) -z 4) z The polar form of the complex number (i 25)3 is
1) cos
+ i sin
2
67.
2) cos + i sin
2
3) cos - i sin
3) the straight line z =
- i sin
2
2
and if 2 z 1 2 z , then the locus of P is
If P represents the variable complex number z 1) the straight line x =
4) cos
1
2) the straight line y =
4 1
1 4
4) the circle x2 + y2 - 4x - 1 = 0
2
68.
69.
1 e 1 e
i i
=
1) cos + i sin 2) cos - i sin 3) sin - i cos n n + i sin then z 1 z 2 z 3 ... z 6 is If z n = cos 3
70. 71.
3
1) 1 2) -1 3) i If z lies in the third quadrant then z lies in the 1) first quadrant 2) second quadrant 3) third quadrant If x = cos + i sin then the value of x n
1 x
72.
73.
If Z1 = 4 + 5i , Z2 = -3 + 2i then 2 13
75. 76. 77.
n
4) –i 4) fourth quadrant
is
1) 2 cos n 2) 2i sin n 3) 2 sin n 4) 2 i cos n 2 2 If a = cos - i sin , b = cos - i sin , c = cos - i sin then( a c - b 2 ) / abc is 1) cos2 ( ) i sin 2 ( ) 2) -2 cos ( ) 3) -2 i sin ( ) 4) 2 cos ( )
1) 74.
4) sin + i cos
22
i
2) -
13
2 13
Z1
is
Z2
22
i
13
3)
2 13
23
i
13
4)
2
13
22
i
13
The value of i + i22 + i23 + i24 + i25 is 1) i 2) -i 3) 1 4) -1 13 14 15 16 The conjugate of i + i + i + i is 1) 1 2) -1 3) 0 4) –i If -i +2 is one root of the equation ax2 - bx + c = 0 then the other root is 1) -i -2 2) i - 2 3) 2 +i 4) 2i + i The quadratic equation whose roots are 1) x 2 + 7 = 0 2) x2 - 7 = 0
E R K HSS –ERUMIYAMPATTI
i 7 is 3) x2 + x +7 = 0
Page 88
4) x2 - x -7 = 0
+2 STUDY MATERIALS
www.tnschools.co.in
78.
The equation having 4 - 3i and 4 + 3i as roots is 1) x2 +8x + 25 = 0 2) x2 + 8x - 25 = 0 3) x2 - 8x + 25 = 0 1 i
is a root of the equation ax2 + bx +1 = 0 ,where a, b are real then ( a, b) is
79.
If
80.
1) ( 1, 1) 2) ( 1, -1) 3) ( 0,1) 2 If –i+3 is a root of x - 6x + k = 0, then the value of k is
1 i
1) 5 81. 82.
4) x2 - 8x - 25 = 0
2)
3)
5
4) ( 1, 0 ) 4) 10
10
If is a cube root of unity then the value of (1 - + 1) 0 2) 32 3) -16 If is the nth root of unity then
2
) + (1 + - 2 )4 is 4) -32 4
1) 1 + 2 + 4 + …. = + 3 + 5 + ….
2) n = 0
3) n = 1
4) = n 1
If is the cube root of unity then the value of (1- ) ( 1- 2 ) ( 1- 4 ) ( 1 - 8 ) is 1) 9 2) -9 3) 16 4) 32 -------------------------------------------------------------------------------------------------------------------------------------------------------4. ANALYTICAL GEOMETRY (THREE QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------84. The axis of the parabola y2 - 2y +8x - 23 = 0 is 1) y = -1 2) x = -3 3) x = 3 4) y = 1 85. 16x2 - 3y2 - 32x - 12y -44 = 0 represents 1) an ellipse 2) a circle 3) a parabola 4) a hyperbola 86. The line 4x + 2y = c is a tangent to the parabola y2 = 16 x then c is 1) -1 2) -2 3) 4 4) -4 87. The point of intersection of the tangents at t1 = t and t2 = 3t to the parabola y2 = 8x is 1) (6t2, 8t) 2) ( 8t, 6t2 ) 3) ( t2, 4t ) 4) (4t, t 2 ) 2 88. The length of the latus rectum of the parabola y - 4x + 4y +8 = 0 is 1) 8 2) 6 3) 4 4) 2 89. The directrix of the parabola y2 = x + 4 is 83.
1) x
15
2) x
4
90. 91. 92.
3) x
4
17
4) x
17 4
4
The length of the latus rectum of the parabola whose vertex is (2, -3 ) and the directrix x = 4 is 1) 2 2) 4 3) 6 4) 8 The focus of the parabola x2 = 16y is 1) ( 4, 0 ) 2) ( 0, 4 ) 3) ( -4, 0 ) 4) ( 0, -4 ) 2 The vertex of the parabola x = 8y -1
1)
93.
15
, 0 8 1
, 0 8 1
2)
3) 0 ,
1 8
4) 0 ,
1 8
The line 2x + 3y + 9 = 0 touches the parabola y2 = 8x at the point 1) (0, -3)
2) ( 2, 4)
3) 6 ,
, 6 2
9 2
9
4)
The tangents at the end of any focal chord to the parabola y = 12x intersect on the line 1) x-3 = 0 2) x + 3 = 0 3) y + 3 = 0 4) y - 3 = 0
The angle between the two tangents drawn from the point (- 4, 4) to y2 = 16x is 1) 45 2) 30 3) 60 4) 90 The eccentricity of the conic 9x2 + 5y2 - 54x - 40y + 116 = 0 is
96.
2
1)
1
2)
3
E R K HSS –ERUMIYAMPATTI
2 3
3)
4 9
Page 89
4)
2 5
+2 STUDY MATERIALS
www.tnschools.co.in
97.
x
The length of the semi-major and the length of semi-minor axis of the ellipse
2
144
98. 99.
( x 4)
2
( y6)
4
3)
( x 4)
2
( y6)
103. 104. 105.
1
1 is
169
( y6)
2
16 1
4)
1
4
( x 4)
2
2
( y6)
2
4
2
1
16
The straight line 2x - y + c = 0 is a tangent to the ellipse 4x2 + 8y2 = 32 then c is 2) 6
4) 4
3) 36 2
2
The sum of the distance of any point on the ellipse 4x +9y = 36 from 5 , 0 1) 4 2)8 3)6 4)18 2 2 The radius of the director circle of the conic 9x + 16y = 144 is
and
5 ,0
is
1) 7 2)4 3)3 4)5 The locus of foot of the perpendicular from the focus to a tangent of the curve 16x2 + 25y2 = 400 is 1) x2 + y2 = 4 2) x2 + y2 = 25 3) x2 + y2 = 16 4) x2 + y2 = 9 2 2 The eccentricity of the hyperbola 12y -4x -24x+48y-127 = 0 is 1) 4 2) 3 3) 2 4) 6 The eccentricity of the hyperbola whose latus rectum is equal to half of its conjugate axis is 3
1)
2)
5
3)
3
2
106.
2)
4
1) 2 3
102.
( x 4)
2
16
16
101.
2
1) 26, 12 2) 13, 24 3) 12, 26 4) 13, 12 The distance between the foci of the ellipse 9x2 + 5y2 = 180 is 1) 4 2) 6 3) 8 4) 2 If the length of major and semi-minor axes of an ellipse are 8,2 and their corresponding equations are y - 6 = 0 and x + 4 = 0 then the equations of the ellipse is 1)
100.
y
3
5
4)
2
2
The difference between the focal distances of any point on the hyperbola
x
2
a
2
y
2
b
2
1 is 24 and the
eccentricity is 2 . Then the equation of the hyperbola is 1)
x
2
144
107.
y
2
1
2)
432
x
2
y
432
2
1
3)
2
y
12
144
8
2) x =
8
5
1
x
4)
2
5
1
8
2
The line 5x - 2y + 4k = 0 is a tangent to 4x - y = 36 then k is 4
2)
2
9
3)
3
81
4)
4
16
The equation of the chord of contact of tangents from (2,1) to the hyperbola
x
2
16
1) 9x -8y -72 = 0
2) 9x + 8y + 72 = 0
3) 8x -9y -72 = 0
The angle between the asymptotes to the hyperbola
x
2
16
1) 2 tan 111.
2
5
4) x=
8
5
9
110.
y
12
12 3
12 3
3) y = 2
1) 109.
2
The directrices of the hyperbola x2 – 4(y - 3)2 = 16 1) y =
108.
x
1
3 4
2) 2 tan
1
4 3
y
y
2
1 is
9
4) 8x + 9y + 72 = 0
2
1 is
9
3) 2 tan
1
3 4
4) 2 tan
1
4 3
The asymptotes of the hyperbola 36y2 - 25x2 + 900 = 0 are 1) y =
6
x
2) y =
5
E R K HSS –ERUMIYAMPATTI
5
x
3) y =
6
36 25
Page 90
x
4) y =
25
x
36
+2 STUDY MATERIALS
www.tnschools.co.in
112.
The product of the perpendiculars drawn from the point (8,0) on the hyperbola to its asymptotes x
2
64
y
2
1
is
36 25
1)
576
2)
576
113.
6
3)
25
4)
25
25
6
The locus of the point of intersection of perpendicular tangents to the hyperbola
x
2
16
114. 115. 116. 117. 118. 119. 120.
121.
y
2
1 is
9
1) x2 +y2 = 25 2) x2 +y2 = 4 3) x2 +y2 = 3 4) x2 +y2 = 7 The eccentricity of the hyperbola with asymptotes x + 2y - 5 = 0 , 2x - y + 5 = 0 is 3) 3 4)2 1) 3 2) 2 Length of the semi-transverse axis of the rectangular hyperbola xy = 8 is 1) 2 2)4 3)16 4)8 The asymptotes of the rectangular hyperbola xy = c2 are 1) x = c , y = c 2) x = 0 , y = c 3) x = c , y = 0 4) x = 0 , y = 0 The co-ordinate of the vertices of the rectangular hyperbola xy = 16 are 1)(4 , 4 ) , (-4 , -4) 2)(2 , 8) , (-2 , -8) 3)(4 , 0) , (-4 , 0) 4)(8 , 0) , (-8 , 0) One of the foci of the rectangular hyperbola xy =18 is 1) (6 , 6) 2)(3 , 3) 3)(4 , 4) 4)(5 , 5) The length of the latus rectum of the rectangular hyperbola xy = 32 is 1) 8 2 2)32 3)8 4)16 The area of the triangle formed by the tangent at any point on the rectangular hyperbola xy = 72 and its asymptotes is 1) 36 2)18 3)72 4)144
The normal to the rectangular hyperbola xy = 9 at 6 ,
3
8
1) , 24
2) 24 ,
3 8
3 meets the curve again at 2
3
, 24 8
3)
4) 24 ,
3 8
-----------------------------------------------------------------------------------------------------------------------------------------------------5. DIFFERENTIAL CALCULUS- APPLICAIONS-I (THREE QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------1. The gradient of the curve y = -2x3 +3x + 5 at x = 2 is 1) -20 2) 27 3) -16 4) -21 2. The rate of change of area A of a circle of radius r is 2) 2 r
1) 2 r
dr
3) r 2
b
2)
x
6.
a
3)
x
x
4)
b
x a
A spherical snowball is melting in such a way that its volume is decreasing at a rate of 1 cm 3/min. The rate at which the diameter is decreasing when the diameter is 10 cms is 1) -
5.
dt
The velocity v of a particle moving along a straight line when at a distance x from the origin is given by a + bv2 = x2 where a and b are constants. Then the acceleration is 1)
4.
dr
4)
dt
dt
3.
dr
1 50
cm / min
2)
1 50
cm / min
3) -
11 75
cm / min
4)-
2 75
cm / min
The slope of the tangent to the curve y = 3x2 + 3sinx at x = 0 is 1) 3 2)2 3) 1 4) -1 The slope of the normal to the curve y = 3x2 at the point whose x coordinate is 2 is 1)
1
2)
13
E R K HSS –ERUMIYAMPATTI
1 14
3)
1 12
Page 91
4)
1 12
+2 STUDY MATERIALS
www.tnschools.co.in
7.
The point on the curve y = 2x2 -6x – 4 at which the tangent is parallel to the x–axis is 5
1)
2
8.
,
17 2
5
2)
2
,
17 2
5
3)
2
The equation of the tangent to the curve y =
x
9.
2) 5y - 3x = 2
4)
2
at the point 1,
3) 3x - 5y = 2
The equation of the normal to the curve
1 t
1) 3 = 27t - 80
3
17 2
3
5
1) 5y + 3x = 2
,
2) 5 = 27 t -80
,
17 2
1 is 5
4) 3x + 3y = 2
at the point 3 ,
3) 3 = 27 t + 80
1 is 3
4)
1 t
10.
x
The angle between the curves
2
25
1)
2)
4
11.
3) tan
13.
14.
1 and
x
9
2
y
8
2
1 is
8
3)
3
4)
6
2
The angle between the curve y = emx and y = e-mx for m> 1 is 1) tan
12.
y
2
1
1
2m 2 m 1
2) tan
2m 2 1 m
4) tan
1
1
2m 2 1 m 2m 2 m 1
The parametric equation of the curve x2/3 + y2/3 = a2/3 are 1) x = a sin3 , y = a cos3 2) x = a cos3 , y = a sin3 3) x = a3 sin , y = a3 cos 4) x = a3 cos , y = a3 sin 2/3 2/3 2/3 If the normal to the curve x + y = a makes an angle with the x-axis then the slope of the normal is 1) -cot 2) tan 3) -tan 4) cot If the length of the diagonal of a square is increasing at the rate of 0.1 cm/sec. What is the rate of increase of its area when the side is
15
cm ?
2
15.
1) 1.5 cm2/sec 2) 3 cm2/sec 3) 3 2 cm2/sec 4) 0.15 cm2/sec What is the surface area of a sphere when the volume is increasing at the same rate as its radius? 1) 1
16.
18. 19.
20. 21.
1
3) 4
2
4)
4 3
For what values of x is the rate of increasing of x3 - 2x2 + 3x + 8 is twice the rate of increase of x 1)
17.
2) , 3 3
1
1
3
2) , 3
3)
, 3 3
1
1
3
4) , 1
The radius of a cylinder is increasing at the rate of 2cm/sec and its altitude is decreasing at the rate of 3 cm/sec. The rate of change of volume when the radius is 3cm and the altitude is 5 cm is 1) 23 2) 33 3) 43 4) 53 3 If y = 6x - x and x increases at the rate of 5 units per second, the rate of change of slope when x = 3 is 1) -90 units/sec 2)90 units/sec 3)180 units/sec 4)-180 units/sec If the volume of an expanding cube is increasing at the rate of 4cm3/sec then the rate of change of surface area when the volume of the cube is 8 cubic cm is 1) 8 cm2/sec 2) 16 cm2/sec 3) 2 cm2/sec 4) 4 cm2/sec 2 The gradient of the tangent to the curve y = 8+4x-2x at the point where the curve cuts the y-axis is 1. 8 2. 4 3. 0 4. -4 The angle between the parabolas y2 = x and x2 = y at the origin is 1) 2 tan
1
3 4
2) tan
E R K HSS –ERUMIYAMPATTI
1
4 3
3)
2
Page 92
4)
4
+2 STUDY MATERIALS
www.tnschools.co.in
22.
For the curve x =etcost; y = etsint , the tangent line is parallel to the x-axis when t is equal to 1. -
2.
4
3. 0
4.
4
2
23.
If a normal makes an angle with positive x-axis then the slope of the curve at the point where the normal is drawn is 2) tan 3) -tan 4) cot 1) -cot
24.
The value of „a‟ so that the curves y = 3ex and y =
a
e
x
intersect orthogonally is
3
1) -1
2) 1
3)
1
4) 3
3
25.
If s = t3 - 4t2 + 7, the velocity when the acceleration is zero is 1)
32
2)
m / sec
16
3
26.
27.
m / sec
3)
3
16
4)
m / sec
32
3
m / sec
3
If the velocity of a particle moving along a straight line is directly proportional to the square of its distance from a fixed point on the line. Then its acceleration is proportional to 1) s 2) s2 3) s3 4) s4 2 The Rolle‟s constant for the function y = x on [ -2, 2] is 1)
2 3
2) 0
3) 2
4) -2
3
28.
The „c‟ of Lagranges Mean Value Theorem for the function f (x) = x2 + 2x -1; a= 0 , b= 1 is 1) -1
2) 1
3) 0
4)
1 2
29.
The value of c in Rolle‟s theorem for the function f (x) = cos
x 2
2) 2
1) 0
3)
on [ , 3 ] is
4)
3 2
2
30.
The value of „c‟ of Lagranges Mean Value Theorem for f (x) = 1)
9
2)
4
31.
lim
x
x
2
e
x
a b
x
d
x
x
lim
x 0
c
1)
x
3)
1
4)
2
2
4
2) 0
3)
4) 1
2) 0
3) log
is =
1) 2 32.
3
x when a = 1 and b = 4 is 1
= ab
4)
cd
log ( a / b ) log ( c / d )
33.
If f (a) = 2; f ' (a) = 1 ; g (a) = -1 ; g ' (a) = 2, then the value of lim
34.
1) 5 2) -5 3) 3 Which of the following function is increasing in ( 0, ) ?
g ( x) f ( a) g ( a ) f ( x)
x a
1) e x
2)
1
3) - x 2
xa
is
4) -3 4) x -2
x
35. 36.
The function f (x) = x 2 - 5x + 4 is increasing in 1) ( - , 1) 2) ( 1 , 4 ) 3) ( 4 , ) The function f (x) = x2 is decreasing in 1) ( - , ) 2) ( - , 0 ) 3) ( 0, )
E R K HSS –ERUMIYAMPATTI
Page 93
4) everywhere 4) ( -2, )
+2 STUDY MATERIALS
www.tnschools.co.in
37.
The function y = tan x - x is 1) an increasing function in 0 , 2
2) a decreasing function in 0 , 2
3) increasing in 0 , and decreasing in , 4 4 2
4) decreasing in 0 , and increasing in , 4 4 2
;
38.
In a given semi circle of diameter 4 cm a rectangle is to be inscribed.The maximum area of the rectangle is 1) 2 2) 4 3) 8 4) 16 39. The least possible perimeter of a rectangle of area 100 m2 is 1. 10 2. 20 3. 40 4. 60 40. If f (x) = x2 - 4x +5 on [0, 3 ] then the absolute maximum value is 1) 2 2) 3 3) 4 4) 5 41. The curve y = - e -x is 1) concave upward for x > 0 2) concave downward for x > 0 3) everywhere concave upward 4) everywhere concave downward 42. Which of the following curves is concave down? 1) y = - x 2 2) y = x 2 3) y = e x 4) y = x2 + 2x - 3 4 43. The point of inflexion of the curve y = x is at 1) x = 0 2) x = 3 3) x = 12 4) nowhere 44. The curve y = ax3 + bx2 + cx + d has a point of inflexion at x = 1 then 1) a + b = 0 2) a + 3b = 0 3) 3a + b = 0 4) 3a + b = 1 -----------------------------------------------------------------------------------------------------------------------------------------------------6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (TWO* QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------45.
1) yxy-1 46.
u
If u = x y then
x
2) u log x x
If u = sin - 1
4
y
x y 2
1) 0 47.
is equal to
x y 2
1
2
and f = sinu, then f is a homogeneous function of degree
2) 1 1
If u = 1)
4
, then x 2
3) 2 u x
y
u y
3)
2
48.
49. 50.
51.
4) 4
2) u
u
4) xyx-1
3) u log y
3
u
4) - u
2
The curve y2(x-2) = x2(1+x) has 1) an asymptote parallel to x-axis 3) asymptotes parallel to both axes If x = r cos , y = r sin , then
r x
2) an asymptote parallel to y-axis 4) no asymptotes
1) sec 2) sin 3) cos 4) cosec Identify the true statements in the following: (i) If a curve is symmetrical about the origin, then it is symmetrical about both axes. (ii) If a curve is symmetrical about both the axes, then it is symmetrical about the origin. (iii) A curve f (x, y) = 0 is symmetrical about the line y = x if f (x, y) = f (y, x). (iv) For the curve f (x, y) = 0, if f (x, y) = f (-y, - x), then it is symmetrical about the origin. 1. (ii), (iii) 2. (i), (iv) 3. (i), (iii) 4. (ii), (iv) x2 y2 u u , then x y xy y x
If u = log 1) 0
2) u
E R K HSS –ERUMIYAMPATTI
is 3) 2u
Page 94
4) u-1
+2 STUDY MATERIALS
www.tnschools.co.in
52.
The percentage error in the 11th root of the number 28 is approximately _________ times the percentage error in 28. 1
1)
1
2)
28
53.
54.
3) 11
The curve a2y2 = x2 (a2-x2) has 1. only one loop between x =0 and x = a. 2. two loops between x =0 and x = a. 3. two loops between x = -a and x = a. 4. no loop An asymptote to the curve y2 (a + 2x) = x2 (3a - x) is 1) x = 3a
a
2) x = -
3) x =
2
55.
4) 28
11
a
4) x = 0
2
In which region the curve y2 (a + x ) = x2 (3a - x) does not lie? 1) x > 0 2) 0 < x < 3a 3) x -a and x > 3a u
4) -a < x < 3a
2
56.
If u = y sinx, then
=
x y
1) cos x
2) cos y
3) sin x
4) 0
57.
If u = f
1) 0 2) 1 3) 2u The curve 9y2 = x2(4-x2) is symmetrical about 1. y-axis 2. x-axis 3. y = x
4) u
58.
u u y is equal to y then x y x x
4. both the axes
The curve ay2 = x2 (3a-x) cuts the y-axis at 1) x = -3a, x = 0 2) x = 0, x = 3a 3) x = 0, x = a 4) x = 0 -------------------------------------------------------------------------------------------------------------------------------------------------------7. INTEGRAL CALCULUS (THREE QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------59.
/2
60.
cos
The value of
cos
0
1)
2)
dx x
is 3) 0
4)
4 /2
The value of
sin x cos x 1 sin x cos x
0
1)
5/3
2
61.
x
x sin
5/3
5/3
dx
is
2) 0
c)
2
d)
4 1
62.
The value of
x (1 x )
4
is
dx
0
1)
1
2)
12
1 30
/2
63.
The value of
3)
sin x 2 cos x / 2
1) 0
2) 2
dx
1
4)
1
24
20
3) log 2
4) log 4
is
64.
The value of
sin
4
x dx
is
0
E R K HSS –ERUMIYAMPATTI
Page 95
+2 STUDY MATERIALS
www.tnschools.co.in
1.
3
3
2.
16
3. 0
4.
3
16
8
/4
65.
cos
The value of
3
is
2 x dx
0
1.
2
2.
3
1
3. 0
4.
2
3
3
66.
The value of
sin
2
x cos
3
is
xdx
0
1)
2)
3)
The area bounded by the line y = x, the x-axis, the ordinates x = 1, x = 2 is 1)
3
2)
2
68.
4) 0
4
2
67.
5
3)
2
1
4)
7
2
2
The area of the region bounded by the graph of y = sinx and y = cosx between x = 0 and x=
is
4
1) 2 1 69. 70.
The area between the ellipse
4
2)
3
76. 77. 78.
y b
2
4) 2 2 2
1 and its auxillary circle is
3)
6
2)
2
4)
3
100
x
2
y
2
1 about the minor axis is
16
3) 32 3 x
8 3
2
9
2) 64
4) 128
from x = 0 to x = 4 is rotated about x- axis is 3)
100
4)
100 3
3
The volume generated when the region bounded by y = x, y = 1, x = 0 is rotated about y-axis is 2 1) 2) 3) 4) 4
75.
1
9
74.
a
2
The volume, when the curve y = 1) 100
73.
x
2
The volume of the solid obtained by revolving 1) 48
72.
3) 2 2 2 2
2) 2 a (a - b) 3) a (a - b) 4) 2 b ( a - b) 1) b(a - b) The area bounded by the parabola y2 = x and its latus rectum is 1)
71.
2 1
2)
2
3
Volume of solid obtained by revolving the area of the ellipse
3 x
2
a
2
y
2
b
2
1 about major and minor axes are
in the ratio 1) b2 : a2 2) a2 : b2 3) a : b 4) b : a The volume generated by rotating the triangle with vertices at (0 , 0) , (3 , 0) and (3 , 3) about x-axis is 1)18 2) 2 3) 36 4) 9 2/3 2/3 The length of the arc of the curve x + y = 4 is 1) 48 2) 24 3) 12 4) 96 The surface area of the solid of revolution of the region bounded by y = 2x , x = 0 and x = 2 about x-axis is 1) 8 5 2) 2 5 3) 5 4) 4 5 The curved surface area of a sphere of radius 5, intercepted between two parallel planes of distance 2 and 4 from the centre is 1) 20 2) 40 3)10 4) 30
E R K HSS –ERUMIYAMPATTI
Page 96
+2 STUDY MATERIALS
-----------------------------------------------------------------------------------------------------------------------------------------------------8. DIFFERENTIAL EQUATIONS www.tnschools.co.in (THREE QUESTIONS FOR FULL TEST) ----------------------------------------------------------------------------------------------------------------------------------------------------79.
The integrating factor of
dy
2
y
dx
4x
is
x
2) x2
1) log x 80.
e
3) ex
4) x dy
If cos x is an integrating factor of the differential equation
+ Py = Q , then P =
dx
81. 82.
1) -cot x 2) cot x 3) tan x The integrating factor of dx + xdy = e-y sec2 y dy is 1) ex 2) e-x 3) ey The integrating factor of
dy dx
1) ex
1
.y
x log x
2 x
4) e-y
is
2
2) log x
4) -tan x
3)
1
4) e-x
x
83.
The solution of
dx
+ mx = 0 , where m < 0 is
dy
84.
1) x = cemy 2) x = ce-my 3) x = my + c 2 y = cx - c is the general solution of the differential equation 2) y = 0 3) y = c 1) (y)2 - xy + y = 0
85.
dx 1/ 3 5 y The differential equation x is dy
4) x = c 4) (y)2 + xy + y = 0
2
86.
1. of order 2 and degree 1 2. of order 1 and degree 2 3. of order 1 and degree 6 4. of order 1 and degree 3 The differential equation of all non-vertical lines in a plane is 1)
dy
2
=0
2)
dx
87. 88.
d y dx
2
0
3)
2
dy
=m
4)
dx
dx
The differential equation of all circles with centre at the origin is 1) x dy + y dx = 0 2) x dy - y dx = 0 3) x dx + y dy = 0 dy
The integrating factor of the differential equation
d y
m
2
4) x dx - y dy = 0
+ py = Q is
dx
1) 89.
90.
p dx
2)
3) e
Q dx
4) e
Qdx
The complementary function of (D2 + 1 ) y = e2x is 1) (Ax + B)ex 2) A cos x +B sinx 3) (Ax + B)e2 x
4) (Ax + B)e-x
A particular integral of (D2 - 4D +4)y = e2x is 1)
x
2
e
2x
2) x e 2 x
3) x e 2 x
4)
e
2 x
The differential equation of the family of lines y = mx is 1)
dy
2
=m
2) y dx - xdy = 0
dx
92.
x 2
2
91.
pdx
d y dx
The degree of the differential equation 1) 1
3)
2) 2
E R K HSS –ERUMIYAMPATTI
2
dy 1 dx
0 1/3
4) ydx + xdy = 0 2
3) 3
Page 97
d y dx
2
is 4) 6
+2 STUDY MATERIALS
www.tnschools.co.in
93.
The degree of the differential equation c=
3 dy 1 dx
2/3
, where c is a constant is
3
d y 3
dx
94.
1) 1 2) 3 3) -2 4) 2 The amount present in a radioactive element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is ( k is negative) 1)
dp
dt
95.
k
2)
3)
dt
p
dp
kp
4)
dt
dp
kt
dt
2
dy
= a constant
2.
dx
d y dx
0
2
dy
3. y+
2
=0
4.
dx
d y dx
y0
2
If y = kex then its differential equation is 1)
dy
=y
2)
dx
97.
kt
The differential equation satisfied by all the straight lines in xy-plane is 1.
96.
dp
dy
= ky
3)
dx
dy
+ ky = 0
4)
dx
dy
= ex
dx
The differential equation obtained by eliminating a and b from y = ae3x + be-3x is 2
1)
d y dx
2
2
ay 0
2)
d y dx
2
2
9y 0
3)
d y dx
2
9
dy
2
0
4)
dx
d y dx
9x 0
2
98.
The differential equation formed by eliminating A and B from the relation y = ex (A cos x +Bsinx) is 1) y2 + y1 = 0 2) y2 - y1 = 0 3) y2 - 2y1 +2y = 0 4) y2 - 2y1 -2y = 0
99.
If
dy
=
dx
x y x y
then
1) 2xy + y2 + x2 = c 100.
2) x2 + y2 - x + y = c
3) x2 + y2 - 2xy = c
4) x2 - y2 - 2xy = c
If f (x) =
x and f (1) = 2, then f (x) is 2 3 1) ( x x 2 ) 2) ( x x 2 ) 3 2
3)
2
( x x 2)
4)
3
2
x( x 2)
3
101.
On putting y = vx, the homogeneous differential equation x2dy +y(x + y)dx=0 becomes 1) xdv + (2v+v2)dx = 0 2) vdx + (2x+x2)dv = 0 2 2 3) v dx - (x+x )dv = 0 4) vdv + (2x+x2)dx = 0
102.
The integrating factor of the differential equation
dy
- y tan x = cos x is
dx
103.
1) sec x 2) cos x 3) etanx 2 The particular integral of (3D + D-14)y = 13e2x is
4) cotx
1) 26xe2x
4)
2)13xe2x
3) xe2x
x
2
e
2x
2
104.
The particular integral of the differential equation f (D) y = eax , where f (D) = (D-a) g(D), g(a) 0 is 1) meax
2)
e
ax
3) g(a)eax
g (a )
4)
xe
ax
g (a )
-----------------------------------------------------------------------------------------------------------------------------------------------------9. DISCRETE MATHEMATICS (THREE QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------105. Which of the following are statements? (i) May God bless you (ii) Rose is a flower (iii) Milk is white (iv) 1 is a prime number 1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) 4) (ii) , (iii) , (iv)
E R K HSS –ERUMIYAMPATTI
Page 98
+2 STUDY MATERIALS
www.tnschools.co.in
106.
107.
108. 109. 110. 111. 112. 113. 114. 115. 116.
117. 118. 119. 120. 121.
122. 123. 124.
If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 If p is T and q is F, then which of the following have the truth value T ? (i) p q (ii) ~ p q (iii) p ~q (iv)) p ~q 1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) 4) (ii) , (iii) , (iv) The number of rows in the truth table of ~ [p(~q)] is 1) 2 2) 4 3) 6 4) 8 The conditional statement p q is equivalent to 2) p ~q 3) ~ p q 4) p q 1) p q Which of the following is a tautology? 1) p q 2) pq 3) p ~p 4) p ~p Which of the following is a contradiction? 2) pq 3) p ~p 4) p ~p 1) p q p q is equivalent to 1) p q 2) q p 3) (p q ) ( q p) 4) (p q ) ( q p) Which of the following is not a binary operation on R? 1) a * b = ab 2) a * b = a- b 3) a * b = ab 4) a * b = a 2 b 2 A monoid becomes a group if it also satisfies the 1. closure axiom 2. associative axiom 3. identity axiom 4.inverse axiom Which of the following is not a group ? 1) (Zn , +n) 2) (Z , +) 3) (Z , .) 4) (R , +) In the set of integers with the operation * is defined by a * b = a + b- ab, then the value of 3 * (4 * 5) is 1) 25 2) 15 3)10 4) 5 The order of [7] in (Z9 , +9) is 1) 9 2) 6 3) 3 4) 1 2 The multiplicative group of cube root of unity, the order of is 1)4 2)3 c)2 d)1 The value of [3] +11 ( [5] +11 [6] ) is 1. [0] 2)[1] 3)[2] 4)[3] In the set of real numbers R, an operation * is defined by a * b =
a b 2
2
Then the value of (3 * 4)* 5 is
1)5 2)5 2 3)25 4)50 Which of the following is correct ? 1. An element of a group can have more than one inverse. 2. If every element of a group is its own inverse, then the group is abelian. 3. The set of all 2x2 real matrices forms a group under matrix multiplication. 4. (a * b )-1= a -1* b -1, for all a, b G. The order of -i in the multiplicative group of 4th roots of unity is 1) 4 2) 3 3) 2 4) 1 In the multiplicative group of nth roots of unity, the inverse of k is (k < n) 2) -1 3) n-k 4) n/k 1) 1/k In the set of integers under the operation * defined by a * b = a + b -1, the identity element is 1)0 2)1 3)a 4)b
E R K HSS –ERUMIYAMPATTI
Page 99
+2 STUDY MATERIALS
www.tnschools.co.in
-----------------------------------------------------------------------------------------------------------------------------------------------------10. PROBABILITY DISTRIBUTIONS (THREE QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------125.
kx 2 , 0 x 3 f ( x) elsewhere 0,
If
is a probability density function then the value of k is 1)
1
2)
3
1
3)
4)
9
6
1
A
1
12
, x is a p.d.f of a continuous random variable X, then the value of A is
126.
If f (x) =
127.
1) 16 2) 8 3) 4 A random variable X has the following probability distribution
16 x
2
1
X
0 1
P(X=x)
4
4) 1
1
2
3
4
2a
3a
4a
5a
5 1 4
Then P(1 x 4) is 1)
10
2)
3)
7
21
128.
2
1
4)
14
2
A random variable X has the following probability mass function as follows: 2
X P(X=x)
3
6
Then the value of is 1) 1 2) 2 129.
1
1
4
12
3) 3
4) 4
X is a discrete random variable which takes the values 0, 1, 2 and P(X=0) =
144
, P(X=1) =
169
1
, then
169
the value of P(X =2 ) is 1)
145
2)
169
130.
24
2
3)
169
4)
169
143 169
A random variable X has the following p.d.f. X P(X =x)
0 0
1 k
2 2k
3 2k
4 3k
5 k2
6 2k2
7 7k2 + k
The value of k is 1)
1
2)
132. 133. 134. 135.
3) 0
1
4) -1 or
10
8
131.
1
10
Given E(X+ c) = 8 and E(X-c) = 12 then the value of c is 1) -2 2) 4 3) -4
4) 2
X is a random variable taking the values 3, 4 and 12 with probabilities
1 1 , 3 4
and
5
Then E (X) is
12
1)5 2)7 3) 6 4) 3 Variance of the random variable X is 4. Its mean is 2. Then E(X2) is 1)2 2)4 3)6 4)8
2 = 20, 2 = 276 for a discrete random variable X. Then the mean of the random variable X is 1)16 2)5 3)2 4)1 Var (4X + 3) is 1)7 2)16 Var(X) 3)19 4)
E R K HSS –ERUMIYAMPATTI
Page 100
+2 STUDY MATERIALS
www.tnschools.co.in
136.
In 5 throws of a die, getting 1 or 2 is a success. The mean number of successes is 1.
5
3
2.
3
137.
5
4
, 25 5
1
2.
4 5
5
1
5
3. , 25
1
3.
3
1
2)
20
4. 25 ,
2
4.
3
18
3)
125
1
2)
2
4
4)
145. 146.
3 10
26
3)
25
4)
51
25 102
If in a Poisson distribution P(X = 0) = k, then the variance is 1
2) log k
3) e
4)
k
144.
1 4
25
51
1) log 143.
1 5
If 2 cards are drawn from a well shuffled pack of 52 cards, the probability that they are of the same colours without replacement, is 1)
142.
9
In 16 throws of a die getting an even number is considered a success. Then the variance of the successes is 1. 4 2. 6 3. 2 4. 256 A box contains 6 red balls and 4 white balls. If 3 balls are drawn at random, the probability of getting 2 white balls without replacement is 1)
141.
9
2. 25 ,
2
140.
4.
If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is 1.
139.
5
The mean of a binomial distribution is 5 and its its standard deviation is 2. Then the value of n and p are 1.
138.
3.
1 k
2
If a random variable X follows Poisson distribution such that E(X ) = 30, then the variance of the distribution is 1. 6 2. 5 3. 30 4. 25 The distribution function F(X) of a random variable X is 1. a decreasing function 2. a non-decreasing function 3. a constant function 4. increasing first and then decreasing For a Poisson distribution with parameter =0.25 the value of the 2nd moment about the origin is 1)0.25 2)0.3125 3)0.0625 4)0.025 In a Poisson distribution if P(X = 2) =P(X=3) then the value of its parameter is 1) 6 2) 2 3) 3 4) 0
147.
If f ( x ) is a p.d. f. of a normal distribution with mean , then
f ( x ) dx is
1. 1
2. 0.5
3. 0
4. 0.25 1 2 ( x 100 )
148.
The random variable X follows normal distribution f(x) = c e 1)
2
2)
1 2
3) 5 2
2
25
. Then the value of „c‟ is 4)
1 5 2
149.
If f ( x ) is a p.d. f. of a normal variate X and
X ~ N( , 2 ), then
f ( x ) dx is
1. undefined 150.
2. 1
3. 0.5
4. -0.5
The marks secured by 400 students in a Mathematics test were normally distributed with mean 65. If 120 students got more marks above 85, the number of students securing marks between 45 and 65 is 1)120 2)20 3)80 4)160
E R K HSS –ERUMIYAMPATTI
Page 101
+2 STUDY MATERIALS
www.tnschools.co.in
www.tnschools.co.in
+2 MATHEMATICS Chapter wise Collection of Public out Of Book Questions From 26 Public Question Papers [2006-14] 2006: March/June/October 2007: March/June/October 2008: March/June/October 2009: March/June/October 2014: March/June.
E R K HSS –ERUMIYAMPATTI
2010: March/June/October 2011: March/June/October 2012: March/June/October 2013: March/June/October
Page 102
+2 STUDY MATERIALS
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: I MATRICES AND DETERMINANTS
www.tnschools.co.in
--------------------------------------------------------------------------------------------------------------------------------------------------------
PART-B [SIX MARKS] 1.
2.
3. 4. 5.
Verify that (A-1 )T = (AT )-1 for the matrix 2 0 1
Find the rank of the matrix
Find the rank of the matrix
1
3
1
1
3
4
4 0 4
4
2 A 5
3 6
4 2 7
12
12
4
8
4
8
4 4 0
Solve the system of linear equations by determinant method. Solve the system of linear equations by determinant method.
2 x 3 y 7 ; 4 x 6 y 14
x y 3 z 4 ; 2 x 2 y 6 z 7 ; 2 x y z 10
6.
Solve the system of linear equations by determinant method. x y 2z 0 ; x y z 5 ; 2x z 6
7. 1.
Solve by determinant method.
x 2 y 2 ; 2x 4 y 4
PART-C [TEN MARKS] Solve the following non- homogeneous system of linear equations by using determinant method. x +2 y + z =2; 2x +4 y +2z =4; x -2 y - z =0
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 2 VECTOR ALGEBRA --------------------------------------------------------------------------------------------------------------------------------------------------------
PART-B [SIX MARKS]
1.
For any three vectors
a , b , c
prove that a b , b c , c a 0
2.
For any three vectors
a , b , c
prove that
3. 4.
If a i j , b j k , c k i then find Show that the points A (1,2,3) , B (3,-1,2) , C (-2,3,1) and D (6,-4,2) are lying on the same plane. PART-C [TEN MARKS] Find the vector and Cartesian equation of the plane through the point (-1,2,1) and perpendicular to the plane x 2 y 4 z 7 0 . and 2 x y 3 z 3 0 . Find the vector and Cartesian equation of the plane through the point (1, 2, -2) and parallel
1. 2.
to the line 3.
x2 3
y 1 2
z4 4
a b c , b c , c a a b , b c , c a
b c
and perpendicular to the plane 2 x + 3 y + 3
Find the vector and Cartesian equation of the plane which contains the line
z
x 1 2
= 8.
y 3
z 1 1
and
Perpendicular to the plane x -2 y + 3 z -2=0. --------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 3 COMPLEX NUMBERS --------------------------------------------------------------------------------------------------------------------------------------------------------
PART-B [SIX MARKS] E R K HSS –ERUMIYAMPATTI
Page 103
+2 STUDY MATERIALS
www.tnschools.co.in
1.
P represents the variable complex number z. Find the locus of P, if
2.
P represents the variable complex number z. Find the locus of P, if
3.
P represents the variable complex number z. Find the locus of P, if
3z 5 3 z 1
4.
If P represents the variable complex number z. Find the locus of P, if
z 1 Re 0 zi
5.
Solve the equation
x 4 0 if one of the root is 1 i
6.
Solve the equation
x 4 x 6 x 4 0 if 1 i is a root PART-C [TEN MARKS]
1.
If P represents the variable complex number z. Find the locus of P, if
2z 1 z 2
z 3i z 3i
4
3
2
2z i Im 1 iz 1
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT:4 ANALYTICAL GEOMETRY --------------------------------------------------------------------------------------------------------------------------------------------------------
1. 2. 3. 4.
PART-C [TEN MARKS] Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the 2 following parabolas and hence sketch their graphs. y 8 x 2 y 1 7 0 Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the 2 following parabolas and hence sketch their graphs. y 4 y 4 x 8 0 Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the 2 following parabolas and hence sketch their graphs. x 4 x 4 y 0 Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram: 16 x 9 y 32 x 36 y 92 0 2
5.
2
Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram: 9 x 25 y 18 x 100 y 116 0 2
6.
2
Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams. 12 x 4 y 24 x 32 y 127 0 2
7.
2
Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams. 9 x 7 y 36 x 14 y 92 0 2
8.
2
Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams. 16 x 9 y 32 x 36 y 164 0 2
2
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT:6 DIFFERENTIAL CALCULUS – APPLICATIONS - II --------------------------------------------------------------------------------------------------------------------------------------------------------
PART-C [TEN MARKS] 2
1. 2.
Verify Prove
u x y
u y x
for the functions: u sin
1 y x y 2 x u
x y
2
u
E R K HSS –ERUMIYAMPATTI
x y x
cos y
x y x
y
u sin if
Page 104
x y x
y +2 STUDY MATERIALS
www.tnschools.co.in
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 7 INTEGRAL CALCULUS --------------------------------------------------------------------------------------------------------------------------------------------------------
1. 2. 3.
PART-C [TEN MARKS] 2 Find the area between the curves y x 2 x 3, x-axis and the lines x 3 and x 5 Find the common area enclosed by the parabolas y x and x y Find the volume of the solid obtained by revolving the area of the triangle whose sides are having the equation y=0 ; x=4 and 3x-4y = 0 , about x-axis [OR] Find the volume of the cone obtained by revolving the area of the triangle whose vertices are (0,0) ; (4,0) and (4,3) about x-axis 2
2
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 8 INTEGRAL CALCULUS --------------------------------------------------------------------------------------------------------------------------------------------------------
1.
3 D
Solve:
2
D 14 y 13 e
PART-B [SIX MARKS] 2x
10 e
x
2.
Solve: D 5 D 4 y sin 5 x PART-C [TEN MARKS]
1.
Solve : dy x dy 3 x ydx sec x[sec x tan x ]dx
2.
3 Solve : 1 2 x
3.
3 Solve : 1 x
4.
Solve: D 5 D 6 y sin x 2 e
5. 6.
2
3
2
dy
6 x y cos ec x 2
2
dx dy 2 3 x y sec dx
2
x
2
3x
Solve: D 2 D 2 y sin 2 x 5 2
Solve: D 13 D 12 y x 5 e 2
x
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 9 DISCRETE MATHEMATICS --------------------------------------------------------------------------------------------------------------------------------------------------------
PART-B [SIX MARKS] 1. Use the truth table to determine whether the statement ~ P q p ~ p is a tautology. 2. 3. 4.
With usual symbol, prove that Z 5 0 , 5 forms a group. Find the order of all elements of the group (Z6 ,+6) Find the order of all elements of the group Z 7 0 , 7
--------------------------------------------------------------------------------------------------------------------------------------------------------
UNIT: 10 PROBABILITY DISTRIBUTIONS --------------------------------------------------------------------------------------------------------------------------------------------------------
1.
Find k , f
x
and ke
2x
2
PART-B [SIX MARKS] of the normal distribution whose probability function is given by
4 x2
X
.
******** E R K HSS –ERUMIYAMPATTI
Page 105
+2 STUDY MATERIALS
www.tnschools.co.in
SOME IMPORTANT TRIGONOMETRICAL FORMULA’S TRIGONOMETRICAL IDENTITIES: opp opp adj tan = sin = cos = hyp hyp
cosec = sin =
sin θ
cos2
+ =1 = 1- cos2 cos2 = 1- sin2 sin2
cos =
cosec θ
cosec = sin2
sec =
opp
TRIGONOMETRICAL TABLES: π O 6 1 0 sin 2 √3 1 cos 2 1 0 tan √3
sec = 1+tan2
hyp hyp
cot =
adj
tan =
sec θ
cot =
cos θ
sec2
= - tan2 = 1 tan2 = sec2 -1 sec2
π 4 1
cot θ tan θ
adj adj
opp
(OR)
tan =
(OR) cot =
1+cot2
cosec2
= - cot2 = 1 cot2 = cosec2 - 1 cosec2
sinθ cos θ cosθ sin θ
π 2
π
3π 2
2π
1
0
-1
0
√2
π 3 √3 2 1 2
0
-1
0
1
1
√3
∞
0
−∞
0
√2 1
ASTC RULE: I Quadrant
(90˚ − θ)
A→ All the trigonometric functions are positive
II Quadrant (90° + θ) (OR) (180˚ − θ)
S→ sin & cosec only positive, remains are negative
III Quadrant (180° + θ) (OR) (270˚ − θ)
T→ tan & cot only positive, remains are negative
IV Quadrant (270° + θ) (OR) (360˚ − θ)
C→ cos & sec only positive, remains are negative
T-Ratios of (90˚ ± θ) (OR) (270˚ ± θ)
sin↔cos ; tan↔cot ; cosec↔sec (T-functions Changes)
T-Ratios of (180˚ ± θ) (OR) (360˚ ± θ)
No change in trigonometric functions
SOME SPECIAL PROPERTIES OF TRIGONOMETRICAL FUNCTIONS: Even Function: Replace x =- x in f(x), then we get f(-x) = f(x) Odd Function: Replace x =- x in f(x), then we get f(-x) = -f(x)
E R K HSS –ERUMIYAMPATTI
Page 106
+2 STUDY MATERIALS
www.tnschools.co.in
S NO
Compound Angles for: sin(A±B) & cos(A±B)
1
sin(A+B) = sinA cosB + cosA sinB
2
sin(A-B) = sinA cosB – cosA sinB
3
cos(A+B) = cosA cosB – sinA sinB
4
cos(A-B) = cosA cosB + sinA sinB
5
2 sinA cosB = sin(A+B) + Sin(A-B)
6
2 cosA sinB = sin(A+B) - Sin(A-B)
7
2 cosA cosB = cos(A+B)+ Cos(A-B)
8
-2 sinA sinB = cos(A+B)- Cos(A-B)
S NO
Multiple Angles for: sin2A & cos2A
(OR)
2 sinA sinB = cos(A-B)- cos(A+B)
A
A
2
2
1
sin2A = 2SinA cosA
(OR) sinA = 2sin cos
2
cos2A = cos2A- sin2A
(OR) cosA = cos2 - sin2
3
cos2A = 2 cos2A – 1
(OR) cos2A =
4 S NO 1 2 S NO
cos2A = 1 - 2 sin2A (OR) sin2A = Multiple Angles for: sin3A & cos3A
A
A
2
2
+cos2A 2
2
−cos2A
sin3A = 3sinA – 4sin3A (OR) sin3A =
A
(OR) cosA = 2 cos2 - 1 A
(OR) cosA = 1 - 2 sin2
2
2
3sinA−sin3A 4 cos3A+3cosA
cos3A = 4cos3A – 3cosA (OR) cos3A = 4 Properties of Inverse trigonometric functions:
1
sin− x + sin− y = sin− [x√1 − y 2 + y√1 − x 2 ]
2
sin− x - sin− y = sin− [x√1 − y 2 - y√1 − x 2 ]
3
tan− x + tan− y = tan− ⌈
4
tan− x - tan− y = tan− ⌈
E R K HSS –ERUMIYAMPATTI
x+y −xy
x−y +xy
⌉
⌉ Page 107
+2 STUDY MATERIALS
www.tnschools.co.in
IMPORTANT DIFFERENTIATION & INTEGRATION FORMULA S:NO 1.
DIFFERENTIATION (
−
)= n
:
n
(x) = 1
2.
∫ ∫(
3.
(log x) =
∫
4.
(k) = 0 : where k is an constant
∫
5.
(
∫
6.
(
7.
(
8.
(
9.
(
10.
(
)= −
11.
(
)=
12.
(
13.
(
)=
14.
(
)= −
)= )= )=
20.
=
∫
: where
−1
+
−
+
=
+
=
∫
+
=
∫
( =
∫
)+ (
−
∫
=
(
+
∫
=
(
)+
∫ ∫ ∫ ∫
2
= 2
)+ )+
+ =− + = + =−
+
Integration by parts:
)=
+
(OR) )=
+v
∫
=
∫
=
Where
, ,
+
− , 2,
+
2
−
3
+
,… →successive derivative of u , 3 …→Repeated integrals of v
-
−
E R K HSS –ERUMIYAMPATTI
−∫
Bernoulli’s formula:
Quotient rule:
( )=
+c
+
=
)= −
2
( + )
+
=
)=−
+ )
+
∫
-
(
=
)=
x
(
−1
dx = log x + c
=−
2
: where
= x+ c
∫
Product rule:
19.
+c
+
+ ) dx =
∫
15. 16. 17. 18. (
dx =
∫1
−
( ) =
INTEGRATION
Page 108
+2 STUDY MATERIALS
www.tnschools.co.in
21.
(
−
22.
(
−
23.
(
−
24.
(
25.
(
−
26.
(
−
)= )=
∫√
√ −
) = −
∫√
√ − −
∫
+ −
) = ) =
√
√ −
)=
√
∫
−
=
−
+
=
−
+
−
+
=
−
=
−
−
=
√ − −
∫
+
− + −
∫
−
− −
+
−
=
+
+
+
Some important standard results of integrals ( )
[ ( )] +
=
27.
∫
28.
∫
29.
∫
30.
∫
31.
∫
32.
∫√
+
33.
∫√
−
34.
∫√
−
35.
∫√
2
36.
∫√
2
37.
∫√
2
38.
∫
sinbx
=
39.
∫
cosbx
=
40.
Gamma Integral: ∫
( ) ( )
= 2 [√ ( )] +
√ ( )
−
=
+
=
−
=
−
+
2
−
2
−
2
( )+c −
(
2
=
[x+√
2
+
2
]+c
=
[x+√
2
−
2
]+c
=
−
( )+c 2
= √ 2
2
= √ 2
2
+ +
=
+
−
+
)+c
−
= √ 2
∫
)+c
+ +
(
2
(OR)
(OR)
+
2
−
2
−
2
+ − +
2 2
2
+
2
]+c
[x+√
2
−
2
]+c
=
−
+
( )+c
[
−
]+ c
[
+
]+ c
−
−
[x+√
− 2
∫√
= Reduction formulae
=∫
41.
=∫
42. 43. 44.
, then , then =∫
∫
=∫
∫
=
−
− −
= = =
−
+ −3 −5 −2 −4
−
−
+
−3 −5 −2 −4
−
−2 −2
2
… .1 (when n is odd) 3
…
2 2
(when n is even)
Relationship Between Exponential & Logarithmic Functions
45. 46.
o x
e =x loge = 1
E R K HSS –ERUMIYAMPATTI
e =1 log1 = 0
e =∞ log∞ = ∞ Page 109
e− = 0 log0 = -∞ +2 STUDY MATERIALS
www.tnschools.co.in
COME BOOK PUBLIC ONE WORD (MAR/JUN/SEP: 2006-14) 1. 2. 3.
4.
5. 6.
7.
8.
9.
10.
11.
1. MATRICES AND DETERMINANTS (AT)-1 is equal to 1) A-1 2) AT 3) A 4) (A-1)T Which of the following is not elementary transformation? 2) Ri 2Ri+Rj 3) Ci Cj+Ci 4) Ri Ri+Cj 1) Ri Rj In echelon form, which of the following is incorrect? 1. Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. 2. The first non-zero entry in each non-zero row is 1. 3. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row. 4. Two rows can have same number of zeros before the first non-zero entry In the system of 3 linear equations with three unknowns, if 0 and one of x , y , or z is non-zero then the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. Every homogeneous system(linear) 1. is always consistent 2. has only trivial solution 3. has infinitely many solution 4. need not be consistent If ( A ) A , B then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent If ( A ) A , B = the number of unknowns then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent ( A ) A , B then the system is 1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent In the system of 3 linear equations with three unknowns, ( A ) A , B = 1, then the system 1.has unique solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent In the homogeneous system with three unknowns, ( A ) = number of unknowns then the system has 1. only trivial solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent In the system of three linear equations with three unknowns, in the non-homogeneous system ( A ) A , B = 2, then the system 1. has unique solution 2. reduces to two equations and has infinitely many solutions
E R K HSS –ERUMIYAMPATTI
Page 110
+2 STUDY MATERIALS
www.tnschools.co.in
3. reduces to a single equations and has infinitely many solutions 4. is inconsistent 12.
13.
14.
In the homogeneous system ( A ) < the number of unknowns then the system has 1. only trivial solution 2. trivial solution and infinitely many non-trivial solutions 3. only non-trivial solutions 4. no solution If (A) = r , then which of the following is correct? 1. all the minors of order r which do not vanish. 2. A has atleast one minor of r which does not vanish and all higher order minors vanish. 3. A has atleast one (r+1) order minor which vanishes. 4. all (r+1) and higher order minors should not vanish. Cramer’s rule is applicable only (with three unknowns) when 2. 0 3. 0 , x 0 4. x y z 0 1. 0 2. VECTOR ALGEBRA
1.
The angle between the vectors i j and j k is
1. 2.
2.
2
3. -
4.
2
3 3 3 3 If a , b , c are mutually perpendicular unit vectors, then a b c
4. 3 2. 9 3. 3 3 Let u , v and w be vector such that u v w 0 . If u 3 , v 4 and w 5 , then u v v w w u is 1. 3
3. 4. 5.
1. 25
2. -25
3. 5
4.
1. 0
2. 1
3. -1
4. 2
The projection of i j on z-axis is
The projection of i 2 j 2 k on 2 i j 5 k is
1.
10
10
2.
3.
30 The projection of 3 i j k on 4 i j 2 k is 9
1.
9
2. -
21
7.
The angle between the two vectors a and
2.
3. 27
if a b = a b is
3.
3
to the point B, with
4. 28
4.
6
2
The unit normal vectors to the plane 2 x y 2 z 5 are
1. 26
1
2 i
j 2k
2.
26 169
3.
3. 2
The distance from the origin to the plane r .
E R K HSS –ERUMIYAMPATTI
1
2 i j 2k
2 i j 2k 3 3 3 The length of the perpendicular from the origin to the plane r . 3 i 4 j 12 k 26 is
1. 2 i j 2 k 2.
11.
81
4. -
21
2. 26
4
10.
81
21 The work done in a moving particle from the point A with position vector 2 i 6 j 7 k position vector 3 i j 5 k by a force F i 3 j k is
1. 9.
3.
30
21
1. 25 8.
10
4.
3
30
6.
1
5
4.
4.
2 i j 5k
Page 111
7 is
1
1 2
+2 STUDY MATERIALS
www.tnschools.co.in
7
1.
30
2.
7
30
12.
13. 14.
Chord AB is a diameter of the sphere
16.
17.
7 r 2i j 6k
30
18 with coordinate of A as (3, 2, -2). Then the
3. (-1, 0, 10)
1. (2, -1, 4) and 5
2. (2, 1, 4) and 5
3. (-2, 1, 4) and 6
4. (1, 0, -10)
The centre and radius of the sphere r ( 2 i j 4 k ) 5 are
4. (2, 1, -4) and 5
The centre and radius of the sphere 2 r ( 3 i j 4 k ) 4 are
3
2
, 2 ,4 2
1
,
3
2
2.
,
, 2 and 2 2
1
3
2
3.
,
, 2 ,6 2
1
4.
3
,
, 2 and 5 2
1
2 The vector equation of a plane passing through a point whose position vector is a and perpendicular to a vector n is 1. r n a n 2. r n a n 3. r n a n 4. r n a n The non-parametric vector equation of a plane passing through a point whose position vector is a and parallel to u and v is 1. r a , u , v 0 2. r u v 0 3. r a u v 0 4. a u v 0
The non-parametric vector equation of a plane passing through the points whose position vectors are a . b and parallel to v is
a
b a
b
v
v 0
r
b a v 0 a b 0
a
a b 0 b c 0
2. r
0 3. 4. The non-parametric vector equation of a plane passing through three non-collinear points whose position vectors are a , b , c is
r
b a
1. r a 19.
7
4.
2. (-1, 0, -10)
1. r a 18.
30
coordinates of B is 1. (1, 0, 10)
1.
15.
3.
b
c
ca 0
2. r
0 3. 4. The vector equation of a plane passing through the line of intersection of the planes r . n 1 q 1 and r . n 2 q 2 is
q 1 r . n2 q2 0 3. r n 1 r n 2 q1 q 2
1. r . n
2. r . n 1 r . n 2 q1 q
1
2
20.
4. r n 1 r n 2 q1 q 2 The vector equation of a plane whose distance from the origin is p and perpendicular to a unit vector nˆ is 1. r . n p 2. r . nˆ q 3. r n p 4. r . nˆ p
21.
The angle between the line r a t b and the plane r n q is connected by the relation
1. cos
a n q
1. 2.
3.
a b 3. sin n
b n 2. cos b n
b n 4. sin b n
3. COMPLEX NUMBERS If a ib ( 8 6 i ) ( 2 i 7 ) then the values of ‘a’ and ‘b’ are 1. 8 , 15 2. 8 , 15 3. 15 , 9 4. 15 , 8 The cube roots of unity are 1. in G.P. with common ratio , 2. in G.P. with common difference 3. In A.P. with common difference 4. in A.P. with common difference th The arguments of n roots of a complex number differ by 2 3 4 1. 2. 3. 4. n
E R K HSS –ERUMIYAMPATTI
n
n
Page 112
2
2
n
+2 STUDY MATERIALS
www.tnschools.co.in
4.
5.
Which of the following statements is correct? 1. negative complex numbers exist 2. order relation does not exist in real numbers 3. order relation exist in complex numbers 4. (1 i ) > (3 2 i ) is meaningless The value of is zz 1. | z |
6.
2. | z | 2
If | z z
1
3. 2 | z | 4. 2 | z | 2
| | z z 2 | then the locus of z is
1. a circle with centre at the origin 2. a circle with centre at z 1 3. a straight line passing through the origin 4. is a perpendicular bisector of the line joining z 1 and z 2 7. 8. 9. 10.
p
The number of values of cos i sin q where p and q are non-zero integers prime to each other, is 1. p 2. q 3. p+q 4. p-q i i The value of e e is 1. sin 2. 2 sin 3. i sin 4. 2 i sin Polynomial equation P(x)=0 admits conjugate pairs of imaginary roots only if the coefficients are 1. imaginary 2. complex 3. real 4. either real or complex Which of the following is incorrect? 2. Im ( z ) | z | 3. z z | z | 4. Re ( z ) | z | 1. Re ( z ) | z | Which of the following is incorrect regarding nth roots of unity? 1. The number of distinct roots is n 2 2. the roots are in G.P. with common ratio cis 2
11.
n
3. the arguments are in A.P. with common difference
2
.
n 4. product of the roots is 0 and the sum of the roots is 1 .
12.
Which of the following are true? n 1. If n is a positive integer then cos i sin cos n i sin n 2. If n is a negative integer then cos i sin cos n i sin n n
3. If n is a fraction then cos n i sin n is one of the values of ( cos n i sin ) n . 4. If n is a negative integer then cos i sin cos n i sin n 1. (i), (ii), (iii), (iv) 2. (i), (iii), (iv) 3. (i), (iv) The principal value of arg z lies in the interval n
13.
1. 0 ,
14.
15.
2
2. ( , ]
Which of the following is incorrect? 1. | z 1 z 2 | | z 1 | | z 2 |
0 ,
3.
4. ( , 0 ]
2. | z 1 z 2 | | z 1 | | z 2 |
3. | z 1 z 2 | | z 1 | | z 2 | 4. | z Which of the following is incorrect? 1. z is the mirror image of z on the real axis
1
z
2
| | z1 | | z 2 |
3. –z is the point symmetrical to z about the origin 16.
4. (i) only
2. The polar form of z is r , 4. The polar form of -z is r ,
Which of the following are correct? a. Re ( z ) z
b. I m ( z ) z
c. z z
n n d. z z
1. (a), (b)
2. (b), (c)
3. (b), (c) and (d)
4. (a), (c) and (d)
E R K HSS –ERUMIYAMPATTI
Page 113
+2 STUDY MATERIALS
www.tnschools.co.in
17.
If Z = 0, then the arg(Z) is 2.
1. 0
3.
4. Indeterminate
2
4. ANALYTICAL GEOMETRY 1.
x
The equation of the major and minor axes of
2
y
9
1. x 3 , y 2 2.
2. x 3, y 2
3, y 2
3. x 3 , y 2
4. y 0 , x 0
The length of the major and minor axes of 4 x 3 y 12 are 2
2. 2 ,
3. 2 3 , 4
2. x
7
The foci of the ellipse
x
2
4
1. 5 , 0
2
3
x
The equation of the latus rectum of 1. y
6.
4. y 0 , x 0
2
2. x 0 , y 0
2
16
5.
3. x 0 , y 0
The equation of the major and minor axes of 4 x 3 y 12 are
1. 4 , 2 3 4.
1 are
4 2
1. x 3.
2
y
y
4.
3, 2
2
1 are
9
3. x 7
7
4. y 7
2
1 are
9
2. 0 ,
5
3. 0 , 5
4.
5, 0
The equations of transverse and conjugate axes of the hyperbola 144 x 2 25 y 2 3600 are 1. y 0 ; x 0
2. x 12 ; y 5
3. x 0 ; y 0
4. x 5 ; y 12
7.
The equation of chord of contact of tangents from the point (-3, 1) to the parabola y 2 8 x is 2. 4 x y 12 0 3. 4 y x 12 0 4. 4 y x 12 0 1. 4 x y 12 0
8.
The equation of chord of contact of tangents from the point (5, 3) to the hyperbola 4 x 2 6 y 2 24 is 1. 9 x 10 y 12 0 2. 10 x 9 y 12 0 3. 9 x 10 y 12 0 4. 10 x 9 y 12 0
9.
The point of contact of the tangent y m x c and the parabola y 2 4 ax is a
1.
m
10.
2a m
2a
2.
m
2
,
a m
a
3.
,
m
a
2a 2 m
4.
m
The point of contact of the tangent y m x c and the ellipse b2
1.
c
11.
,
2
,
a 2m
2 a m c
2.
c
,
2 b c
a 2m
3.
c
,
x
2
a
2
y
2
b
2
2
,
2a m
1 is a 2m
2 b c
4.
,
c
2 b c
If ' t 1 ' , ' t 2 ' are the extremities of any focal chord of a parabola y 2 4 ax then ; t 1 t 2 is 1. 1
2. 0
3. 1
4.
1 2
12.
1. t 2 13.
14.
2
is The normal at ' t 1 ' on the parabola y 2 4 ax meets the parabola at ' t 2 ' then t 1 t 1
2. t 2
3. t 1 t 2
4.
1 t2
The chord of contact of tangents from any point on the directrix of the hyperbola
x
2
a
2
its 1. vertex 2. focus 3. directrix 4. latus rectum The point of intersection of tangents at ' t 1 ' and ' t 2 ' to the parabola y 2 4 ax is
E R K HSS –ERUMIYAMPATTI
Page 114
y
2
b
2
1 passes through
+2 STUDY MATERIALS
www.tnschools.co.in
1. ( a ( t 1 t 2 ), at 1 t 2 ) 15. 16.
17.
3. ( at 2 , 2 at )
4. ( a t1 t 2 , a ( t1 t 2 ))
If the normal to the R.H. xy c 2 at ' t 1 ' meets the curve aagain at ' t 2 ' then t 1 t 2 1. 1 2. 0 3. -1 4. -2 The locus of the foot of perpendicular from the focus on any tangent to the parabola y 2 4 ax is 3
1. x 2 y 2 a 2 b 2
2. x 2 y 2 a 2
3. x 2 y 2 a 2 b 2
4. x 0
The condition that the line lx my n 0 1. al 3.
18.
2. ( a t1 t 2 , a ( t1 t 2 ))
2 alm
3
a
2
l
2
b
2
m
2
2
may be a normal to the ellipse 2.
m n 0 2
(a
b )
2
2
n
2
4.
2
a
2
l
2
a
2
l
2
b
2
m
2
b
2
m
2
(a
2
(a
2
2
a
2
b ) 2
y
2
b
2
1 is
2
2
n
x
b ) 2
n
2
2
The condition that the line lx my n 0 may be a normal to the parabola y 2 4 ax is 1. al 3 2 alm 3.
a
2
l
2
b
2
m
2
2
2.
m n 0 2
(a
2
b ) 2
n
2
4.
2
a
2
l
2
a
2
l
2
b
2
m
2
b
2
m
2
(a
2
b ) 2
2
n
(a
2
2
b ) 2
n
2
2
19.
The locus of the point of intersection of perpendicular tangents to the parabola y 2 4 ax is 1. latus rectum 2. directrix 3. tangent at the vertex 4. axis of the parabola
20.
The locus of the point of intersection of perpendicular tangents to the ellipse 1. x 2 y 2 a 2 b 2
1.
2.
2. x 2 y 2 a 2
l ' H oˆ pital ' s rule cannot be applied to
lim x 0
4. 5.
2
a
2
y
2
b
2
1 is
3. x 2 y 2 a 2 b 2
4. x 0
5. DIFFERENTIAL CALCULUS- APPLICATIONS-I The law of the mean can also be put in the form 0 1 1. f ( a h ) f ( a ) hf ' ( a h ), 0 1 2. f ( a h ) f ( a ) hf ' ( a h ), 0 1 3. f ( a h ) f ( a ) hf ' ( a h ), 4. f ( a h ) f ( a ) hf ' ( a h ), 0 1 x 1 x3
1. not continuous 3. not in the indeterminate form as x 0 3.
x
x tan x
as x 0 because f(x) = x+1 and g(x) = x+3are 2. not differentiable 4. in the indeterminate form as x 0
is
1. 1 2. -1 3. 0 4. f is differentiable function defined on an interval I with positive derivative. Then f is 1. increasing on I 2. decreasing on I 3. strictly increasing on I 4. strictly decreasing on I If f has a local extremum at a and if f ' ( a ) exists then 1. f ' ( a ) <0 2. f ' ( a ) >0 3. f ' ( a ) =0 4. f " ( a ) =0
6.
x x 0 is a root of even order for the equation f ' ( x ) = 0, then x x 0 is a
7.
1. maximum point 2. minimum point 3. inflection point 4. critical point The statement: “ If f has a local extremum (minimum or maximum) at c and if f ' ( c ) exists then f ' ( c ) 0 ” is
E R K HSS –ERUMIYAMPATTI
Page 115
+2 STUDY MATERIALS
www.tnschools.co.in
1. the extreme value theorem 2. Fermat’s theorem 8.
9.
10.
11.
3. Law of mean 4. Rolle’s theorem
Identify the false statement: 1. all the stationary numbers are critical numbers 2. at the stationary point the first derivative is zero 3. at critical numbers the first derivative need not exist 4. all the critical numbers are stationary numbers Identify the correct statements: a. Every constant function is an increasing function b. Every constant function is a decreasing function c. Every identity function is an increasing function d. Every identity function is a decreasing function 1. a, b and c 2. a and c 3. c and d Which of the following statement is incorrect? 1. Initial velocity means velocity at t = 0 2. Initial acceleration means acceleration at t = 0 3. If the motion is upward, at the maximum height, the velocity is not zero 4. If the motion is horizontal, v = 0 when the particle comes to rest. Which of the following statements are correct ( m 1 and m 2 are slopes of two lines)
4. a, c and d
a. If the two lines are perpendicular then m 1 m 2 = -1 b. If m 1 m 2 = -1, then two lines are perpendicular c. If m 1 = m 2 , then the two lines are parallel d. If m 1 12.
13. 14.
15.
1.
1
then the two lines are perpendicular
m2
1. b, c and d 2. a, b and d 3. c and b 4. a and b The statement “If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c ) and an absolute minimum value f ( d ) at some number c and d in [a, b] ” is 1. The extreme value theorem 2. Fermat’s theorem 3. Law of mean 4. Rolle’s theorem The point that separates the convex part of a continuous curve from the concave part is 1. the maximum point 2. the minimum point 3. the inflection point 4. critical point Identify the correct statement: a. a continuous function has local maximum then it has absolute maximum b. a continuous function has local minimum then it has absolute minimum c. a continuous function has absolute maximum then it has local maximum d. a continuous function has absolute minimum then it has local minimum 1. a and b 2. a and c 3. c and d 4. a, c and d One of the conditions of Rolle’s theorem is 1. f is defined and continuous on (a, b) 2. f is differentiable on [a, b] 3. f ( a ) f ( b ) 4. f is differentiable on ( a , b ] 6. DIFFERENTIAL CALCULUS- APPLICATIONS-II The curve y 2 ( x 1) ( x 2 ) 2 is not defined for 1. x 1 2. x 2 3. x 2 4. x 1
2.
The curve y 2 ( x a ) ( x b ) 2 , a , b 0 and a b does not exist for 1. x a 2. x b 3. b x a 4. x a
3.
The curve y 2 ( 2 x ) x 2 ( 6 x ) exists for 1. 2 x 6 2. 2 x 6 3. 2 x 6
4.
The curve y
2
(1 x ) x
E R K HSS –ERUMIYAMPATTI
2
(1 x )
4. 2 x 6
is defined for
Page 116
+2 STUDY MATERIALS
1. 1 x 1
2. 1 x 1
3. 1 x 1
4. 1 x 1
7. INTEGRAL CALCULUS 1.
If I n sin 1
1.
n 1
3.
n 1
sin
n 1
sin
xdx , then I n
n
x cos x x cos x
n 1 n n 1 n
n
1
I n2
2.
I n2
4.
sin
n 1
x cos x
n 1
sin
n 1
n
n 1
I n2 n n 1 In x cos x n
2a
2.
f ( x ) dx
= 0 , if
0
1. f(2a-x) = f(x) 2. f(2a-x) = -f(x)
3. f(x) = -f(x)
4. f(-x) = f(x)
b
3.
f ( x ) dx
is
a b
a
1. 2 f ( x ) dx
2.
b
f ( b x ) dx
4.
a
f ( a b x ) dx
a
The arc length of the curve y = f(x) from x=a to x=b is b
1.
dy 1 dx
a
b
3. 2
a
5.
3.
a
0
4.
b
f ( a x ) dx
2
dx 1 dy
d
2.
dx
c
dy y 1 dx
2
d
4.2
dx
c
2
dx
dx y 1 dy
2
dx
The surface area obtained by revolving the area bounded by the curve y= f(x), the two ordinates x=a, x=b and xaxis, about x-axis is b
1.
dy 1 dx
a
b
3. 2
a
2
dx 1 dy
d
2.
dx
c
dy y 1 dx
2
d
4.2
dx
c
2
dx
dx y 1 dy
2
dx
6.
5
x e
4 x
dx
is
0
1.
6 4
2.
6
6 4
3.
5
5 4
4.
6
5 4
5
7.
e
mx
7
x dx
is
0
1.
m 7
8.
2.
m
m
3.
7
m 7
4.
m 1
7 m
8
The area bounded by the curve x = f(y) to the left of y-axis and between the lines y=c and y=d is d
1.
d
xdy
d
2. - xdy
c
9.
7
c
3.
d
4. - ydx
ydx
c
c
The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then the volume of the solid is d
d
1. x dy 2
d
2. x dx
c
E R K HSS –ERUMIYAMPATTI
2
c
3.
y c
Page 117
d 2
dx
4.
y
2
dy
c
+2 STUDY MATERIALS
a
10.
a
f ( x ) dx
+
f ( 2 a x ) dx =
0
0 a
1.
a
2. 2 f ( x ) dx
f ( x ) dx
0
4.
f ( x ) dx
0
f ( a x ) dx
0
= 2 f ( x ) dx , if
0
0
2a
a
f ( x ) dx
1. f(2a-x) = f(x) 12.
3.
0
2a
11.
2a
6
x e
2. f(a-x) = f(x)
3. f(x) = -f(x)
4. f(-x) = f(x)
x 2
dx
is
0
1.
6 2
2.
7
6 2
3. 2 6 6
6
4. 2 7 6
8. DIFFERENTIAL EQUATIONS 2
1.
The order and degree of the differential equation
d y dx
2
2 dy 4 dx
3 4
are
2.
1. 2, 1 2. 1, 2 3. 2, 4 4. 4,2 2 2 The order and degree of the differential equation (1 y ' ) y ' are 1. 2, 1 2. 1, 2 3. 2, 2 4. 1, 1
3.
dy d 3 y d y The order and degree of the differential equation y 2 3 dx dx dx
3 2
1. 2, 3
2. 3, 3
2 0 are
3. 3, 2
4. 2,2 2
4. 5. 6. 7.
The order and degree of the differential equation y " ( y y ' 3 ) 3 are 1. 2, 3 2. 3, 3 3. 3, 2 4. 2,2 2 2 The order and degree of the differential equation y ' ( y " ) x ( x y " ) are 1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 1 The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1. 1, 1 2. 0, 0 3. 1, 2 4. 2, 1 The differential equation corresponding to xy = c2 where c is an arbitrary constant, is 1. xy " x 0 2. y " 0 3. xy ' y 0 4. xy " x 0 2
8. 9. 10.
dx dy 2 The order and degree of the differential equation x are x dy dx
1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 3 The order and degree of the differential equation y ' ( y " ) 2 ( x y " ) 2 are 1. 1, 1 2. 1, 2 3. 2, 1 4. 2,2 In finding the differential equation corresponding to y = emx where m is the arbitrary constant, then m is 1.
y
2.
y'
1.
2.
y'
3. y’
4. y
y
9. DISCRETE MATHEMATICS Which of the following are not statements? (i) Three plus four is eight. (ii) The sun is a planet. (iii) Switch on the light. (iv) Where are you going? 1. (i) and (ii) 2. (ii) and (iii) 3. (iii) and (iv) 4. (iv) only Which of the following are statements?
E R K HSS –ERUMIYAMPATTI
Page 118
+2 STUDY MATERIALS
www.tnschools.co.in
3.
4. 5. 6. 7. 8. 9. 10. 11. 12.
13. 14. 15. 16. 1.
2. 3. 4. 5.
(i) 7+2<10. (ii) The set of rational numbers is finite. (iii) How beautiful you are! (iv) Wish you all success. 1. (iii), (iv) 2. (i), (ii) 3. (i), (iii) 4. (ii), (iv) Let p be “Kamala is going to school” and q be “There are twenty students in the class”.“Kamala is not going to school or there are twenty students in the class” stands for 2. p q 3. p 4. p q 1. p q If p is true and q is unknown then 2. p ( p ) is false 3. p ( p ) is true 4. p q is true 1. p is true If p is true and q is false then which of the following is not true? 2. p q is true 3. p q is false 4. p q is true 1. p q is false ‘+’ is not a binary operation on 1. N 2. Z 3. C 4. Q-{0} ‘-’ is a binary operation on 1. N 2. Q-{0} 3. R-{0} 4. Z ‘ ’ is a binary operation on 1. N 2. R 3. Z 4. C-{0} In congruence modulo5, { x z / x 5 k 2 , k z } represents 1. 0 2. 5 3. 7 4. 2 . In the group (G, .), G = {1, -1, i, -i} the order of -1 is 1. -1 2. 1 3. 2 4. 0 In the group ( Z 4 , 4 ) , order of [3] is 1.4 2. 3 3. 2 4. 1 In S , , x y x , x , y S then ‘ ’ is 1. only associative 2. only commutative 3. associative and commutative 4. neither associative nor commutative In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1. only closed 2. only semi group 3. only monoid 4. a group In the group ( Z 5 0 , 5 ) , O [ 4 ] is 1. 5 2. 3 3. 4 4. 2 In the group ( Z 5 0 , 5 ) , O [ 2 ] is 1. 5 2. 3 3. 4 4. 2 In the group (G, .), G = {1, -1, i, -i} the order of i is 1. 2 2. 0 3. 4 4. 3 10. PROBABILITY DISTRIBUTIONS A discrete random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values If X is a discrete random variable then P(X a) = 1. P(X
E R K HSS –ERUMIYAMPATTI
Page 119
+2 STUDY MATERIALS
6. 7.
8.
9.
10.
1. a, b only 2. b,d only 3. a.b,c only 4. all For a standard normal distribution the mean and variance are 1. , 2 2. , 3. 0, 1 4. 1, 1 If X is a discrete random variable then which of the following is correct ? 2. F ( ) 0 and F ( ) 1 1. 0 F ( x ) 1 3. P [ X x n ] F x n F ( x n 1 ) 4. F(x) is a constant function If X is a continuous random variable then which of the following is incorrect ? 1. F ' ( x ) f ( x ) 2. F ( ) 1 and F ( ) 0 3. P [ a x b ] F ( b ) F ( a ) 4. P [ a x b ] F ( b ) F ( a ) Which of the following are correct? (i) E(aX+b) = aE(X)+b (ii) 2 2 ' ( 1 ' ) 2 (iii) 2 var iance (iv) var(aX+b) = a2var(X) 1. all 2. (i), (ii), (iii) 3. (ii), (iii) Which of the following is not true regarding the normal distribution? 1. skewness is zero. 2. mean = median = mode 3. the points of inflection are at X =
11.
12.
4. (i), (iv)
4. maximum height of the curve is
A continuous random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values If X is a continuous random variable then P(X a) = 1. P(Xa)
3. P(X>a)
1 2
4. 1-P(X a-1)
**************
E R K HSS –ERUMIYAMPATTI
Page 120
+2 STUDY MATERIALS
www.tnschools.co.in
MODEL QUESTION PAPER -I PART III- MATHEMATICS (English Version)
Time Allowed: 3 hours
Maximum Marks: 200
SECTION- A NOTE: Choose the most suitable answer from the given four alternatives 1. 2. 3.
4. 5. 6. 7. 8.
If
10.
dx
=
x y x y
then
1) 2xy + y2 + x2 = c 2) x2 + y2 - x + y = c 3) x2 + y2 - 2xy = c 4) x2 - y2 - 2xy = c The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1) 1, 1 2) 0, 0 3) 1, 2 4) 2, 1 If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 Which of the following is a tautology? 1) p q 2) pq 3) p ~p 4) p ~p The value of [3] +11( [5] +11 [6] ) is 1. [0] 2)[1] 3)[2] 4)[3] In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1)only closed 2) only semi group 3) only monoid 4) a group Given E(X+ c) = 8 and E(X-c) = 12 then the value of c is 1) -2 2) 4 3) -4 4) 2 If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is 1)
9.
dy
1
2)
2
1
3)
3
2 3
12.
13.
4)
1 4
For a Poisson distribution with parameter =0.25 the value of the 2nd moment about the origin is 1)0.25 2)0.3125 3)0.0625 4)0.025 Which of the following is not true regarding the normal distribution? 1)skewness is zero. 2)mean = median = mode 3)the points of inflection are at X =
11.
40 x 1= 40
4) maximum height of the curve is
The least possible perimeter of a rectangle of area 100 m 2 is 1) 10 2) 20 3) 40 4) 60 The law of the mean can also be put in the form 0 1 1) f ( a h ) f ( a ) hf ' ( a h ), 2) f ( a h ) f ( a ) hf ' ( a h ), 0 1 3) f ( a h ) f ( a ) hf ' ( a h ), 4) f ( a h ) f ( a ) hf ' ( a h ), If x = r cos , y = r sin , then 1) sec
2) sin u
r x
1 2
0 1 0 1
3) cos
4) cosec
3) sin x
4) 0
2
14.
If u = ysinx, then 1) cosx
x y
=
2) cosy
E R K HSS –ERUMIYAMPATTI
Page 121
+2 STUDY MATERIALS
www.tnschools.co.in /2
sin x 2 cos x / 2
dx
15.
The value of
16.
1) 0 2) 2 3) log 2 4) log 4 The area bounded by the line y = x, the x-axis, the ordinates x = 1, x = 2 is 1)
17.
3
2)
2
5
3)
2
1
4)
2
The length of the arc of the curve x2/3 + y2/3 = 4 is 1) 48 2) 24 3) 12 a
18.
is
f ( x ) dx
+
4) 96
f ( 2 a x ) dx
=
0
0 a
a
1) f ( x ) dx
2a
2) 2 f ( x ) dx
0
0
dx
2a
3) f ( x ) dx
0
The solution of
20.
1) x = cemy 2) x = ce-my 3) x = my + c The differential equation of all non-vertical lines in a plane is dy dx
dy
+ mx = 0 , where m< 0 is
2
=0
2)
d y dx
2
0
3) 1 x iy
dy dx
=m
If x2 + y 2 = 1, then the value of
22.
1) x –iy 2) 2x 3) -2iy If z represents a complex number then arg (z) + arg ( z ) is
23. 24.
25. 26. 27. 28. 29. 30.
2)
4
dx
2
m
is
3) 0
2
4)
d y
4) x + iy 4)
4
The conjugate of i13 + i14 + i15 + i16 is 1) 1 2) -1 3) 0 4) –i The cube roots of unity are 1) in G.P. with common ratio 2) in G.P. with common difference 3) In A.P. with common difference 4) in A.P. with common difference 2 The length of the latus rectum of the parabola y - 4x + 4y +8 = 0 is 1) 8 2) 6 3) 4 4) 2 The tangents at the end of any focal chord to the parabola y2 = 12x intersect on the line 1) x-3 = 0 2) x + 3 = 0 3) y + 3 = 0 4) y - 3 = 0 2 2 The eccentricity of the hyperbola 12y -4x -24x+48y-127 = 0 is 1) 4 2) 3 3) 2 4) 6 The equation of chord of contact of tangents from the point (-3, 1) to the parabola y 2 1) 4 x y 12 0 2) 4 x y 12 0 3) 4 y x 12 0 4) 4 y x 2 The slope of the tangent to the curve y = 3x + 3sinx at x = 0 is 1) 3 2)2 3) 1 4) -1 2 2 The angle between the parabolas y = x and x = y at the origin is 3
1) 2 tan 1 4
31.
1 x iy
4) x = c 2
21.
4) f ( a x ) dx
0
19.
1)
2
a
1)
7
4
2) tan 1 3
2
4)
2
2
8 x is 12 0
4
T
If A = [2 0 1], then rank of AA is 1)1 2)2
E R K HSS –ERUMIYAMPATTI
3)
3)3 Page 122
4)0 +2 STUDY MATERIALS
www.tnschools.co.in
32. 33.
If A is a matrix of order 3, then det(kA) 1) k3det (A) 2) k2det (A) 0
If A =
0
0 0
34.
35.
0
0 12 5
0
1 0
0 0
3) 0
4) 0 1
2) taking transposes 4) taking finite number of elementary transformations
If a b c 0 , a 3, b 4 , c 5 , then the angle between a and b is 1)
36.
0
2)
Equivalent matrices are obtained by 1)taking inverses 3)takingadjoints
4) det (A)
0 12 , then A is 5
0 60
1)
3) k det (A)
If
2)
6
2
3)
3 PR 2 i j k , QS i 3 j 2 k
4)
3
2
then the area of the quadrilateral PQRS is
2) 10 3
1) 5 3
5
3) 5 j k
3
4)
2
2
37.
If the magnitude of moment about the point
38.
is 8 then the value of a is 1) 1 2) 2 3) 3 4) 4 The work done by the force F i j k acting on a particle, if the particle is displaced from A (3,3,3) to the point B(4,4,4) is 1) 2 units 2) 3 units 3) 4 units 4) 7 units
39.
If a =3, b 4 and a b =9, then a b is
40.
2) 63 3) 69 1) 3 7 The distance from the origin to the plane r . 2 i j 5 k
i j
1)
7
2)
30
of a force
i aj k
3
acting through the point
30
3)
7
7 is
30 7
4) 69
4)
7 30
--------------------------------------------------------------------------------------------------------------------------------------------------------
SECTION- B NOTE: Answer any ten questions. Question No.55 is compulsory and choose nine from remaining questions. 10*6=60 41.
1
2
42.
If A= and verify that the result A (adj A) = (adj A)A = A . I2. 1 4 State and prove reversal law for inverses of matrices.
43.
With usual notations prove that
44. 45. 46.
Show that diameter of a sphere subtends a right angle at a point on the surface. P represents the variable complex number z. Find the locus of P, if Im (z2) = 4 The tangent at any point of the rectangular hyperbola xy = c2 makes intercepts a, b and the normal at the point makes intercepts p, q on the axes. Prove that ap+bq = 0. Find the equations of the tangent and normal to the curve y x 2 x 2 at the point (1, -2).
47.
a sin A
b sin B
c sin C
1
48.
Find the intervals of concavity and the points of inflection of the function f ( x ) ( x 1) 3
E R K HSS –ERUMIYAMPATTI
Page 123
+2 STUDY MATERIALS
www.tnschools.co.in
49.
If V ze ax by and z is a homogeneous function of degree n in x and y prove that x
V x
y
V y
( ax by n )V .
2
50.
Evaluate
sin 0
51.
52. 53.
54.
9
x
dx
4
i) Solve. (D2+D+1)y = 0. ii) Form the differential equations for the given function y = ax2+bx+c by eliminating arbitrary constants {a, b}. Show that (pq ) (p q ) is a tautology 20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii) Poisson distribution. [e-2 = 0.1353]. A discrete random variable X has the following probability distributions. X 0 1 2 3 4 5 6 7 8 P(X=x) a 3a 5a 7a 9a 11a 13a 15a 17a (i) Find the value of a (ii) Find P(x<3) (iii) Find P(3
55.
a) i)
Find the least positive integer n such that
1 i 1 . 1 i
5
ii)
5
1 i 3 1 i 3 1 Prove that if 1, then 2 2 3
(OR) b)
Show that (R – {0}, . ) is an infinite abelian group. Here . denotes usual multiplication
--------------------------------------------------------------------------------------------------------------------------------------------------------
SECTION- C NOTE: Answer any ten questions. Question No.70 is compulsory and choose nine from remaining questions. 10*10 =100 --------------------------------------------------------------------------------------------------------------------------------------------------------
56. 57. 58. 59. 60.
61.
Discuss the solutions of the system of equations for all values of . x+y+z = 2, 2x+y-2z =2, x+y+4z =2. Prove that cos(A-B) = cosAcosB +sinAsinB Find the vector and Cartesian equation to the plane through the point (-1, 3, 2) and perpendicular to the planes x+2y+2z=5 and 3x+y+2z=8. If and are the roots of the equation x 2 2 x 4 0 , then prove that n n i 2 n 1 sin
n 3
and
deduct 9 9 . Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground? A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him if the distance between
E R K HSS –ERUMIYAMPATTI
Page 124
+2 STUDY MATERIALS
www.tnschools.co.in
63.
the poles is 6m. Show that the line x-y+4=0 is a tangent to the ellipse x2+3y2=12. Find the co-ordinates of the point of contact. Evaluate lim ( x ) sin x
64. 65.
Use differentials to find an approximate value for the given number y = Find the area of region enclosed by y 2 = x and y = x-2.
66. 67.
Solve: D 2 2 D 1 y e 2 x cos 3 x A cup of coffee at temperature 100C is placed in a room whose temperature is 15C and it cools to 60C in 5 minutes. Find its temperature after a further interval of 5 minutes.
68.
Show that the set G of all matrices of the form
62.
x 0
3
1 . 02
4
1 . 02
x x , where x R 0 , is a group under matrix x x
multiplication. 69.
The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31 psi and standard deviation 0.2 psi. (i)What is the probability that the pressure for a randomly selected tyre (a) between 30.5 and 31.5 psi
(b) between 30 and 32 psi.
(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi 70.
[ Area table: P(0
LIFE IS GOOD FOR ONLY TWO THINGS, DISCOVERING MATHEMATICS AND TEACHING MATHEMATICS
E R K HSS –ERUMIYAMPATTI
Page 125
+2 STUDY MATERIALS
www.tnschools.co.in
MODEL QUESTION PAPER -II PART III- MATHEMATICS (English Version) Time Allowed: 3 hours
Maximum Marks: 200 PART-A
NOTE: i) All questions are compulsory. 40 x 1 =40 ii) Choose the most suitable answer from the given four alternatives and write the option code and the corresponding answer. --------------------------------------------------------------------------------------------------------------------------------------------------------
1.
2.
3. 4. 5. 6. 7. 8. 9.
The curve a 2 y 2 x 2 ( a 2 x 2 ) has 1) a loop between x = a and x = -a 2) two loops between x = -a and x = 0; x = 0 and x = a 3) two loops between x = 0 and x = a 4) no loop The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then the volume of the solid is d
d
d
d
1) x 2 dy
2) x 2 dx
3) y 2 dx
4) y 2 dy
c
c
c
c
c2
The differential equation corresponding to xy = where c is an arbitrary constant, is 1) xy " x 0 2) y " 0 3) xy ' y 0 4) xy " x 0 In congruence modulo5, { x z / x 5 k 2 , k z } represents 1) 0 2) 5 3) 7 4) 2 If X is a continuous random variable then P(X a) = 1) P(Xa) 3) P(X>a) 4) 1-P(X a-1) If the equation -2x + y + z = l ; x - 2y +z = m ; x + y -2z = n such that l+m+n = 0, then the system has 1) a non-zero unique solution 2) trivial solution 3) infinitely many solution 4) No solution 2 The line 4x + 2y = c is a tangent to the parabola y = 16 x then c is 1) -1 2) -2 3) 4 4) -4 2 2 The distance between the foci of the ellipse 9x + 5y = 180 is 1) 4 2) 6 3) 8 4) 2 2 2 The asymptotes of the hyperbola 36y - 25x + 900 = 0 are 1) y =
10.
6
x
5
12. 13.
5
3) y =
x
6
The equation of the normal to the curve 1) 3 = 27t - 80
11.
2) y =
1
36
x
25
4) y =
25
x
36
1
at the point 3 , is 3 t
2) 5 = 27 t -80
3) 3 = 27 t + 80
4)
1 t
If the volume of an expanding cube is increasing at the rate of 4cm3/sec then the rate of change of surface area when the volume of the cube is 8 cubic cm is 1) 8 cm2/sec 2) 16 cm2/sec 3) 2 cm2/sec 4) 4 cm2/sec th The order of -i in the multiplicative group of 4 roots of unity is 1) 4 2) 3 3) 2 4) 1 The curve y2(x-2) = x2(1+x) has 1) an asymptote parallel to x-axis 2) an asymptote parallel to y-axis
E R K HSS –ERUMIYAMPATTI
Page 126
+2 STUDY MATERIALS
www.tnschools.co.in
3) asymptotes parallel to both axes /2
14.
The value of
0
1)
5/3
2
0
16.
x sin
sin x cos x 1 sin x cos x
4) no asymptotes is
dx x
2)
The value of 1)
cos
x
/2
15.
cos 5/3
5/3
3) 0
4
is
dx
2) 0
2
4)
3)
1 1
2
4 4
8
4)
4
The rank of the matrix 2 2 4 is 1)1
2) 2
3)3
4)4
17.
If I is unit matrix of order n, where k≠0 is a constant, then adj(kI)= 1) kn(adj I) 2) k(adj I) 3) k2(adj I))
18.
The quadratic equation whose roots are i 7 is 1) x2 + 7 = 0 2) x2 - 7 = 0 3) x2 + x +7 = 0 If a b c 0 , a 3, b 4 , c 5 , then the angle between a and b is
19.
1) 20.
2)
6
22.
3
5
4)
3
2) y 0
4) x2 - x -7 = 0
2
3) x and y are parallel 4) x 0 (or) y 0 (or) x and y are parallel The curved surface area of a sphere of radius 5, intercepted between two parallel planes of distance 2 and 4 from the centre is 2) 40 3)10 4) 30 1) 20 The differential equation of all non-vertical lines in a plane is 1)
23.
3)
If a ( b c ) b ( c a ) c ( a b ) x y then 1) x 0
21.
2
4) kn-1(adj I)
dy dx
2
=0
2)
d y dx
2
3)
0
The degree of the differential equation c =
3 dy 1 dx 3
dy dx
2
=m
4)
d y dx
2
m
2/3
, where c is a constant is
d y dx
24.
1) 1 2) 3 3) -2 The differential equation satisfied by all the straight lines in xy-plane is 1)
25.
26.
3
dy dx
2
= a constant
2)
d y dx
2
3) y+
0
dy dx
4) 2
=0
4)
dx
Which of the following are statements? (i) May God bless you (ii) Rose is a flower (iii) Milk is white (iv) 1 is a prime number 1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) In the homogeneous system ( A ) < the number of unknowns then the system has 1) only trivial solution
E R K HSS –ERUMIYAMPATTI
Page 127
2
d y 2
y0
4) (ii) , (iii) , (iv)
+2 STUDY MATERIALS
www.tnschools.co.in
27.
2) trivial solution and infinitely many non-trivial solutions 3) only non-trivial solutions 4) no solution The non-parametric vector equation of a plane passing through the points whose position vectors are a . b and parallel to v is 1) r a 2) r 3) a b v 0 4) b a v 0 b a v 0
r
28.
29. 30. 31. 32.
34.
36.
37.
1
2)
20
18
3)
125
4
4)
25
3 10
For a Poisson distribution with parameter =0.25 the value of the moment about the origin is 1)0.25 2)0.3125 3)0.0625 4)0.025 If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is 2nd
1) 35.
b 0
If a line makes 450, 600 with positive direction of axes x and y then the angle it makes with the z-axis is 1) 300 2) 900 3) 450 4) 600 If a , b , c are non-coplanar and [ a b b c c a ] [ a b b c c a ] then [ a , b , c ] is 1) 2 2) 3 3) 1 4) 0 If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 The point of inflexion of the curve y = x 4 is at 1) x = 0 2) x = 3 3) x = 12 4) nowhere A box contains 6 red balls and 4 white balls. If 3 balls are drawn at random, the probability of getting 2 white balls without replacement is 1)
33.
a
1
2)
2
1
3)
3
If the amplitude of a complex number is
2
2
4)
3
1 4
, then the number is
1) purely imaginary 2) purely real 3) 0 4) neither real nor imaginary The equation of the plane passing through the point (2 , 1 , -1) and the line of intersection of the planes r .( i 3 j k ) 0 ; and r .( j 2 k ) 0 is 1) x + 4y -z = 0 2) x + 9y +11z = 0 3) 2x + y - z +5 =0 4) 2x -y +z = 0 If x 2 + y 2 = 1, then the value of
1 x iy 1 x iy
is
39.
1) x –iy 2) 2x 3) -2iy 4) x + iy Polynomial equation P(x) = 0 admits conjugate pairs of imaginary roots only if the coefficients are 1) imaginary 2) complex 3) real 4) either real (or) complex The condition that the line lx my n 0 may be a tangent to the rectangular hyperbola xy c 2 is
40.
1) a 2 l 2 b 2 m 2 n 2 2) am 2 ln 3) a 2 l 2 b 2 m 2 n 2 4) 4 c 2 lm n 2 If a real valued differentiable function y = f (x) defined on an open interval I is increasing then
38.
1)
dy
0
dx
E R K HSS –ERUMIYAMPATTI
2)
dy
3)
0
dx
Page 128
dy dx
0
4)
dy
0
dx
+2 STUDY MATERIALS
--------------------------------------------------------------------------------------------------------------------------------------------------------
PART-B NOTE: i) Answer any ten questions. 10 X 6 = 60 ii) Question No. 55 is compulsory and choose any nine questions from the remaining: --------------------------------------------------------------------------------------------------------------------------------------------------------
41. 42.
Prove that (AB)-1 = B-1A-1, where A and B are two non-singular matrices. Find the rank of the matrix
4 6 2
2
1
3
4
1
0
3 7 1
and a c b d , show that a d and b c parallel.
43.
(i) If a
44.
(ii) Find the angle between 2 x y z 4 and x y 2 z 4 Find the vector and Cartesian equation of the sphere on the join of the points A and B having position
45.
b c d
vectors 2 i 6 j 7 k radius of the sphere.
and 2 i 4 j 3 k respectively as a diameter. Find also the centre and
(i) Graphically prove that
z1 z 2 z 3
z1
z2
z3
cos 2 i sin 2 3 cos 3 i sin 3 3 (ii) Simplify : cos 4 i sin 4 6 cos i sin 8 46.
The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatest possible distance between the earth and the sun.
47.
Prove that sin x x tan x , x 0 ,
48.
Resistance to motion F, of a moving vehicle is given by F
2 5
100 x .
x
Determine the minimum value of
50.
resistance. A circular template has a radius of 10 cm 0 . 02 . Determine the possible error in calculating the area of the templates. Find also the percentage error. Find the area of the region bounded by x 2 36 y , y - axis , y = 2 and y = 4.
51.
(i) Construct the truth tables for the given statement p q ~ q
49.
(ii) Show that 52.
a
1 1
a
a G,
a group.
1
0 1 , 1 1
1 0 , 0 1
Show that the set G = {
0
1 } is a group under of matrix multiplication. 1
Determine whether G forms an abelian group? 53.
2 A continuous random variable X has p.d.f. f x 3 x , 0 x 1 , Find a and b such that.
54.
(i) P X a P X a and (ii) P X b 0 . 05 In a gambling game a man wins ` 10 if he gets all heads or all tails and loses ` 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.
55.
2 2 a) For what values of x and y, the numbers 3 ix y and x y 4 i are complex conjugate of each other? [OR]
b) Solve: y
x
dy
a
2
dx
E R K HSS –ERUMIYAMPATTI
Page 129
+2 STUDY MATERIALS
www.tnschools.co.in
--------------------------------------------------------------------------------------------------------------------------------------------------------
PART-C NOTE: i) Answer any ten questions. 10 X 10 = 100 ii) Question No. 70 is compulsory and choose any nine questions from the remaining. --------------------------------------------------------------------------------------------------------------------------------------------------------
56.
3 3 4 Find the adjoint of the matrix A 2 3 4 and verify the result 0 1 1 A ( adj A ) ( adj A ) A A .
57.
Find the vector and Cartesian equation of the plane containing the line x 1
and parallel to the line x 1
Show that the lines
59.
intersection. If a cos i sin and
61. 62. 63.
y 1
3
58.
60.
1
2 y 1 1
z 3
z 1
x2
y2
2
z 1
3
3
.
1
and
x2
1
y 1
z 1
2
intersect and find their point of
1
b cos i sin , then Prove that
(i) cos ( + ) = * + + (ii) sin ( - ) = * − + 2 2 The girder of a railway bridge is in the parabolic form with span 100 ft. and the highest point on the arch is 10 ft, above the bridge. Find the height of the bridge at 10 ft, to the left or right from the midpoint of the bridge. The arch of a bridge is in the shape of a semi –ellipse having a horizontal span of 40 ft and 16 ft high at the centre. How high is the arch, 9 ft from the right or left of the centre. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line x 2 y 5 0 and passes through the points (6,0) and (-3,0). Find the equations of those tangents to the circle
x
2
y
2
52
, which are parallel to the straight line
2x 3y 6 .
64.
Find the intervals of concavity and the points of inflection for the given functions x
2
a
2
y
2
b
2
y 12 x
2
2x
3
x
4
1
65.
Find the area of the region bounded by the ellipse
66.
Find the perimeter of the circle with radius a.
67.
Solve:
68.
A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient is administered 10 mg of drug at time t = 0 , which is excreted at a Rate of 3t 1 2 mg/h. (i) What is the general equation for the amount of drug in the patient at time t > 0 ? (ii) When will the patient be drug free?
69.
Show that the set G a b 2 / a , b Q
2
d y dx
2
3
dy
2y 2e
dx
E R K HSS –ERUMIYAMPATTI
3x
when x log 2 , y 0 and
x 0 , y 0
is an infinite abelian group with respect to addition.
Page 130
+2 STUDY MATERIALS
www.tnschools.co.in
70.
a) If w u
2
e
v
where u
x y
and v y log x , find
w x
and
w y
[OR] b) The mean score of 1000 students for an examination is 34 and S.D. is 16. (i) How many candidates can be expected to obtain marks between 30 and 60 assuming the normally of the distribution and (ii) Determine the limit of the marks of the central 70% of the candidates.
[ Area table : P(0
TIME IS ALWAYS RUNNING, ONCE LAPSED IS LAPSE FOREVER, YOU CANNOT RE-CAPTURE TIME. TIME IS PRECIOUS! TIME IS GOD!
E R K HSS –ERUMIYAMPATTI
Page 131
+2 STUDY MATERIALS