UP/217/1
2088
Question Booklet No ...................... .
(To be filled up by the candidate by bluel black ball-point pen)
Roll Xo. Roll ~D. flVrire the digits in words) ........................................................................................................ .
Serial No. of OMR Answer Sheet .......................................... . Day and Date ..................................................................... .
(Signature of Invigilator)
INSTRUCTIONS TO CANDIDATES (ese only blue/black ball-point pen in the space above and on both sides of the Answer Sheet) 1.
Within 10 minutes of the issue of the Question Booklet, check the Question Booklet to ensure that it contains all the pages in correct sequence and that no page/question is missing. In case of faulty Question Booklet bring it to the notice of the Superintendent/lnvigilators immediately to obtain a fresh Question Booklet.
2.
Do not bring any loose paper, written or blank, inside the Examination Hall except the Admit Card without its envelope.
3.
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4.
Write your Roll Number and Serial Number afthe Answer Sheet by pen in the space provided above.
5.
On the front page of the Answer Sheet, write by pen your Roll Number in the space provided at the top, and by darkening the circles at the bottom. Also, wherever applicable, write the Question Booklet Number and the Set Number in appropriate places.
6.
No overwriting is allowed in the entries of Roll No., Question Booklet No. and Set No. (if any) on OMR sheet and also Roll No. and OMR Sheet No. on the Question Booklet.
7.
Any change in the aforesaid entries is to be verified by the invigilator, otherwise it will be taken as unfair means.
8.
Each question in this Booklet is followed by four alternative answers. For each question, you are to record the co1Tect option on the Answer Sheet by darkening the appropriate circle in the corresponding row of the Answer Sheet, by ball-point pen as mentioned in the guidelines given on the first page of the Answer Sheet.
9.
For each question, darken only one circle on the Answer Sheet. If you darken more than one circle or darken a circle partially, the answer will be treated as incorrect.
10.
,r-..'ote that the answer once filled in ink cannot be changed. If you do not wish to attempt a question, leave all the circles in the corresponding row blank (such question will be awarded zero mark).
11.
For rough work, use the inner back page of the title cover and the blank page at the end of this Booklet.
12.
Deposit only the OMR Answer Sheet at the end of the Test.
13.
You are not permitted to leave the Examination Hall until the end of the Test.
14.
Ifa candidate attempts to use any form of unfair means, he/she shall be liable to such punishment as the University may determine and impose on him/her. [No. of Printed Pages: 40+2
llP/217/1 No. of QueotloDll{5lT-if "" ~ : 150 Full Markst'rlJ1f.. : 450
Time/11Iltl·: 2% Hours/l:fUZ Koter=n-z:
(I)
Attempt as many questions as you can. Each question carries 3 marks. One mark will be deducted for each incorrect answer. Zero mark will be awarded for each unattempted question.
~
full; 'l'fi (2)
1.
-mt -.it
at<.
~ -.;B ifiT ""'" 'Ii( I ~ _
ifiT2J """-'"
I ~ ~ _
at<.
3 .
ifiT JIIlliq; ~
ifiT
~ I ~ TfffiI om: iii
WIT I
If more than one alternative answers seem to be approximate to the correct answer, choose the closest one.
Let a relation R be defmed over the set of rational numbers Q by aRb if a> b. Then this relation R is
(1) reflexive, but not symmetric and transitive (2) symmetric, but not reflexive and transitive (3) transitive, but not reflexive and symmetric (4) not transitive, but reflexive and symmetric
(254)
1
(P.T.D.)
llP/217/1
ill; W
1lFI1
to
(1) ~
~
'!fu "ffi;moi\ it "!!"f'I
wrllrn om:
(2) wrllrn~. ~ ~ (3)
'It!
~
~o;'" ..
'It!
~
om
~ ~
"!!R
a> b.
~o;'" .. ~. ~ ~
(4) ~o;lq ..
2.
om:
~o;lq..
Q '" aRb
'It!
Which of the
t.
~
om:wrllrn 'It! t ~ om: wrllrn ~
follo~g
is not an equivalence relation?
(1) The relation R defined on N x N by (a, b) R (e. d) if a+ d
~
b +c
(2) The relation of 'brotherhood' over the set of men
(3) The relation R defIned over the set of non-zero rational numbers by aRb if ab = : (4) The relation R defined over the set of integers by aRb if a - b is divisible
~ '" iBllild
11 it 'Iii'! ~"tl"d I W
(1) N x N '" (a, b) R (e. d) (2) ~
3.
"!!R
a+d ~ b+c
it "!!"f'I '" 'Iffi!l'l' ""
'!fu "ffi;moi\ it ~ '"
(4) ~
it "!!"f'I '"
"!!R
om ~ W
R
W
(3) mp!
aR b
t?
"!!R
aRb
a- b, 7
it
ab ~ 1 om ~ W
~
t om
~ W
Which is not necessarily a normal subgroup of a group G? (1) G
(2) {e}, where e is the identity element of G (3) The centre Z of G
(4) The normaliser of an element a E G (254)
2
,,:::~.-
7
llP/2l7/l
(2) {e) ,
(1) G
..m
e, G 'lit
(3) G 'lit ~ Z
4.
The number of elements in the alternating group As
(I)
5.
(2)
15
30
(3)
IS
60
(4)
120
.Let R be the additive group of real numbers and R+ be the multiplicative group of
positive real numbers. Then the mapping f: R ----) R+ given by f{x) = eX 'if x
E
R is
(1) one-one, onto, but not homomorphism (2) one-one, homomorphism, but not onto (3) onto, homomorphism, but not one-one (4) one-one, onto and homomorphism llRT f.f; R CllfdklCfl ~m CfiT ~l.tlfI'lCfl ~ x
t
(1) ~, 311""'1<;",
(2)
(3) 311""'1<;10,
(4) ~, 311""'1<;10
"'Ii' ~I
6.
"" !(x)=e
'
om
JIG'!
.nil""1
f.n.1j "q","dl -;rtf "",,,qcll, f.n.1j ~ -;rtf
~ R+ 'El"llfl'lCfl CI!fdIilCfl
i&:rran
CfiT ~O'''ilf'lCfl
!:R-->R+ ~
~,
6Ji'«\Qdl,
~
afu
"q",qdl
Let n be the order of an element a of a group G. Then which of the following elements of G has order different from n ?
(1) a P , where p is relatively prime to n (2) x-lax, where x (3)
G
a-I
(4) ax, where x (254)
E
E
G
3
(P.T. 0.)
llP/217/1
llT'!T ~
"'l!!
G i\; """'" a ""
]Ii1!
t?
n
tI
i\;
f.I"I f€I ro."
~ it ..
mm
(4) ax, ~ XEG
7.
An infinite cyclic group has exactly (1) one generator (3)
three generators
(I) 11!Ii
8.
(2)
two
(4) four generators
(2)
(3)
Which group is not Abelian? (I) A cyclic group (2) Symmetric· group So (3) A group of 4 elements (4) A group G for which (ab)' ~ a'b' 't a; bEG
'Iit.!-m "'l!! (I) ~
~ ~
t?
"'l!!
(2) ~ "'l!! So
~3) 4 ~ ""
(254)
gen~rators
4
(4) "".
]Ii1!
n i\ Nil
llP/217/1
9.
Let Z be the additive group of integers and H be the subgroup of Z obtained on multiplying each element of Z by 3. Then the number of distinct cosets ofH in Z is
am
~ "" 41'11"''' "'li' t H. Z "" ... ~ t it Z iii ~ """'" -.it 3 "i\ T" -.i-\ '" >Ill! ~ tl "" Z il Hili 1'ffi-1'ffi «~«~,",
ft
'IR1 Z
(1)
10.
11.
2
(2) 3
(3)
(4)
1
00
Let H and K be fInite subgroups of a group G. Then o{HK) is equal to
(1)
o(H)+o(K)
(2) o(H)'o(K)
(3)
o(H)·o(K) o(HnK)
(4) o(H)·o(K)-o(HnK)
How many elements of the cyclic group of order 8 can be used as generators of the group?
(1)
12.
2
(3) 4
(2) 3
(4)
1
Which statement in not correct?
(1) The polynomials over a ring form a ring (2) The polynomials over an integral domain form an integral domain (3) The polynomials over a field form a field (4) A field has no zero divisors
""" -m ..,.., _
'I1ff t?
oR • ~ {I. 1% ""'" iI'Iffi ~ 1% ~ :IIR! '" oR .~{I. 1% ~ :IIR! iI'Iffi 1% $ '" oR .~q{l. 1% $ iI'Iffi ~ 1% $ iii 'lFI ~I~" 'I1ff ~ ~
(1) 1% ""'" '" (2)
(3)
(4)
(254)
5
~
(P. T. D.}
llP/217jl
13.
Let f(x) and g(x) be two non-zero polynomials over a ring R. Then l!l'IT
f.\; f (x)
>itt
9 ( x) 1%
(1) deg (f(x) + g(x))
~
max (deg f(x). deg g(x))
(2) deg (f(x) + g(x))
~
max (deg f(x), deg g(x))
(3) deg (f(x)+ g(x))< max (deg f(x). deg g(x)) (4) deg (f(x) + g(x)) > max (deg f(x), deg g(x))
14.
If A is a square matrix of order n, then 1adj A I is equal to
(2) IAl n -'
(1) (Al n - 2
15.
(3) lAin
(4) lAin,'
If H is a subgroup of G and N is a normal subgroup of G, then H n N is a normal
subgroup of (1) H
(2)
~ f.\;
N
(3) H +N
(4) G
(3) H+N'"
(4) G'"
3'lW!i' >itt
WIT (1) H ...
16.
(2)
N'"
If)..r (r == 1, 2, "', n) are the characteristic roots of a non-singular matrix A of order n,
then the characteristic roots of adj A are
-.fu
jiil!
n ~ f.\;
~fiil" '@ (1) IAIA,
(254)
m. madj A ~
o1i\ (2) I AI
(3)
A,
1
A, 6
(4)
1 I AI A,
llP/217/1
17.
If the characteristic values of a square matrix of third order are 2, 3, 4, then the value of its determinant is
(1) 6
18.
(3) 24
(2) 9
(4)
54
Let Tl and T2 be linear operators on R2 defmed as follows: 1!FIl
'R ~ ~ T,
iln R2
a1tt T2
~ ~ ~ ,
TJia, b) =(b, a), T2 (a, b) = (a, 0)
rhen T,T2 defined by T,T2 (a, b) = Td T2 (a, b)) maps
(1)
19.
(0, I)
(2)
If the matrix · [6:
8~ 5~]
~ T,T2
o:m
iii
(~2)
into
~ (~2) '" .faill... ;Wrr
(3) (0, 2)
(1, 0)
is expressed as A + B. where A
(4)
IS
(2, 0)
symmetric and B
18
skew-symmetric, then B is equal to
[6 8
-.f?; 311"!i' 4 2
;] q;) A + B
iii
<"!
iI 3f!'l.."" WIT _, ;;roi A Wlflra a1tt B m..-Wlflra
9 7 ~,
mB
r0
(1)
(254)
l-~
0RJiR ~
-2] o2 -2 2
0
(2)
6
U ~] -2 0 -2
(3)
7
[~ ~] 2 5
6 (4)
[;
2
5
~] (P. T.O.)
llP/217/l
20.
2 -3 4 -1 a 1 5 -4 are The characteristic values of the matrix a a 3 7 a a a 4 2 -3 4 a 1 5 -1] ~ 1lF! a a 3 -:
~ 3lf'r.n~
a a (1)
2l.
1, -2, 3, -4
(2)
The rank of the matrix
~l~ (1)
22.
°
1,2,3,4
-1 1 1
~
_:]
l~
(3) -1, 2, -3, 4
(4) -1,
(3) 4
(4)
-2, -3, -4
2]
-1 1 . 4 is 1 -1
~~t
2
(2) 3
1
If T is a linear transformation from an n-dimensional vector space U to an m-dimensional vector space V. then the sum of the rank of T and the nullity of T is
equal to
'If!.
T ll,'f; n -flf$I •
~ ~ ~ ~ ~ (1) n
23.
m u 'fiT
it
ll,'f; m flf$I •
m
V
i!
ll,'f; ~ \'<4HH'1 ~,
T
-,m ""'" t (3) n +m
(2) m
(4) n-m
The rank. and nullity of T, where T is the linear transfonnation from R2 to R3 defined by T (0; b) ~ (a+ b, a-b, b) are respectively R2
it
R3 q;) T (0; b) ~ (a+ b, a-b, b) IDU ~ ~ \'<4H,(01 T ~ ~
~ (1)
(254)
1, 1
(2) 2,
a
(3) 0, 2
8
(4) 2, 1
3i'R 1!f'«I1
JI'Im,
llP/217/1
24.
For the matrix- A
=
r~ ~ ~]. A-I is equal to
II ~ =l~ ~ ~] ~ ~ ~ t 0 0
A
A-I
(2)
(I) I
25.
(2) sin (l/x 2 )
The coefficient of
I (I) - 24
If we expand cos
(I)
(254)
(4) 2A
(3) tan-I (l/x)
I
../2
(4) tanx
then the nth derivative of e""cos{bx+c) is
e~sin{bx+c+n$)
(4) e=cos{bx+c+!n$)
in the Maclaurin's expansion of log cos x is
X4
log cos x ~ ~
28.
=tan-I (bla),
If r='{a 2 +b 2 ) and $
(3)
27.
-A
Which function is continuous at x = 0 ?
(I) sin (l/x)
26.
(3)
A
=
it
X4
'fiT ~
t
I 12
I
(3) - -
(2) - -
45
(4)
I
6
(.!:4 +8) in powers of 9, the coefficient of ~ is 31
(2)
I
(3)
.J2 9
!
2
(4)
I
2 (P. T. D.}
llP/217fl
29.
The infmite series expansion of log ( 1 + x 1 is valid for (I) x>-lonly
(2) x< I only
log (1+x) "" 3FR! ~:sTaR~"
30.
i\or.r
(3)
i\or.r I xld
(I)
~ l!FI
.."
x< 1 ~ ~
3
if
~
m
q;')
~
i
1 (2) 2
I
-
2 (3) 5
(4)
'"
~
2 3
-
The angle between the radius vector arid the tangent at any point on the cardioid r=a(1+cos8) is
~
e
(I) -+2 4
~
e
(2) -+-
(3)
2 2
~
~
e
(4) -+-
2 3
2
The pedal equation of a curve is a relation between (1) P and r
am: r ~ ,1r.,.11""< ~ (I) P
(254)
i\or.r
Writing mean value theorem as f(b)-f(a)=(b-a)f'(c), a
c""
32.
(2)
(4) -i
only
(4) -i
~ ~
f(b)-f(a)=(b-a)f'(c), a
31.
1x 1<1
t
x> -I ~ ~
(I)
(3)
(2) s and 'I'
(2) s
(3) rand
am: 'I'
(3)
10
r
e
am: 8
(4) x and y
(4)
x
am:
y
lIP/217/1
33.
For any curve ds is equal to dB
(2)
rp
(3)
r/ p
34 . For any curve rdB. - IS equaJ to ds
(I)
35.
(2) sin ~
cos~
For the curve p2
'=
(3) cos IjI
(4) sin IjI
ar, the radius of curvature is
(3) 2p/a2
36.· Which curve has no asymptotes?
(3)
A B y=mx+c+-+-
(I) 2
(254)
x
x2
(3) 4
(2) 3
11
(4) 1 (P. T. D.)
llP/217/1
38.
If ex is a root of the equation f(9) =0, then an asymptote of the curve!
lIfu
39.
40.
a eiftOM'1
f
(8)
~,
q,n
! ~f r
(8) "" 1(ifi
".",,,,,ff ~
(2)
(3)
rco.(8-a)~f'(a)
(4) rcos (8 -a) ~I/f'(a)
The curve r (I)
I loop
(I)
I
=
rsin(8-a)~I/f'(a)
a sin 56 has
~
(2) 3 loops
(3) 5 loops
(4)
10 loops
(2) 3 ~
(3) 5 ~
(4)
10 ~
•2
(4)
x~-
An asymptote of the curve y
(2)
=
tan x is
•3
(3)
x~-
x~-
x 3 10g (y/x) is a homogeneous function of x and y of degree 3
x 10g (y/x), x (I) 0
(254)
"" 1(ifi '@
r.in(8-a)~f'(a)
•4
42.
=0
(I)
(I) x =-
41.
=
r
am y "" lIlI'mI '""'" t (2)
f.m
I
'lR!
(3) 2
If f(x,y)=O, ~(y,z)~O, then the value of dz is dx
lIfu
f(x,y)=O,
(I)
fAy f~z
~(y,z)~O,
dz "" 1!H
dx
~
12
~ (4) 3
3. 4
f(9) is
llP/2l7/l
43.
a2 u + ya-2 uax 2 ax ay
If u is a homogeneous function of x and y of degree n, then the value of x IS
ou
(I) (n-l)ox
44.
If x =
T
(2)
n ou
(3) (n_l)Ou
oy
ox
sin 8 cos $, y == r sin 9 sin $, z = r cos 8, then the value
~ x=rsin8coscp, y = r sin e sin 4>,
(I) sina
45.
0
ou
n-
oy
r o{x,y,z).18
°
(r,a,$)
r cos 8, cIT a (x, y, z)
(2) r sin a
a
~
(4) sin a sin $
The envelope of the family of curves Aa 2 +Ba. +C =0, where A, B, C are functions of x and y variables, is
(I) B' -AC =0
46.
Z =
(4)
(3) C' -4AB =0
(2) B'-4AC=0 x2
(4)
A'-4BC=0
y2
The evolute of the ellipse - + - 2 = 1 is a2 b
(I) {ax)'/3 +{by)'/3 =(a' _b')'/3
(2)
x'/3 +y'/3 = {a' _b')'/3
(4) (ax)'/3 +{by)'/3 =a'/3 b'/3
(254)
13
(P. T.O.)
llP/217/1
47.
The function x 3 +y3 -3axy has a maximum or minimum at the point (1) (o;a)
(2) (0, 0)
(3)
(0; 0)
(4) (0, a)
(1) (0; a)
(2) (0, 0)
(3) (0; 0)
(4) (0, a)
'R ~ 3N
48.
f.Ifu
The minimum value of x 2 +y2 +2/x+2jy is attained at
(2) (1, 1)
(1) (2,2)
49.
(3)
x
= t
3
(2)
x =t'
The value of the integral
_
r,/2
Jo
log sin x
(3)
(4)
(%, %)
(4)
~log2
x =1 12
J:/ 210g sin x dx is
ax "" l!H t ~
(3) -log 2 2
(1) --1tlog2
(254)
(~,~)
An appropriate substitution for the integral
(1)
50.
~I
14
llP/217/1
51.
If m and n are integers and n -m is odd, then the value of the integral
I:
cos mx sin nx dx is
2nm
(3) 0
52.
If In
0:::
Icotn
X
dx, then In +1n_2 is equal to
"-1
(I) _cot x n-I
53.
(2)
S;cos
The value of the integral fl14ICf1t:1
J~cos 4 X dx
31t ( I) -
(3)
8
r
(254)
16
(2)
cot n - 2 x n-I
lIR
3~
4
(4)
31t 32
xsinx dx'IS
o 1+cos 2
2
~2
(4)
x dx is
3~
x sinx dx CfiT """"" ro 1+cos x
cot n- 2 x n -I
~
CfiT lJR
The value of the integral
(I)
4
(3)
(2) -
16
54.
cot n - 1 x n-I
X
t
~2
(3)
8
15
~2
4
(4)
~2
2
(P. T. 0.)
llP/217/1
55.
lim
~.!.
n-+oo r",1 n
f(n +r) is equal to
~ n-T
~
(1) -+1
2
2
56.
(254)
~a2
(2)
""*
4
~a2
2
=
T =
i\ 1'ro
~ ~
a cos 48 is
2a cos 9 is
(2) na
~a
3l1m<
(3) ~a'
2
The perimeter of the curve r
(1)
59.
alR
The whole area of all the loops of the curve
( 1)
58.
(4) - +2
The area included between the cycloid x ::: a (8 - sin e ), y::: a ( 1- cos e) and its base is ..,..., x=a(8-sin8), y=a(1-cos8)
57.
~
~
(3) --1 2
(2) "-
(3)
4~a
(4)
8~a
The length of the arc of the catenary y =ccosh{xjc) from the vertex (0, c) to the point (x" yd is
i\
"$I
(0, c)
(1)
yf -c'
flr%. (x" yd "'" ~ (2)
y = ccosh (x/c) ~ ;m -.!it ~ ~
~Yf -c'
(3) yf+c'
16
(4)
~y~ +c 2
llP/217/1
60.
For the parabola 2a=1+cos6. the value of ds is
2
.
""""" --'" ~ 1 + cos 8 r
(1)
61.
d~
r
ds
iii fuo: -
2a
d~
(2)
sin \If
',r-d
2a
(3)
. 2 8m \jJ
2a sin 3 ,+,
(1) s=asin\ll
(2) s
~2asin ~
~
(3) s
t (4) s
4asin ~ 2
a
y-3!1\I
b
iii 'lfu\, ~ ~ ""'" im
(1) 4 ~ab2 3
(3)
'fiT
=
~6asin ~
2
The volume of the solid generated by revolving the ellipse x 2 + Y 2 a b
~ ~2 + y22 ~ 1 iii
63.
sin 3 \II
The intrinsic equation of the cycloid x=a(8+sin8), y=a{1-cos8) is "fsIi"! x~a(8+sin8). y~a(I-case) 'fiT ~ "qj
62.
a
(4)
:=
1 about the y-axis is
t
.4:~a3
(4)
3
4 ~b3
3
The volume of the solid generated by revolving about the y-axis the area bounded by the curve, the lines y = a, y == b, and the y-axis, is equal to
.... lmall
y ~ a, y ~ b.
>it<
y-3!1\I ~
(2) •
S>2 dy
flR I$i iii
y-3!1\I
iii 'lfu\,
~ ~
""'" im
'fiT
=
""'" t
64.
fax dy b
The surface area of the anchor-ring generated by the revolution of a circle of radius a about an axis in its own plane distant b from its centre (b > a) is
ilI"'II iii "'" VI iii 31
(1) 4< 2ab
(254)
(4) 2~
l(1«r.I
i! ~ ~ ~
b (b > a)
'l!l '"
~ 311\1 iii 'lfu\, ~ ~
(2) ~2ab
17
(P.T.O.)
llP/217/1
65.
r2 ry / 2y dy dx
The value of the integral Jl Jo
II}
66.
~
12}
24
1lR
qjf
12}
1 3
t
~
13} 7
S; S:
13} 3
1
4
On changing the order of integration in the integral
II}
fix, y} dx dy
if "",.."'.,
q;r .,.. ""'"
12}
fix, y} dy dx
14}
fix, y} dy dx
68.
The integral
4a
Io
14}
1
'l"( " "
f; s: f; S;
J: S: f(x,
y) dx dy. it becomes
it "'"" ~
fix, y} dydx fix, y} dx dy
S2<,=, dx dy represents the area of the region enclosed by :2
x /4a
{Il the parabola y2
==
4ax and the lines y =0, x == 4a
(2) the p~abola y2
=
4ay and the lines x
13} the parabolas y 2
= 40x
and x 2
=
4ay .
14} the lines x=O, x=4a, y=o, y=4a 1254}
1 6
(e-Y/y) dx dy is
2
,,",""" f; r f; f: 13} f; f;
14}
6
12
The value of the integral
II}
67.
2
r / y dy dx .hr2 Jo y
€I'1l
is
18
=
0, y
=
4a
llP/217/1
""'ail y ~ O. y2 ~ 4ay ai'I< ""'ail x ~ O.
(I) ~ y2 ~ 4ax
(2) ~ (3) q{q~<:(l (4)
69.
""'ail
y2 =
$
4ax
l!H
q;J
4ay GTU
t (2)
15,[,; 8
3n 4
(3)
(4)
15n 8
The value of integral J~ x m- 1 (l~xt-l dx (m > 0, n > 0) is tI'lICfliil
J~ x m - 1 (1-xt- 1 dx(m>O,n>Oj
(I) r(m)+r(n)
x
fHv x
2:
1 1 -
(I)
0, Y
2:
CfiT 1fR
0,
ym-l zn-l
r(l)r(m)r(n) r(l+m+n)
~
0, Z 2: 0,
0, x + y +z S; 1 dx dy dz CfiT lfR 'irTTT
Z ~
t (4) r(m)r(n) r(m+n)
(3) r(m+n)
(2) r(m)r(n)
If V is the region given by x 2: 0, Y ffJv X I - 1 ym-l zn-l dx dy dz is
'If<:
(254)
=
Y ~ 4a ""
x~O. x~4a, y~O. y~4a ""
(I) 3,[,; 4
71.
x2
x ~ 4a ""
The value of r(~) is r(~)
70.
ai'I<
""
(2) r(l)r(m)r(n) r(l+m+n+1)
19
X
+ Y + z :<;; 1, then the value of the integral
~
(3)
"'"
I$J
r(l)r(m)r(n) r(l+m+n+2)
V
(4)
-.T.
m
'H'l'CfI~
r (I) r(m) r (n) r(l+m+n+~)
(P.T.D.)
llP/217/1
72.
The sum of B(m+L n) and B(m, n+I) is B(m+L n)
73.
aft-.:
B(m, n+I) 'IiI
~
(I) B(m, n)
(2) B(m+Ln+I)
(3) B(2m+L2n+I)
(4) 2B(m, n)
The order of the differential equation
{I+(~)
2}3/2 =p
~;
1S
{I+(~) (I) 1 74.
(2)
2}3/2
2
=p
~;
(4) 3
(3) 4
2
The number of arbitrary constants in the general solution of the differential equation
[~;r +SioX(~)' +IogX~+9Y=COSX will be
[
(I) 2 (254)
d3 )2 +sin x::: (d )3 +log xdd!x dx;
(2) 3
(3) 5
20
+9y
=
cos x
(4) 6
llP/217/1
75.
The general solution of the differential equation sec 2 x tany dx + sec 2 y tan x dy
=
0 is
~ l;'lIilCfl(U1 sec 2 xtany dx+sec 2 ytan x dy =0 CfiT ~ ~ ~ (I) tan x tany
76.
=c
(2) tan x +tany
=c
(3) tan(xy)
=c
x
(2) log(y-x)=c+---.lL y-x
x
(4) log(y-x)=c+ Y
(3) log(y-x)=c+-
x
Y
An integrating factor of the differential equation (1 +X2) dy +2xy;= cos x is dx
(I) log(l+x2)
78.
(2) l+x2
(3)
x2
(4)
x
Equations of the form dy + Py = Qyn , where P and Q are functions of x alone, can be dx reduced to the linear form by dividing by yn and putting
dy + Py dx
=
f,l",Rtflgij
(I)
(254)
tan (x/y) = c
The general solution of the differential equation x+y dy =2y is dx
(I) log(y-x)=c+-y-x
77.
(4)
I
Qy" ""'" 11; .
if "it iiI;"i\
--=V
yn-l
.Ijj,,(,n
ll§
(2)
~
-.it, ""
if
-.m P om Q i\;
I
(3)
-=V
y"
21
x 11; ~
t
"it ~,f.1ij
y"
<0<
om
t1 I
--=V
yn+l
(4)
I
--=V
yn-2
(P.T.O.)
llP/21 7 / 1
79.
The differential equation M dx + N dy exact if
=:0
0, where M and N are functions of x and y, is
(4) aM=aN ax ay
80.
If
~[aN ~
8M)
is a function of ax ay equation M dx. + N dy = 0 is M
-.fl\ ~[aN _aM) ~ M
ax
ay
yalone, say f{y), then an integrating factor of the
y "" '""'"
~.
1lRT f.I; fly)
-m.
cit
.1i10;(01
Mdx+Ndy=O ""
~lil
(1)
81.
fly)
(1)
(3) elflY)dY
px + alp. p ~ dy "" ~ OR
y2 =ax
(4)
dx
e-I fly)dy
(2)
i
y2 =2ax
(3) y2
=
4ax
(4) x 2 = 4ay
The general solution of the equation p = log (px -y), p == dy is dx , dy ~ iE'l"iiCfl(ll1 p=log{px-y), p = - CfiT 'IDll'RJ ~ t> dx
(1) c=log(cx-y)
(254)
Jfly) dy
The singular solution of the equation y = px + a/ p, p == dy is dx .1i10;(U1 Y =
82.
(2)
(2)
y=cx
(3) y=x+c
22
llP/217/1
4
83.
The general solution of the equation d y +m4y:=0 is dx 4
d
4
*'l'llCfl{U1 -..-.1!..+m4y =0
dx 4
(1) Y
84.
=
cle mxl .J2 cos (mx/J2 +c2J +C3e-mxl.J2 cos
The particular integral of the differential equation (D 2 + D - 2) Y = eX, where D denotes d . -
dx'
lS
1 x ( 1) -e 3
85.
(mx./.J'i +C4J
Putting
(2) xe"
X
=e t
and
2
denoting
.!. xe x
(3)
d dt
(4)
3
by
D,
the
1 -x -xe 3
differential
equation
x 2 d y + 7x dy + 13y = log x is transformed into dx' dx
x ~ e'
.il!iWi
(254)
m
'R
",ql'i1~d
>i'R !!:.. -.i\ D i\ ~ -.i.\ dt
€I ""'"
'R _
Wft,.""
t?
(1) (D'+6D+13)y~t
(2) (D' +6D +13) y ~ e'
(3) (D'+8D+13)y~t
(4) (D' +8D+13)y=e'
23
(P.T.O.)
llP/217/1
86.
The solution of the simultaneous equations dx - Y
;0:=
dt
841,,;{O, ~ dx _y =~ dy +x = 1,," .., dt dt
t, dy +x=l is dt
t
(Il x=c1cost+c2sint+2, y=-clsint+c2cost-t (2) X=CICOSt+C2Sint,
(3) x (4)
87.
=
Y=-CISint+C2coSt-t
cl cost+c2 sint+2,
Y = -cl sint+C2 cost
y=-clsint-t
x=c1cost+2,
2 If 2_Px+Qx2 =0, then a particular integral of d y +P dy +Qy =0 is
dx'
'!fit
2-Px+Qx' =0,
d'y dy m --+P -+Qy =0,," "'" ~ dx' dx
1
8'<1'1;;",
t
(3) Y = eX
(2) y=-
x
88.
dx
Choosing z such that dz = e - I P dx and changing the independent variable from x to z,
dx
2
the second-order linear differential equation d y +P dy +Qy =R is transformed into
dx'
dx
the equation
z 'fiT "It'! V
WIT"
d'y +P dy +Qy = R
dx'
dx ",",,,,It! ;;ni\arr?
,
(1)
-.;G 'R fi<;
if .,..m,.
"!(
.:t
dz = e - JP dx
dx
x " z
~
'R ..-;;
l!CI!
i\; ~ ~ 841 ..,0,
f.I .... ~""d if " ~
m
d Y+
dz'
Q
y=
(:)'
R
(2)
dx
d'y --+ dz'
(:)'
d'y Q R (3) - + - y = dz' dz dz
(254)
if
afn: ~
d'y (4) --+ dz'
dx 24
P
dy _ -
( : ) ' dz
P
dy
( : ) ' dz-
R (:)'
R
(:)'
841,",0,
if
llP/2I7/1
2
89.
To solve the linear differential equation d y +P dy +Qy =R by the method of variation
dx 2
dx
of parameters we need two independent solutions of the equation
d2 (I) ~+QY~O
(2) d y2 +P dy ~O dx dx
d2 d (3) ~+P ....!t+QY ~O
(4) d y2 +Q dy +Py
2
dx 2
dx 2
90.
2
dx
dx
~O
dx
The solution of the partial differential equation x (y -z) p +y (z-x) q
=
(x -y) z is
~ """'"" ~
91.
(I)
~
(3)
~(x+y+z,yz/x)~O
(x+y+z,
xyz)~O
(2) ~ (x+y+z, xy/z)~O
(4)
The complete solution of the partial differential equation p2 +q2 =n 2 is
(I)
(2) z~ax+v(n2 _a2 )·y+c
z~ax+ny+c
(4) z~V(n2 _a2 ).x+a2y+c
(3) z = nx +ay +c
92.
(254)
~(x+y+z,zx/y)~O
The complete solution of the partial differential equation z
25
=
px + qy + c..J (1 + p2 + q2) is
(2)
z~ax+by+c
(4)
z~ax+by+c/ab
(P.T.O.)
llP/2l7/l
93.
If
~l
and
«>2
are arbitrary functions, the solution of the partial differential equari::::
r-4s+t;;=O is
94.
z=$dy+2x)+X~2(y+2x)
(1) z=~dy+2x)+~2(y+2x)
(2)
(3) z=~dY+X)+~2 (y+x)
(4) z=~dY+X)+X~2(Y+x)
The solution of the partial differential equation s;;= e X + Y is
(1) z=~dX)+~2(Y)+e"
Z=~, (X)+$2 (y)+e Y
(2)
(4) z=$dX)+~2(Y)+XY
95.
The particular integral of {D 3 -2D 2D'-DD,2 +2D,3)z=e x+ y is
(2)
the
!xe x + Y
(3)
2
96.
Putting x = eU • y = elJ, and denoting
partial
differential
_2. yeX+ Y 2
~ and ~ by D and D' respectively, the equation
au
au
x 2 r_4xys+4y 2 t+6yq=x 3 y 4 is transformed into the equation
(254)
equation
(1) (D-2D')(D-2D'-1)z=e 3u + 4 ,
(2) (D_2D')(D_2D'+1)z=e 3u + 4 ,
(3) (D_D')(D_2D'_1)z=e 3U +4,
(4)
26
(D+2D')(D_2D'_1)z=e 3u + 4 ,
llP/217/1
97.
If R, 8, T, U and V are functions of x, y, Z, p and q variables, the Monge's subsidiary equations for the partial differential equation Rr+Ss+Tt+U(rt-S2)=V are R dp dy+T dqdx+U dp dq-V dx dy =0 and ~
..u
1('i V
R, S, T, U
Rr+Ss+Tt+U(rt-s')=V
x, y,
Z,
,.
~
1('i q
p
~
e41Cfi{U1
:aift
R dp dy +T dq dx+U dp dq- V dx dy =0 1('i (1) Rdy'-Sdydx+Tdx'+Udpdx+Vdqdy=O
(2) Rdy'+Sdydx+Tdx'+Udpdx+Vdqdy=O (3) R dx' -S dx dy+Tdy' +U dp dx+ V dqdy =0 (4)
98.
R dx' +S dx dy+Tdy' +U dp dx+ V dqdy =0
The solution of Brachistochrone problem is
(1) a catenary
(2) a cycloid
(3) an inverted cycloid
(4) a hyperbola
( 1) '"" i\;r.rt\
99.
(2) '"" ""'"
A necessary condition for the functional
fab F{x. y, y') dx to have an extremum for a
given function y (x) is that y (x) satisfies the equation
""""'" J: F(x, y, y') dx _
(254)
mf .....
t
-.it '"" ~ ~
ill; y(x)
f,l..,r.,ltId
'RH y(x) ,.
~,,1"{u1
27
~ ~
"Ill
"l:ffi"! -.t.!
-.it Wi!! ...t
(P,T.O,)
llP/217/1
100.
The extremal of the functional
rl
Jl/2
x 2y'2 dx subject to the conditions y (lf2)
=
1, y{1)
0:=
2
1S
mT
y (1/2) =1, y(I)=2
I (I) y =--
x
101.
( I) 1(iIi
(4) Y =_x 2 +3
(3) y=-x+3
x
tffoI
(2) a cone
(2) 1(iIi
(3) an ellipsoid
~
(4) a sphere
(4) 1(iIi
>i'r.rr
x~
(2) 21t
~
(3) 3.
~
(4)
~
411
For a particle falling under gravity in a resisting medium, if the law of resistance be mku n , the terminal velocity will be 1(iIi
l!flm>fi '""'" if ~ t >1$r ilRa ~
~ ~ (I)
(254)
I (2) y=--+3
The periodic time of a cycloidal pendulum is
(I)
103.
~
The solid of revolution which, for a given surface area, has maximum volume is
(I) a cylinder
102.
t >1$r
9 k
1(iIi q;1lJ
t
~,
-.ft\ l!flm!
it'll (2)
(~ )
'/2 (3) (
28
~)
'/2n
'/n (4)
9
(k)
mkv n
it,
llP/217/1
104.
A body, consisting of a cone and a hemisphere of radius r on the same base, rests on a rough horizontal table, the hemisphere being in contact with the table. The greatest height of the cone, so that the equilibrium may be stable, is
"'" t\ :;mm ~ r ili"'lT ~ "'" ~ "it f.ifif: "'"
~
"'"
iPft (2)
105.
'!>:
r,!2
(3) 2r
(3) X
(2)
N
~O
(4) LX + MY + NZ
~Y~Z ~L~M~N~O
L
M
N
X Y
Z
-~-+-
~0
X
Y
Z
(2) - + - + L M N
The moment of inertia of a hollow sphere of mass M and radius a about a diameter is
(2)
(254)
M~L ~
Which quantity is an invariant for any given system of forces?
(I)
107.
r
The general conditions of equilibrium of a rigid body are
(l)X~Y~Z~O
106.
(4)
~Ma'
(3)
3
29
?. Ma' 5
(4)
~ Ma' 3
(P. T. D.}
llP/217/1
108.
A uniform tetrahedron of mass M is kinetically equivalent to four particles, each of
mass M, at the vertices of the tetrahedron, and a futh particle placed at its centre of 20 inertia of mass "" 11.'" M ~,
M _ "" _
(1)
20 4M
~'Ng"iil" ~ "" " WI¥'! ~ ~ -wi! 'R W "" 'h"i't, f.RiI " """" 3'it< 11.'" ~ 'h"T i\;, -.i\ ~ ~ i\;;j: 'R WI s,31T ~ 3'it< t.m..T _ i
109.
!
Mk2
2
(4)
5
dedt
(2)
!
2
Mk2
(de) dt
2
(3)
k h
(2)
k2
(3)
h
M
5
Mk2
dedt
h
k
The kinetic energy of a body moving in two dimensions is
(1) !Mu 2 +!Mk 2 8 2
2
(2)
2
(3) !Mu 2 2 (254)
2M
The periodic time of a compound pendulum is the same as that of a simple pendulum of length
(1)
111.
5
In-the motion of a body about a fixed axis, the moment of momentum of the body about the fixed axis is
(1)
110.
(3)
(2) 3M
5
30
Mu 2 + Mk 2 82
(4)
~2 k
llP/2l7/l
112.
Forces P, Q, R act along the sides of the triangle x + y = L y - x = 1, y = 2. The magnitude of their resultant is
oooil
x-y~Ly-x~Ly~2
'lfturpfi ""
113.
'lliJnur
';P!
the
~oil if; ~ iffi P,Q,R ~
lines
tl
~
t (2) '(P' +Q' +R' -R(P+Q)}
(3) '{ P' -Q' +R' -2R(P +Q)}
(4) '{P' +Q 2 +R2 -R(P+Q)/,/2}
Let T be the tension in the string AB of length 1. If the string AB is displaced to the position A'B', where A'B'=1+8l, the work done by the tension T is
!if. 1 ~
iHT'! Ttl
om A'B' ~ 1~61, iii iHT'! T -.m 'l'" q;1'f t (1) T -01
(2) -T-51
AB qit filIfu A' B'
(3) 2T ·51
il
~ f.\;'l1 'ill'!,
(4) -2T ·51
Let T be the tension at any point P of a catenary, and To that at the lowest point C. If W is the weight of the arc CP of the catenary, the value of r 2 is
-rg
'!HT
!if. ~ if; M
foR;,
(1)
\\,'
ai'I< ;mil; 'If.! t
P 'l< iHT'! T,
"'" CP "" "" W ;it,
115.
by
(1) ,{ P' ~Q' _R' -R(P +Q),/2}
'!HT
114.
formed
~
foR;,
(2) 2W'
~ if;
t I
C 'l< iHT'! To
(4) 3W'
If a txxiy is slightly displaced from its position of equilibrium and the forces acting on it
in its displaced position are in equilibrium, the body is said to be in
{Il stable equilibrium
(2) unstable equilibrium
(3) neutral equilibrium
(4) limiting equilibrium
11.'1'
m
"""'" il >fl (I)
(254)
-...l'ft
•
qit
ow.li\
H '''' ,.... ,
H'''''.<"'IT
H'''' ,.~
m'II'fi HI"'I.~ 31
(3) ~ HI"'I.... I
q;1'f -.;f.t
-.T.\
iffi
~
(4) ~ H''''I.f1I<1
(P. T. D.)
llP/217/1
116.
The transverse component of acceleration of a particle moving in a plane is
(2) 2fli
(I) 5
117.
(2) 2fli
. uP e=-
(2)
r
(!if
-,.!
Hii
(3)
52
(4)
p
e. =uP-
(4)
r2
3l1'ffi '1ffi oft"!
>l'OR llW!
!f'lRl
~
~ ~ ~ ~
(3) a+~
r-rli 2
r _r1i 2
e. =uP
a-~
~ u, u
0i'R WRr
<"R"T a, ~ ~ I
u-u
a
+~
a+~
A particle coming from rest from infmity will reach the el!rth's surface with a velocity
(I) .J9r (254)
(4)
p
A point executes simple hannonic motion such that in two of its positions the velocities are U, u and the corresponding accelerations are cr., 13. Then the distance between the positions is
11.'> ~ ~
120.
52
The formula for angular velocity of a particle P about a point 0 is
(I)
119.
(3)
The normal component of acceleration of a particle moving in a plane is
(I) 5
118.
Hii
(2) J2gr
(3) J3gr
32
(4) 2.J9r
llP/217/1
121.
A particle is projected from the lowest point with velocity u and moves along the inside of a smooth vertical circle of radius r. The particle will make complete revolutions if the pressure at the lowest point is greater than
wre ~ it "'" _
T fI
mg
(I)
122.
(2) 2mg
'f" i\;
I ..-;; _
u i\>T it "51~ iln'lT W
~ 'I1Ii< <'1'11"11
(3) 4mg
(4) 6mg
A uniform solid cylinder is placed with its axis horizontal on a plane, whose inclination to the horizon is u. The least coefficient of friction between it and the plane, so that it may ro]2 and not slide, is
~ " a ituT 'R ~ "'" 'ffi \! ~ ~ mq; " -..i\;, ~ f.n( ~ "'"
m ,
(2) -tan ex
3
(4) -tanex
4
- -- -
3
---)
~
~
bxc
(II
(2)
~~~
[a bel _
•
"A
"";',',1\"
(3)
~~~
---)
---)
---)
~
axb
(4)
~~~
[a bel
---)
~~~
[a bel
---)
r=1 r I. then the value of div(rn r) is -+
'~x i - Y j +z k "'" r ~ [ r [,
(I) 0
~
~
cxa [a bel
If,.. =xi-yj+zk and
>rR
(254)
2
(3) -tanex
If d , b", c' be a system of vectors reciprocal to the system a, b. c. then a' is equal to
~
124.
1
1
1 2
-tan a
(1)
123.
"'"
tl
---)
iIt div (rn r)
(2) nr n -'
(3)
33
'lIT l!1'!
nrn
t (4) (n +3) rn
(P,T.D.)
llP/217/1
125.
If
~ is a scalar invariant, then 0$, are components of
(1)
ox'
a contravariant vector
(2)
(3) a contravariant tensor of order 2
(4) a covariant tensor of order 2
(1) "'" q;'~I~Il:~'l •
(2) "'" ,,"~j{l(1ll •
~
(3) JP! 2 ~ "'" q;'~I~ll:l(1ll ~ ~
126.
~
~ "'" "'l~Il:~'l ~ ~
(4) JP! 2
The equation of the cone reciprocal to ax 2 +by2 +cz 2 =0 is
(1)
127.
a covariant vector
ayz+bzx+cxy~O
How many points are there on the paraboloid ax 2 +by2 =2z the normals drawn at which pass through a given point (n, ~ y)? q""", • ., ax 2 + by2 ~ 2z 'R f4;
'Ujt ~
m
'R mO;
'!it ~ "'" ~ '!it ~ (ex, iJ, y)
"i\ ~ ~? (1) 3
128.
(2) 4
(3) 5
(4) 6
The equation of the right circular cone whose vertex is the origin, the axis is the z-axis and semi-vertical angle is ~. is 4 "<1"! ~ ~ 'lIT ,,>i\q;"I, f.mq;r ~
'l.<'IflI"J,
tWIT (254)
34
to
f.mq;r
3l", z-3l",
to
am f.mq;r
3l":,~M., "4
llP/217/1
129.
A plane passes through a fixed point ( a, h, c) and cuts the axes in A, B, C. The locus of the centre of the sphere OABC is 1('0 "!I'I1Rl 1('0
OABC
fu«! ~
(a, b, c) j\ ~
* iM: "" ~ t
130.
om
",Pit -.it
(2) -+-+- =2
ax~by+cz=1
(4) ax+by+cz=2
i!
-.mIT
t I 'l'ta
abc x y z
The director sphere of the central conicoid ax 2 + by2 + cz2
(2) x
(4)
131,
~31T A, B, C
abc x y z
(I) ---+- =1
(3)
t
==
1 is
222111 +y +z =-+-+-
abc
2
2
2
1
1
1
x +y +z =--+-+a 2 b 2 c2
which pass through the point
The generators of the hyperboloid
tacosa,bsina,O) are
( I]
(3)
(254)
x-acosa
y-bsina b sino.
=-
z ±c
(2)
x-acosn y-bsina z == -asina -b cos a ±c
(4)
acosn
=
35
x-acosa
asina
=
y-bsina -beosn
z ±c
=-
x-acosex y-bsina. z = = asina ±c bcosa
(P.T.O.)
llP/217/l
132.
The number of paraboloids confocal with a given paraboloid and passing through a given point is
(I) 2 133.
(2)
(3)
3
4
(4)
I
IfT is a linear transformation from a vector space U into a vector space V, then [R(Tl]o
is equal to
(I)
134.
(2)
N(T)
N (T)
(3) R(T)
In an inner product space V (F) "'" "<",,It
(I)
~"H"'''
m
V (F)
if (2)
l(a,~))=llallll~1I
(3) l(a,~)I=llall+II~11
135.
Oil
lIed
",...,IIl.. ~
(4) )(a,~)I
~"H"'''
m
t •
(2) Iiall
(I) 0
(3)
II == II 1311, then the value of
~ '"'"' ~ flI; Iia II =11 ~II,
211a II
(4)
If the function f(z) =u(x. y)+iu(x, y) is analytic, then
-.f<\
'!iff.! f(z)
=u(x, y) +iu(x, y)
au au au
(254)
)(a,~)I
If a, f3 be vectors of a real inner product space such that II ex (a+jl,a-~) is
-.f<\ a, ~ "'"
136.
(4) R(T)
~,~fiI ..
to m
av
au au au au
(I) . - = - , - = - ax ay ay ax
(2) - = - , - = ax ax ay ay
au au au au (3) - = - - , - = - ax ax ay ay
(4) _au ax
36
=__ au,
_au ay ay
=_au ax
m (a +jl, a -~)
Ilallll~11
llP/217/1
137.
If S is a subset of an inner product space V, then S.LU- is equal to
138.
(3) SH
(2) S"
(1) S
(4) V
The infinite series
111 1+-+-+-+··· 2 P 3P 4P is convergent if
(I!
139.
(2) p=l
p< 1
(3)
<1
(4)
P >1
A function f is defmed in [0,11 as follows: f(xl
=
f{x)
=
plq, when x is any non-zero rational number plq in its lowest terms and 0, when x is irrational or O.
Then the Riemann integral of
I
11!> '&'R 1,[0, I] iI ~ ~ J( x I =
-.itf "" [0, I] (1) 0
(254)
P
piq,
~
"'Ii! x
t :
~ «'!
oN-i
llT 0
in [ 0, I] is
iI -.itf mJ:" '!M'!
(x) = 0 "'Ii! x
t1
iI f "" ft'li ~q I.. ", t (2)
(3) -1
1
37
(4)
! (P.T.O.)
llP/217/1
140.
If
f
{ (xl
f."
then a I fIx) I dx
.
1S
=;
1 • when x is rational -1, when x is irrational
equal to
fIx) = { 1 , "I'! -1, 'ili!
(1) -(b-a)
141.
142.
(2)
(3) 0
(b-a)
b-a 2
The test for convergence of an alternating series was given by (1) Cauchy
(2) D'A1embert
(1) ~
(2)
t\-3!R'
(3) Raabe
(4) Leibnitz
(3) ""'
(4)
For an Einstein space
1
(1) Rij =-gij
(3) Rij
n
(254)
(4)
38
=-R gij n
H.41~
llPf217/l
~
143.
~
The value of curl tux u) is ~
~
cur1(uxu) 0fiT 11R ---»
(II
t -')
-+-~
uxcurl u-uxcurl u ---+
---+
---)
--» ---)
---)
---+ -')
(2) (v·V)u-(u·V) v+(div v)u-(divu) v ---?
(3)
-+-+
---)--)
---)---)
---)0
(v-V) u+(u·V) v+ uxcurl u +uxcurl v ---)
---lo
--)0
---)
(4) (v·V)u-(u·V) v
144.
The function
f defmed by f
(x, y) =) x) +) y) is
(1) not continuous at (0,0)
(2) differentiable at (0, 0)
(3) continuous but not differentiable at (0,0) (4) continuous as well as differentiable at (0,0)
om 'IItmfur,.... f
f(x,y)=)x)+)y) (I)
(0,0) 'R """
'It! t
(2) (0,0) 'R 3<.",,,011.
(3) (0,0) 'R """ ~, ~ 3<.",,,011.
145.
'It! t
(4) (0,0) 'R """
If S is the surface of the sphere x 2 +y2 +z2 ffs (oxdydz+bydzdx+czdxdy) is
~ S ~ x 2 +y2
+Z2 =1
m,
.q,,,,,,
=
aIR
i 3<.",,,011.
t
1, then the value of the integral
Ifs (oxdydz+bydzdx+czdxdy) ""
'lHWrr 4 (2) -(a+b+c)
(I) • (a+b+c)
146.
3
4 3
-1[
(a+ b + c)
4 (4) -Rube 3
The value of r (a) r(l-a) is r(a) r(l-a) "" l!H (I) sin ax
(254)
(3)
t (2) sin a
(3) 1t sin em
39
(4)
1t
sin a (P. T. D.)
llP/217/1
147.
148.
149.
The correct inequality for the modulus of the difference of two complex numbers zl and Z2 is
(I) Iz,-z21~lzd-lz21
(2)
Iz, -z21 >1 z,l +1 z21
(3) Iz,-z21';lzd-lz21
(4)
Iz,-z21~lzd
The functionf(z)~lzI2 is (1) differentiable everywhere
(2) differentiable nowhere
(3) differentiabie at the origin only
(4) differentiable at z
=
0 and z
=
i
If LIF(t)) ~f(p). then Llt"F(t)) is equal to
~ LIF(t))~f(p),
iii Llt"Flt))
d"
(I) - f ( p )
(2)
dp"
150.
IZ21
iIUOf(
~
dn-1 dp"-' f(p)
(3) (_I)"
~ flp)
(4) (_I)"
dp"
d""-',
flp)
dp
at} .
sin The value of L { - t -
L { 'i~
at} ""
l!1'I
1.
~ (4)
coc'[;)
*** 40
D/1(254)-210
1.
m ~ flrffi i\; 10 fi'r-rI: i\; 3!'i;< tt ~ <'! flI; ~ if <11fi '!l! lit"" toil< '!iTI m 'fP ~ t 1 ~fi<1o;1 ~ ~ "If.t 'R w4il
2.
'!U1;lT
3.
"'"- 'l'! "I<'I'T " fi;'!T '1'IT
'1'!'f
"" "" "it
't<"I'.,,"
3l'l'fT
5.
""'-'l'!
% ~""
...,;-...,; ",,""0;
7.
~ 1 '"
,
fit ~ ail< , "it
W'IT
firt
iR " 3l'l'fT "':lifi"io; f.!'Iffur """ 11 -.m lWT-~ 'iii """" ,,'IT ;)e 'iii
'jl! 'R
am:
~ .;t
'IT " ' "
"",-w "" ~ Jl'l11 ~-'fl!/ 'IT
airo 1('!o amo 'l'! '"
3l:!SfiiiiQ; ~o
m-w
'IiI'1'f mOl
if
'f .,."if 1
m fi;w """'"-
fi1i'Ir 0fliIw I
8I:J,hl/i" "."
4.
6.
if Rtww W
'R
<'!R 'R ~
~ ,,'IT oft
"'"
om ""'" '"
1
'f'if -.it
'1T<;1 '"' ~ 1
imi 1
"':lMio; -.i=. lWT-~ -.i=
alto "Q;qo
~ ~ if ~ .;t
3IT{o -q7f "Ho
qft ~rqf~4f
q ;)e -.i= ('lfi; ~ 11) "'IT lWT-~ if dqRfH!F1 qft ~ ~ 't I
mm ~ f.ril"" &m wnJUrn
;iRT
'R
..m """'IT "" "-'" ~ m"" '" wWT
l!RT ;;nit>n 1
8.
m-~ if """" m % 'Iiffl
9.
tl
"""" lWT % "'" %
'l" -.it ""l,"l 'lB 10.
4"'''0'' "'"
~o;f~qo; "'" ~ 'lit '1'1 J1iW; '"" fT'I'Iit ""'"" oil """"'" Wio"
"lR
t,
"'fR
~ flI; "'"
m~
"ii!T(
~ ~ "-'"
-m 'l" -.it
'R .... "'" ""'" l!RT ;;nit>n
~ 1 "-'"
31f"",
"
'f'if -.it '1Tii1
q;G 'R
"''l'lT "'"
m
~ &m 3iflI;o "'" 0l«RT ~ "IT "!!"'hill ~ 'lfi; 3i1'l M lWT '" "'" hi ~ mqlt ~ 'lit ".;t 'f'if -.it ffiffi
%> %
11.
,,"..,.q % ~
13.
'!U1;lT ""'" mil " 'lOFI '!U1;lT
14.
'lfi; ~ ",,,,off '!U1;lT if ~ mtRf '"
Wrr~'
'1T<;1 1
lWT-~
%~ %
'1'!'f "
=
..,-f\ 'jl! "'IT
aiRrl
'1m 'If.t -.it "':j'lfu ~
wWT ""'" 'I;.
m....
'jl! '"
wWT. ~ 1
Wit,
r.,,,r,,,~," &m
f.!'Iffur ~ ",,..;t. 'Wit
