ekWMy iz’u&i= lsV&III

Mathematics ¼xf.kr½ Time Allowed : 3 Hours

Max. Marks -100 General Instructions :

(i) (ii)

(iii) (iv)

lHkh iz’u vfuok;Z gSAa bl iz’ui= esa 29 iz’u gSa tks rhu [k.Mksa esa foHkkftr gS]a v] c rFkk lA [k.M ^v* esa 10 iz’u gSa ftuesa ls izR;sd ,d vad dk gSA [k.M ^c* esa 12 iz’u gSa ftuesa ls izR;sd pkj vadksa dk gSA [k.M ^l* esa 7 iz’u gSa ftuesa ls izR;sd N% vadks dk gSA [k.M ^v* esa lHkh iz’uksa dk mÙkj ,d 'kCn] ,d okD; vFkok iz’u dh vko’;drk ds vuqlkj fn;s tk ldrs gSAa iw.kZ iz’ui= esa fodYi ugha gSA fQj Hkh pkj vadksa okys 5 iz’uksa esa ,oa N% vadksa okys 4 iz’uksa esa varfje fodYi gSAa ,sls lHkh iz’uksa esa vkidks ,d gh fodYi gy djuk gSA dSydqyVs ds iz;ksx dh vuqefr ugha gSA

(v) General Instruction (i) All question are compulsory. (ii) The question paper consists of 29 questions divided into three sections A, B and C. Section A is comprised of 10 questions of one mark each, section B is comprised of 12 questions of Four marks each and section C is comprised of Six marks each. (iii) All questions in section A are to be answered in one word, one sectence or as per the exact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in 5 questions of section B and 4 questions of section C. you have to attempt only one alternates in all such questions. (v) Use of calculator is not permitted.

[k.M & v Section – A iz'u la[;k 1 ls 10 rd izR;sd iz’u 1 vad dk gSA Question number 1 to 10 carry 1 mark each. (1) leqPp; S = {1, 2,3} ij ifjHkkf"kr Qyu f : S → S

f = {(1,3),(3, 2), (2,1)} O;qRØe

Qyu

f −1

ds fy,] tks fuEu#i esa fn[kk;k x;k gS

fy[ksAa

Write the f −1 for the function f : S → S defined over a set S = {1, 2,3} shown as f = {(1,3),(3, 2), (2,1)} (2)

(3)

 1 sin −1  −   2  1 Find the value of sin −1  −   2 2 X 2 Øeokys vkO;wg A ds fy;s λ dk

eku Kkr djsa

eku Kkr djs]a tgk¡

A = 5 ,oa λ A = 20 gSA

Write the value of λ for a matrix A of order 2 X 2 where A = 5 and λ A = 20 . (4)

 1 x  1 3   y −1 =  2 −1 ] rks x + y dk      1 x  1 3  If  =  then find x + y .  y −1  2 −1

;fn

eku crk;sAa

x (5)

lkjf.kd

y

z

2x 2 y 2z x2 y 2 z 2

dk eku fy[ksAa x

y

z

Write the value of the determinant 2 x 2 y 2 z .

x2

y2

z2

π 2

(6)

eku Kkr djsa ∫ sin 3 xdx π −

2

π 2

Evaluate

∫π sin

−

3

xdx .

2 3

(7)

2

 2  lehdj.k  d y2  +  dy  + sin  dy  + 1 = 0 dk  dx   dx   dx  ¼B½ 2 ¼C ½ 1 ¼ D½

vody ¼ A½ 3

(9)

(

)

(

(10)

vifjHkkf"kr

2

 d y   dy   dy  The degree of the differential equation  2  +   + sin   + 1 = 0 is  dx   dx   dx  (A) 3 (B) 2 (C) 1 (D) Not defined

a vkSj − a ds chp cu jgs dks.k dk eku fy[ksA a

Write the angle between a and − a .


a vkSj a × b ds chp vfn’k xq.kd ds eku fy[ksA a


Write the scaler product of a and a × b . 2

(8)

3

dksfV gS

)

x+3 y −4 z +8 = = ds fy;s fnd~ dksT;k 3 2 6 x+3 y −4 z +8 Write the direction cosine of the state line = = . 3 2 6

fn;s x;s ljy js[kk

dk eku fy[ksAa

[k.M & ^c* Section – B

iz'u la[;k 11 ls 22 rd izR;sd iz’u 4 vad dk gSA Question number 11 to 22 carry 4 marks each. (11) izkd`frd la[;kvksa ds leqPp; N ij ifjHkkf"kr

f}vk/kkjh lafØ; ∗ ds fy,] tgk¡

a ∗b = a

,oa

b

ds y?kqÙke lekioR;Z gSa Kkr djsa & ¼i½ 20 ∗16 ¼ii½ D;k ∗ Øe fofues; xq.k dks /kkj.k djrk gS \ ¼iii½ D;k ∗ lkgp;Z fu;e dk ikyu djrk gS \ ¼iv½ ∗ ds fy;s lRled vo;o For the binary operation ∗ defined on the set of natural number N . Where a ∗ b = LCM of a and b , find (i) 20 ∗16 (ii) Is ∗ commutative (iii) Is ∗ associative (iv) Identify element for ∗

(12)

(13)

5 3 63 + cos −1 = tan −1 13 5 16 5 3 63 Prove that sin −1 + cos −1 = tan −1 . 13 5 16  2 −2 −4  fn;s x;s oxZ vkO;wg A =  −1 3 4   1 −2 −3

fl) djsa fd

sin −1

dks l;fer ,oa fo"k; l;fer vkO;wgksa ds ;ksx ds #i

esa iznf’kZr djsAa  2 −2 −4  Express the square matrix A =  −1 3 4  as sum of symmetric and skened symmetric matrices.  1 −2 −3 OR 3 1 2 ;fn A =   ] rks n’kkZb;s fd A − 5 A + 7 I = 0 ] − 1 2    3 1 2 If A =   , then show that A − 5 A + 7 I = 0 . − 1 2   (14)

(15)

 sin x , if x < 0 fn;s x;s Qyu f ds lR;rk dh tk¡p djs]a tgk¡ f ( x) =  x  x + 1, if x ≥ 0  sin x , if x < 0  Investigate the continuity of function f where f ( x) =  x .  x + 1, if x ≥ 0 dy dx

Kkr djsa tc

Find

;fn

x = a (cos θ + θ sin θ )

,oa

y = a (sin θ − θ cos θ )

gks

dy , if x = a (cos θ + θ sin θ ) and y = a (sin θ − θ cos θ ) . dx OR y = 3cos(log x) + 4sin(log x) gks]a rks n’kkZb;s fd x 2 y2 + xy1 + y = 0

If y = 3cos(log x) + 4sin(log x) , then show that x 2 y2 + xy1 + y = 0 . (16)

fn;s x;s i[ky; ¼i ½

y = ( x − 2) 2

ds fy;s Kkr djsa

dy dx

¼ii½ i[ky; ij fLFkr nks fcUnqvksa (2, 0), (4, 4) dks tksMu+ s okys thok dk <+ky ¼slope½ ¼iii½ dk eku] tc Li’kZ js[kk dk <+ky ¾ mi;qZDr thok dk <+ky ¼iv½ x izkIr fcUnq ds funsZ’kkad ij Li’kZ js[kk dk lehdj.k

(17)

For given Parabola y = ( x − 2) 2 , find dy (i) dx (ii) slope of the chord joining two points (2, 0) and (4, 4) . (iii) value of x , when slope of the tangent = slope of the chord (iv) equation of the tangent at the obtained point. eku Kkr dhft;s ∫ dx x( x 4 + 1)

Evaluate

dx 4 + 1)

∫ x( x

OR

eku Kkr dhft;s ∫ Evaluate (18)

∫

x+3 5 − 4 x + x2

x+3 5 − 4 x + x2

dx

dx

eku Kkr dhft;s ∫ e x  1 − 12  dx x x  1 1  Evaluate ∫ e x  − 2  dx x x  π

(19)

eku Kkr dhft;s

cos5 x dx ∫0 sin 5 x + cos5 x 2

π

cos5 x dx ∫0 sin 5 x + cos5 x 2

Evaluate (20)

(21)

(22)

fn[kkb;s fd fcUnq

A(1, 2, 7), B (2, 6, 3) ,oa C (3,10, −1) laj[ s kh gSAa Show that the points A(1, 2, 7), B (2, 6, 3) and C (3,10, −1) are collinear. OR 'kh"kZfcUnq A(1,1, 2), B(2,3,5) ,oa C (1, 5,5) ls cus f=Hkqt dk {ks=Qy Kkr djsAa Find the area of the triangle with vertices A(1,1, 2), B (2,3,5) and C (1, 5,5) . nks js[kkvksa x + 1 = y + 1 = z + 1 ,oa x − 3 = y − 5 = z − 7 ds chp U;wure nwjh Kkr djsAa 7 1 1 1 −6 −2 x +1 y +1 z +1 x −3 y −5 z −7 Find the shortest distance between two lines = = and = = . 7 −6 1 1 −2 1 ;fn If P( A) = 6 , P( B) = 5 ,oa P( A ∪ B) = 7 gksa rks Kkr djsa Find 11 11 11 (i) P ( A ∩ B ) (ii) P(A B) (iii) P( B A) (iv) P ( A − B ) OR ;fn A ,oa B nks Lora= ?kVuk;sa gksa tgk¡ P( A) = 0.3 ,oa P( B) = 0.4 gksa rks Kkr djsa

¼i ½

P( A ∩ B)

¼ii½

P( A ∪ B)

¼iii½

P(A B)

¼iv½

P( B A)

If A and B are two independent events, where P ( A) = 0.3 and P ( B ) = 0.4 , then find

¼i ½

P( A ∩ B)

¼ii½

P( A ∪ B)

¼iii½

P(A B)

¼iv½

P( B A)

[k.M & ^l* Section – C

iz'u la[;k 23 ls 29 rd izR;sd iz’u 6 vadksa dk gSA Question number 23 to 29 carry 6 marks each. (23) vkC;wgksa dk iz;ksx djds] fuEufyf[kr lehdj.k fudk; dks 3 x − 2 y + 3 z = 8, 2 x + y − z = 1, 4 x − 3 y + 2 z = 4. Using matrics solve the following system of equations ; 3 x − 2 y + 3 z = 8, 2 x + y − z = 1, 4 x − 3 y + 2 z = 4. (24) n'kkZb;s fd fn;s x;s o`Ùk esa vUrfuZfgr vk;rksa esa vf/kdre

gy dhft;s (

{ks=Qy dk vk;r ,d oxZ gSA

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. OR 2 ¼i½ vUrjky Kkr djs]a ftlesa Qyu f ( x) = x + 2 x − 5 o/kZeku Øe esa gSA

¼ii½ eku Kkr djsa ¼vodyu dk mi;ksx djrs gq,½&

25.3 (i) Find the internal, where function f ( x) = x + 2 x − 5 is in increasing order. 2

(25)

(ii) Using differentiation, find the value (approx) i[ky; x 2 = y ] ljy js[kk y = x + 2 ,oa x &

25.3 .

v{k ls f?kjs {ks= dk {ks=Qy lekdyu fof/k ls

Kkr djsAa Find the area of the region enclosed by the parabola x 2 = y , st line y = x + 2 and x - axis. OR 3

;ksx dh lhek fof/k vFkkZr izkjafHkd fof/k ls eku Kkr djsa & ∫ xdx 1 3

Using limit of sum ie. Ab-initia method evaluate -

∫ xdx 1

(26)

;fn

dy x + y = x3 ] dx

rks Kkr djsa

¼i½ vodyu lehdj.k dh dksfV ¼ii½ LFkkfir #i esa ykdj P ,oa Q ds eku ¼iii½ lekdyu xq.kkad vFkkZr e∫ Pdx ¼iv½ O;kid gy ¼v½ izkpy fLFkjkad ds eku ;fn x = 1 ,oa y = 2 gks] ¼vi½ fof’k"V gy mi;qZDr izkpy fLFkjkad ds lanHkZ es]a dy + y = x3 , then evaluate – dx (i) Order of the differential equation. (ii) P and Q after bringing it into standard from. Pdx (iii) Integrating factor ie. e ∫ .

If x

(27)

(28)

(iv) General solution. (v) Value of arbitrary constant if x = 1 and y = 2 . (vi) Particular solution. vfu;ksftr pj x ds fy;s izkf;drk caVu P( x) fuEu gSa  K , if x = 0  2 K , if x = 1  P( x) =   3K , if x = 2 0, otherwise ¼i½ K dk eku Kkr djsas (i) Determine the value of K ¼ii½ Kkr djsa P( x < 2) (ii) Find P ( x < 2) ¼iii½ P( x ≤ 2) (iii) P ( x ≤ 2) ¼iv½ P( x ≥ 2) (iv) P ( x ≥ 2) ¼v½ izkf;drk caVu lkj.kh (v) Probability Distribution Table ¼vi½ ek/; (vi) Mean. (3, −4, −5) ,oa (2, −3,1) dks tksMu + s okyh ljy js[kk ds }kjk lery 2 x + y + z = 7

okys fcUnq dk funsZ’kkad Kkr djsAa

dks cs/kus

(29)

Find the co-ordinates of the point where the line through (3, −4, −5) and (2, −3,1) crosses the plane 2x + y + z = 7 . OR lery ds lehdj.k dks Kkr djsa tks (1,1, 0), (1, 2,1) ,oa (−2, 2, −1) ls xqtjrs gksAa Find the equation of the plane passing through there points (1,1, 0), (1, 2,1) and (−2, 2, −1) . ,d mRiknd la;a= nks mRikn A ,oa B dk mRiknu djrk gSA mRikn A esa fuekZ.k Lrj ij 9 ?kaVs ,oa lqlTtkdj.k esa 1 ?kaVk rFkk mRikn B esa fuekZ.k Lrj ij 12 ?kaVk ,oa lqlTtkdj.k esa

3 ?kaVs yxrs gSAa fuekZ.k ds fy;s vf/kdre 180 ekuo ?kaVs ,oa lqlTtkdj.k ds fy;s vf/kdre 30 ?kaVs miyC/k gSAa izR;sd mRikn A ij mRiknd dks ` 8000 ,oa mRikn B ij ` 12000 feyrs gSAa mRikn A ,oa B fdruh la[;k esa izfr lIrkg cuk;h tk;sa ftlls la;a= dks vf/kdre ykHk gks ldsA vf/kd ykHk dh izfr lIrkg jkf’k Kkr djsAa A manufacturing company makes two models A and B of product. Each piece of model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing the maximum labour hours are available 180 and 30 respectively. The company makes a profit of ` 8000 on each piece of model A and ` 12000 on each piece of model B. How many pieces of model A and B should be manufactured per week to realise a maximum profit ? What is the maximum profit per week ? OR ,d vkgkj foKkuh nks izdkj ds HkksT; inkFkksZa X ,oa Y dks bl izdkj feykdj ,d feJ.k cukuk pkgrk gS ftlesa de ls de 10 bZdkbZ foVkfeu A, 12 bZdkbZ foVkfeu B ,oa 8 bZdkbZ foVkfeu C gksA ,d fdyks HkksT; inkFkZ esa foVkfeu dh ek=k bl izdkj gS & A B C Hkkstu foVkfeu X 1 2 3 Y 2 2 1 X dh dher ` 16@fdxzk ,oa Y dh dher ` 20@fdxzk gSA feJ.k Hkkstu dh U;wure dher dk

fu/kkZj.k djsa ftlesa okafNr foVkfeu dh ek=k iwjk gks ldsA A dietician wishes to mix together two kind of foods X and Y in such a way that the mixture contains at least 10 units of vitamine A, 12 unite of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below :Vitamin A B C Food X 1 2 3 Y 2 2 1 One kg of food X costs ` 16 and one kg of food Y costs ` 20. Find the least cost of the mixture which will produce the required diet.