HBSE Class 12 Mathematics Question Paper 2019

CLASS : 12th (Sr. Secondary) Code No. 4331
Series : SS-M/2019
Roll No. SET : A

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh/Re-appear Candidates)
Time allowed : 3 hours ] [ Maximum Marks : 80
• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr i`"B 16 rFkk
iz'u 20 gSaA
Please make sure that the printed pages in this
question paper are 16 in number and it contains
20 questions.
• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks
Nk= mÙkj&iqfLrdk ds eq[;&i`"B ij fy[ksaA
The Code No. and Set on the right side of the
question paper should be written by the candidate
on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad
vo'; fy[ksaA
Before beginning to answer a question, its Serial
Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMsa+A
Don’t leave blank page/pages in your answer-book.

4331/(Set : A) P. T. O.

(2) 4331/(Set : A)
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr%
vko';drkuqlkj gh fy[ksa vkSj fy[kk mÙkj u dkVsaA
Except answer-book, no extra sheet will be given.
Write to the point and do not strike the written answer.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the
question paper.
• d`i;k iz'uksa dk mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i=
iw.kZ o lgh gS] ijh{kk ds mijkUr bl lEcU/k esa dksbZ Hkh nkok
Lohdkj ugha fd;k tk;sxkA
Before answering the question, ensure that you
have been supplied the correct and complete
question paper, no claim in this regard, will be
entertained after examination.
lkekU; funsZ'k %
(i) bl iz'u-i= esa 20 iz'u gSa] tks fd pkj [k.Mksa % v] c]
l vkSj n esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ,d ç'u gS tks 16 (i-xvi) Hkkxksa
esa gS] ftuesa 6 Hkkx cgqfodYih; gSaA izR;sd
Hkkx 1 vad dk gSA
[k.M ^c* % bl [k.M esa 2 ls 11 rd dqy nl ç'u
gSaA çR;sd ç'u 2 vadksa dk gSA
[k.M ^l* % bl [k.M esa 12 ls 16 rd dqy ik¡p ç'u
gSaA çR;sd ç'u 4 vadksa dk gSA
[k.M ^n* % bl [k.M esa 17 ls 20 rd dqy pkj ç'u
gSAa çR;sd ç'u 6 vadksa dk gSA
(ii) lHkh ç'u vfuok;Z gSaA
(iii) [k.M ^n* ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSa]
muesa ls ,d gh iz'u dks pquuk gSA
4331/(Set : A)

(3) 4331/(Set : A)
(iv) fn;s x;s xzkQ-isij dks viuh mÙkj-iqfLrdk ds lkFk vo';
uRFkh djsaA
(v) xzkQ-isij ij viuh mÙkj-iqfLrdk dk Øekad vo'; fy[ksaA
(vi) dSYD;qysVj ds ç;ksx dh vuqefr ugha gSA
General Instructions :
(i) This question paper consists of 20 questions
which are divided into four Sections : A, B,
C and D :
Section 'A' : This Section consists of one
question which is divided into
16 (i-xvi) parts of which 6 parts
of multiple choice type. Each
part carries 1 mark.
Section 'B' : This Section consists of ten
questions from 2 to 11. Each
question carries 2 marks.
Section 'C' : This Section consists of five
questions from 12 to 16. Each
question carries 4 marks.
Section 'D' : This Section consists of four
questions from 17 to 20. Each
question carries 6 marks.
(ii) All questions are compulsory.
(iii) Section 'D' contains some questions where
internal choice have been provided. Choose
one of them.
(iv) You must attach the given graph-paper along
with your answer-book.
(v) You must write your Answer-book Serial No.
on the graph-paper.
(vi) Use of Calculator is not permitted.

4331/(Set : A) P. T. O.

(4) 4331/(Set : A)
[k.M – v
SECTION – A

1. (i) ;fn f : R → R rFkk g : R → R Qyu Øe'k%
f (x) = cos x vkSj g(x ) = 3x 2 }kjk ifjHkkf"kr gSa] rks
fog Kkr dhft,A 1
Find fog, if f : R → R and g : R → R are given
by f (x) = cos x and g(x ) = 3x 2 .

 1
(ii) sin−1 −  dk eku gS % 1
 2
π π
(A) − (B) −
3 6
π
(C) − (D) buesa ls dksbZ ugha
4
 1
The value of sin−1 −  is :
 2
π π
(A) − (B) −
3 6
π
(C) − (D) None of these
4
(iii) ,d ,sls 3 × 2 vkO;wg dh jpuk dhft,] ftlds vo;o
1
aij = | i − 3 j | }kjk çnÙk gSaA 1
2
Construct a 3 × 2 matrix whose elements
1
are given by aij = | i − 3 j |.
2
4331/(Set : A)

(5) 4331/(Set : A)
3 x 3 2
(iv) = ds fy, x dk eku gS % 1
x 1 4 1

(A) ±2 3 (B) ±3 3

(C) ±2 2 (D) buesa ls dksbZ ugha
3 x 3 2
The value of x for which =
x 1 4 1
is :

(A) ±2 3 (B) ±3 3

(C) ±2 2 (D) None of these

(v) ( ) dk vodyu dhft,A
x ds lkis{k sin x 2 + 5 1

Differentiate sin(x + 5) w. r. t. x.
2

(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj bldh f=T;k r ds
lkis{k tcfd r = 3 cm, gS % 1

(A) 6π cm2 /sec (B) 4π cm2 /sec

(C) 5π cm2 /sec (D) buesa ls dksbZ ugha
The rate of change of the area of a circle
with respect to its radius r when r = 3 cm
is :

(A) 6π cm2 /sec (B) 4π cm2 /sec

(C) 5π cm2 /sec (D) None of these

4331/(Set : A) P. T. O.

(6) 4331/(Set : A)
(vii) x = 2 ij oØ y = x 3 − x dh Li'kZjs[kk dh ço.krk Kkr
dhft,A 1
Find the slope of tangent to the curve
y = x 3 − x at x = 2.

(viii) ∫
(
sin tan−1 x ) dx dk eku Kkr dhft,A 1
2
1+ x

Find the value of ∫
(
sin tan−1 x ) dx .
2
1+ x
1
(ix) ∫ sin5 x cos 4 x dx dk eku gS % 1
−1

(A) 1 (B) −1
(C) 0 (D) buesa ls dksbZ ugha
1
The value of ∫ sin5 x cos 4 x dx is :
−1

(A) 1 (B) −1
(C) 0 (D) None of these

d 2y dy
(x) 2x 2 2
−3 + y = 0, vodyu lehdj.k dh
dx dx

dksfV gS % 1

(A) 2 (B) 0

(C) 1 (D) buesa ls dksbZ ugha
4331/(Set : A)

(7) 4331/(Set : A)
The order of the differential equation
2
2d y dy
2x 2
−3 + y = 0, is :
dx dx

(A) 2 (B) 0

(C) 1 (D) None of these

2
d 3y  d 2y 
(xi) + 2  − dy + y = 0, vodyu lehdj.k
dx 3  2 dx
 dx 
dh ?kkr gS % 1

(A) 3 (B) 2

(C) 1 (D) buesa ls dksbZ ugha
The degree of the differential equation
2
d 3y  d 2y  dy
+ 2 2  − + y = 0, is :
dx 3   dx
 dx 

(A) 3 (B) 2

(C) 1 (D) None of these

(xii) ;fn P (E ) = 0.6, P (F ) = 0.3 rFkk P (E ∩ F ) = 0.2,

rks P (E / F ) Kkr dhft,A 1

If P (E ) = 0.6, P (F ) = 0.3 and P (E ∩ F ) = 0.2,
then find P (E / F ) .

4331/(Set : A) P. T. O.

(8) 4331/(Set : A)
(xiii) ;fn P (A ) = 0.3, P (B ) = 0.6 rFkk A vkSj B LorU=
?kVuk,¡ gSa, rks P (A vkSj B ) dk eku Kkr dhft,A 1

If P (A ) = 0.3, P (B ) = 0.6 and A and B are
independent events, then find the value of
P (A and B ).

(xiv) ,d FkSys esa 4 lQsn vkSj 6 dkyh xsansa gSaA nks xsan
çfrLFkkiu ds lkFk ;kn`fPNd fudkyh x;h gSaA nksuksa xsan
dkyh gksus dh çkf;drk Kkr dhft,A 1

A bag contains 4 white and 6 black balls.
Two balls are drawn at random with
replacement. Find the probability that both
balls are black.

→
(xv) a = 2iˆ + 2 ˆj − 5kˆ vkSj b→ = 2iˆ + ˆj + 3kˆ ds ;ksxQy
ds vuqfn'k ek=d lfn'k Kkr dhft,A 1

Find the unit vector of the sum of the
→ →
vectors a = 2iˆ + 2 ˆj − 5kˆ and b = 2iˆ + ˆj + 3kˆ .

(xvi) ;fn ,d js[kk ds fnd~ vuqikr 2, −1, −2 gSa] rks js[kk ds
fnd~-dksT;k Kkr dhft,A 1

If direction ratio's of a line are 2, −1, −2,
then find the direction cosines of the line.

4331/(Set : A)

(9) 4331/(Set : A)
[k.M – c
SECTION – B

2. eku yhft, fd N esa ,d f}vk/kkjh lafØ;k ∗, a ∗ b = a rFkk
b dk L. C. M. }kjk ifjHkkf"kr gSA 5 ∗ 7 Kkr dhft,A 2

Let ∗ be the binary operation on N given by a ∗ b
= L. C. M. of a and b. Find 5 ∗ 7.

 3π 
3. sin −1  sin  dk eq[; eku Kkr dhft,A 2
 5 

 3π 
Find the principal value of sin −1  sin .
 5 

ekuk A = 
2 4  1 3
4. , B = , 2A + B dk eku Kkr
3 2  − 2 5
dhft,A 2

2 4  1 3
Let A =  , B= , find 2A + B.
3 2  − 2 5

;fn A = 
1 2
5. , rks |2A| = 4|A| n'kkZb,A 2
4 2
1 2
If A =  , then show that |2A| = 4|A|.
4 2

4331/(Set : A) P. T. O.

( 10 ) 4331/(Set : A)

6. lehdj.k 2x + 3y = sin y ls dy Kkr dhft,A 2
dx

dy
Find from the equation 2x + 3y = sin y.
dx

7. ;fn x = 2 at 2 , y = at 4 , rks dy Kkr dhft,A 2
dx

dy
If x = 2 at 2 , y = at 4 , then find .
dx

8. eku Kkr dhft, % 2

∫ x . sin x dx
Evaluate :

∫ x . sin x dx
9. eku Kkr dhft, % 2

3
1
∫ x dx
2

Evaluate :

3
1
∫ x dx
2

4331/(Set : A)

( 11 ) 4331/(Set : A)
x y
10. a rFkk b dks foyqIr djrs gq, oØ + = 1 dks fu:fir
a b
djus okys vodyu lehdj.k Kkr dhft,A 2

Find differential equation corresponding to
x y
+ = 1 , by eliminating a and b.
a b

11. ,d U;k¸; flDds dks 10 ckj mNkyk x;k gSA Bhd N% fpr vkus
ds fy, çkf;drk Kkr dhft,A 2

A fair coin is tossed 10 times. Find the
probability of exactly 6 heads.

[k.M – l
SECTION – C

12. fuEu dks ljyre :i esa O;Dr dhft, % 4

1 + x2 −1
tan−1 ,x ≠0
x
Write the simplest form of the following :

1 + x2 −1
tan−1 ,x ≠0
x

4331/(Set : A) P. T. O.

( 12 ) 4331/(Set : A)
13. k dk eku Kkr dhft, rkfd Qyu %

k cos x
 , ; fn x ≠ π
f (x ) =  π − 2x 2] x = π ij larr gksA 4
π 2
 3 , ; fn x =
 2

Find the value of k, so that the function :
k cos x π
 , if x ≠
f (x ) =  π − 2x 2
π
 3 , if x =
 2
π
is continuous at x = .
2

14. vUrjky Kkr dhft, ftuesa f (x) = 2 x 3 − 3 x 2 − 36 x + 7
ls çnÙk Qyu f fujUrj o/kZeku gSA 4

Find the interval in which the function

f (x) = 2 x 3 − 3 x 2 − 36 x + 7 is strictly increasing.

15. rk'k ds 52 iÙkksa dh ,d lqfefJr xM~Mh ls nks iÙks mÙkjksÙkj
çfrLFkkiuk ds lkFk fudkys tkrs gSaA bDdksa dh la[;k dk
çkf;drk caVu Kkr dhft,A 4

4331/(Set : A)

( 13 ) 4331/(Set : A)
Two cards are drawn with replacement from a
well shuffled pack of 52 cards. Write the
probability distribution of the number of aces
obtained.

→ → → →
16. lfn'k a + b vkSj a − b esa ls çR;sd ds yEcor ek=d

lfn'k Kkr dhft, tgk¡ a→ = iˆ + ˆj + kˆ , b→ = iˆ + 2 ˆj + 3kˆ gSA 4

Find a unit vector perpendicular to each vector
→ → → → →
a +b and a − b, where a = iˆ + ˆj + kˆ,

→
b = iˆ + 2 ˆj + 3kˆ .

[k.M – n

SECTION – D

17. fuEu lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6

x − y + z = 4,

2x + y − 3z = 0,

x + y + z = 2.

4331/(Set : A) P. T. O.

( 14 ) 4331/(Set : A)
Solve the following equations by a matrix method :

x − y + z = 4,

2x + y − 3z = 0,

x + y + z = 2.

x2 y2
18. nh?kZo`Ùk + = 1 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 6
a2 b2

Find the area enclosed by the ellipse :
x2 y2
+ =1
a 2 b2
vFkok

OR

eku Kkr dhft, % 6

π /2
cos5 x
∫ sin5 x + cos5 x
dx
0

Evaluate :

π /2
cos5 x
∫ sin5 x + cos5 x
dx
0

4331/(Set : A)

( 15 ) 4331/(Set : A)
→ →
19. r = iˆ + ˆj + λ(2iˆ − ˆj + kˆ ) vkSj r = 2iˆ + ˆj − kˆ + µ

(3iˆ − 5 ˆj + 2kˆ ) js[kkvksa ds chp dh U;wure nwjh Kkr dhft,A 6

Find the shortest distance between the lines :
→
r = iˆ + ˆj + λ(2iˆ − ˆj + kˆ ) and

→
r = 2iˆ + ˆj − kˆ + µ (3iˆ − 5 ˆj + 2kˆ )

vFkok
OR

lery dk lehdj.k Kkr dhft,] tks fcUnqvksa (1, 1, −1),
(6, 4, −5) vkSj (−4, −2, 3) ls xqtjrk gSA 6

Find the equation of plane passing through the
points (1, 1, −1), (6, 4, −5) and (−4, −2, 3).

20. vkys[k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft, % 6

vf/kdre % Z = 250x + 75y
O;ojks/kksa ds vUrxZr %
5x + y ≤ 100,

x + y ≤ 60,

x ≥ 0, y ≥ 0.

4331/(Set : A) P. T. O.

( 16 ) 4331/(Set : A)
Solve the following linear programming problem
by graphical method :

Maximize : Z = 250x + 75y

subject to the constraints :

5x + y ≤ 100,

x + y ≤ 60,

x ≥ 0, y ≥ 0.

s

4331/(Set : A)

CLASS : 12th (Sr. Secondary) Code No. 4331
Series : SS-M/2019
Roll No. SET : B

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh/Re-appear Candidates)
Time allowed : 3 hours ] [ Maximum Marks : 80
• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr i`"B 16 rFkk
iz'u 20 gSaA
Please make sure that the printed pages in this
question paper are 16 in number and it contains
20 questions.
• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks
Nk= mÙkj&iqfLrdk ds eq[;&i`"B ij fy[ksaA
The Code No. and Set on the right side of the
question paper should be written by the candidate
on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad
vo'; fy[ksaA
Before beginning to answer a question, its Serial
Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMsa+A
Don’t leave blank page/pages in your answer-book.

4331/(Set : B) P. T. O.

(2) 4331/(Set : B)
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr%
vko';drkuqlkj gh fy[ksa vkSj fy[kk mÙkj u dkVsaA
Except answer-book, no extra sheet will be given.
Write to the point and do not strike the written answer.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the
question paper.
• d`i;k iz'uksa dk mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i=
iw.kZ o lgh gS] ijh{kk ds mijkUr bl lEcU/k esa dksbZ Hkh nkok
Lohdkj ugha fd;k tk;sxkA
Before answering the question, ensure that you
have been supplied the correct and complete
question paper, no claim in this regard, will be
entertained after examination.
lkekU; funsZ'k %
(i) bl iz'u-i= esa 20 iz'u gSa] tks fd pkj [k.Mksa % v] c]
l vkSj n esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ,d ç'u gS tks 16 (i-xvi) Hkkxksa
esa gS] ftuesa 6 Hkkx cgqfodYih; gSaA izR;sd
Hkkx 1 vad dk gSA
[k.M ^c* % bl [k.M esa 2 ls 11 rd dqy nl ç'u
gSaA çR;sd ç'u 2 vadksa dk gSA
[k.M ^l* % bl [k.M esa 12 ls 16 rd dqy ik¡p ç'u
gSaA çR;sd ç'u 4 vadksa dk gSA
[k.M ^n* % bl [k.M esa 17 ls 20 rd dqy pkj ç'u
gSAa çR;sd ç'u 6 vadksa dk gSA
(ii) lHkh ç'u vfuok;Z gSaA
(iii) [k.M ^n* ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSa]
muesa ls ,d gh iz'u dks pquuk gSA
4331/(Set : B)

(3) 4331/(Set : B)
(iv) fn;s x;s xzkQ-isij dks viuh mÙkj-iqfLrdk ds lkFk vo';
uRFkh djsaA
(v) xzkQ-isij ij viuh mÙkj-iqfLrdk dk Øekad vo'; fy[ksaA
(vi) dSYD;qysVj ds ç;ksx dh vuqefr ugha gSA
General Instructions :
(i) This question paper consists of 20 questions
which are divided into four Sections : A, B,
C and D :
Section 'A' : This Section consists of one
question which is divided into
16 (i-xvi) parts of which 6 parts
of multiple choice type. Each
part carries 1 mark.
Section 'B' : This Section consists of ten
questions from 2 to 11. Each
question carries 2 marks.
Section 'C' : This Section consists of five
questions from 12 to 16. Each
question carries 4 marks.
Section 'D' : This Section consists of four
questions from 17 to 20. Each
question carries 6 marks.
(ii) All questions are compulsory.
(iii) Section 'D' contains some questions where
internal choice have been provided. Choose
one of them.
(iv) You must attach the given graph-paper along
with your answer-book.
(v) You must write your Answer-book Serial No.
on the graph-paper.
(vi) Use of Calculator is not permitted.
4331/(Set : B) P. T. O.

(4) 4331/(Set : B)
[k.M – v
SECTION – A

1. (i) ;fn f : R → R rFkk g : R → R Qyu Øe'k%
f (x) = cos x rFkk g(x ) = 3x 2 }kjk ifjHkkf"kr gSa] rks

gof Kkr dhft,A 1

Find gof, if f : R → R and g : R → R are given

by f (x) = cos x and g(x ) = 3x 2 .

 3
(ii) cos −1  dk eku gS % 1
2 
 

π π
(A) (B)
3 4

π
(C) (D) buesa ls dksbZ ugha
6

 3
The value of cos −1  is :

 2 

π π
(A) (B)
3 4

π
(C) (D) None of these
6

4331/(Set : B)

(5) 4331/(Set : B)
(iii) ,d ,sls 2 × 2 vkO;wg dh jpuk dhft,] ftlds vo;o
(i + j )2
aij = }kjk çnÙk gSaA 1
2

Construct a 2 × 2 matrix whose elements

(i + j )2
are given by aij = .
2

x 2 6 2
(iv) = ds fy, x dk eku gS % 1
18 x 18 6

(A) 0 (B) –5

(C) 7 (D) ±6

x 2 6 2
The value of x for which =
18 x 18 6

is :

(A) 0 (B) –5

(C) 7 (D) ±6

(v) x ds lkis{k cos(sin x ) dk vodyu dhft,A 1

Differentiate cos(sin x ) w. r. t. x.

4331/(Set : B) P. T. O.

(6) 4331/(Set : B)
(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj bldh f=T;k r ds
lkis{k tcfd r = 4 cm, gS % 1

(A) 6π cm2 /sec (B) 8π cm2 /sec

(C) 4π cm2 /sec (D) buesa ls dksbZ ugha
The rate of change of the area of a circle
with respect to its radius r, when r = 4 cm
is :

(A) 6π cm2 /sec (B) 8π cm2 /sec

(C) 4π cm2 /sec (D) None of these

(vii) x = 4 ij oØ y = 3x 4 − 4x dh Li'kZjs[kk dh ço.krk

Kkr dhft,A 1

Find the slope of tangent to the curve

y = 3x 4 − 4x at x = 4.

(viii) eku Kkr dhft, % 1

2x
∫ 1 + x 2 dx
Evaluate :
2x
∫ 1 + x 2 dx
4331/(Set : B)

(7) 4331/(Set : B)
π /2

∫ (x + x cos x + tan x ) dx dk eku gS %
3 5
(ix) 1
− π /2

π π
(A) (B) −
2 2
(C) π (D) 0
π /2

∫ (x + x cos x + tan x ) dx is :
3 5
The value of
− π /2

π π
(A) (B) −
2 2
(C) π (D) 0

d 2y
(x) + y = 0 ] vodyu lehdj.k dh dksfV gS % 1
dx 2

(A) 1 (B) 0

(C) 2 (D) buesa ls dksbZ ugha
The order of the differential equation

d2y
+ y = 0 , is :
dx2

(A) 1 (B) 0

(C) 2 (D) None of these

4331/(Set : B) P. T. O.

(8) 4331/(Set : B)
2
 dy  dy
(xi)   + − sin2 y = 0, vodyu lehdj.k dh
 dx  dx
?kkr gS % 1

(A) 2 (B) 1

(C) 0 (D) buesa ls dksbZ ugha
The degree of the differential equation
2
 dy  dy
  + − sin2 y = 0, is :
 dx  dx

(A) 2 (B) 1

(C) 0 (D) None of these

(xii) ;fn P(E ) = 0.6, P(F ) = 0.3 rFkk P(E ∩ F ) = 0.2,
rks P (F/E ) Kkr dhft,A 1

If P(E ) = 0.6, P(F ) = 0.3 and P(E ∩ F ) = 0.2,
then find P (F/E ).

(xiii) ;fn P(A ) = 0.3, P(B ) = 0.6 rFkk A vkSj B LorU=
?kVuk,¡ gSa] rks P (A vkSj B ugha ) dk eku Kkr dhft,A 1

If P(A ) = 0.3, P(B ) = 0.6 and A and B are
independent events, then find the value of
P (A and B not).
4331/(Set : B)

(9) 4331/(Set : B)
(xiv) ,d FkSys esa 4 lQsn vkSj 6 dkyh xsansa gSaA nks xsan çfrLFkkiu

ds lkFk ;kn`fPNd fudkyh x;h gSaA igyh xsan lQsn vkSj
nwljh xsan dkyh gksus dh izkf;drk Kkr dhft,A 1

A bag contains 4 white and 6 black balls.
Two balls are drawn at random with
replacement. Find the probability that first
ball is white and second ball is black.

→
(xv) a = iˆ − 2 ˆj lfn'k ds vuqfn'k lfn'k Kkr dhft,
ftldk ifjek.k 7 bdkbZ gSA 1

Find a vector in the direction of a vector
→
a = iˆ − 2 ˆj which has magnitude 7 units.

(xvi) ;fn ,d js[kk x, y vkSj z-v{k ds lkFk Øe'k% 90°,

135° rFkk 45° ds dks.k cukrh gS] rks blds fnd~-dksT;k

Kkr dhft,A 1

If a line makes angles 90°, 135° and 45°
with the x, y and z-axis, then find the
direction cosines of the line.

4331/(Set : B) P. T. O.

( 10 ) 4331/(Set : B)
[k.M – c
SECTION – B

2. eku yhft, fd N ,d f}vk/kkjh lafØ;k ∗] a ∗ b = a rFkk b
dk L.C.M. }kjk ifjHkkf"kr gSA 20 ∗ 16 Kkr dhft,A 2

Let ∗ be the binary operation on N given by a ∗ b =
L. C. M. of a and b. Find 20 ∗ 16.

 13π 
3. cos −1 cos  dk eq[; eku Kkr dhft,A 2
 6 

 13π 
Find the principal value of cos −1 cos .
 6 

ekuk A = 
2 4  1 3
4. , B= , 3A – 2B dk eku Kkr
3 2  − 2 5
dhft,A 2

2 4  1 3
Let A =  , B= , find 3A – 2B.
3 2 − 2 5 

1 0 1
5. ;fn A =  0 1 2 , rks n'kkZb, |3A| = 27|A|. 2
 
 0 0 4 

1 0 1
If A =  0 1 2 , then show that : |3A| = 27|A|.
 
 0 0 4 

4331/(Set : B)

( 11 ) 4331/(Set : B)

6. lehdj.k ax + by2 = cos y ls dy Kkr dhft,A 2
dx
dy
Find from the equation ax + by2 = cos y.
dx

7. ;fn x = a cos θ, y = b cos θ, rks dy Kkr dhft,A 2
dx
dy
If x = a cos θ, y = b cos θ, then find .
dx
8. eku Kkr dhft, % 2

∫ x . sin 3x dx
Evaluate :

∫ x . sin 3x dx
9. eku Kkr dhft, % 2
π/4
∫0 sin 2x dx
Evaluate :
π/4
∫0 sin 2x dx

10. a rFkk b dks foyqIr djrs gq, oØ y 2 = a(b 2 − x 2 ) dks
fu:fir djus okys vodyu lehdj.k Kkr dhft,A 2
Find differential equation corresponding to
y 2 = a (b 2 − x 2 ) , by eliminating a and b.
11. ,d U;k¸; flDds dks 10 ckj mNkyk x;k gSA U;wure N% fpr vkus
ds fy, çkf;drk Kkr dhft,A 2
A fair coin is tossed 10 times. Find the probability
of at least 6 heads.
4331/(Set : B) P. T. O.

( 12 ) 4331/(Set : B)
[k.M – l
SECTION – C

12. fuEu dks ljyre :i esa O;Dr dhft, % 4

1 − cos x
tan−1 ,x <π
1 + cos x
Write the simplest form of the following :
1 − cos x
tan−1 ,x <π
1 + cos x

13. k dk eku Kkr dhft, rkfd Qyu % 4
kx + 1,
f (x ) = 
;fn x ≤π
] x = π ij larr gksA
 cos x , ;fn x >π
Find the values of k, so that the function :
kx + 1, if x ≤ π
f (x ) = 
 cos x , if x > π
is continuous at x = π .

14. vUrjky Kkr dhft, ftuesa Qyu %
f (x) = 2x 3 − 3x 2 − 36x + 7 ls çnÙk Qyu f fujUrj
Ðkleku gSA 4
Find the interval in which the function :
f (x) = 2x 3 − 3x 2 − 36x + 7 is strictly decreasing.

4331/(Set : B)

( 13 ) 4331/(Set : B)
15. iklksa ds ,d tksM+s dks rhu ckj mNkyus ij f}dksa (doublets)
dh la[;k dk izkf;drk caVu Kkr dhft,A 4

Find the probability distribution of the number
of doublets in 3 throws of a pair of dice.

16. lekUrj prqHkZqt dk {ks=Qy Kkr dhft, ftldh layXu Hkqtk,¡¡
→
a = 3iˆ + ˆj + 4kˆ vkSj b→ = iˆ − ˆj + kˆ }kjk nh xbZ gSaA 4

Find the area of parallelogram whose adjacent
→ →
sides are a = 3iˆ + ˆj + 4kˆ and b = iˆ − ˆj + kˆ .

[k.M – n

SECTION – D

17. fuEu lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6

2x + 3y + 3z = 5,

x – 2y + z = –4,

3x – y – 2z = 3.

4331/(Set : B) P. T. O.

( 14 ) 4331/(Set : B)
Solve the following equations by a matrix method :

2x + 3y + 3z = 5,

x – 2y + z = –4,

3x – y – 2z = 3.

2 2
18. nh?kZo`Ùk x + y = 1 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 6
16 9

x 2 y2
Find the area enclosed by the ellipse + = 1.
16 9
vFkok

OR

eku Kkr dhft, % 6

π /2

∫ log sin x dx
0

Evaluate :

π /2

∫ log sin x dx
0

4331/(Set : B)

( 15 ) 4331/(Set : B)
→ →
19. r = (iˆ + 2 ˆj + kˆ ) + λ(iˆ − ˆj + kˆ ) vkSj r = 2iˆ − ˆj − kˆ +

µ(2iˆ + ˆj + 2kˆ ) js[kkvksa ds chp dh U;wure nwjh Kkr dhft,A 6

Find the shortest distance between the lines :
→
r = (iˆ + 2 ˆj + kˆ ) + λ(iˆ − ˆj + kˆ ) and

→
r = 2iˆ − ˆj − kˆ + µ (2iˆ + ˆj + 2kˆ )

vFkok
OR

lery dk lehdj.k Kkr dhft,] tks fcUnqvksa (1, 1, 0),
(1, 2, 1) vkSj (−2, 2, –1) ls xqtjrk gSA 6

Find the equation of plane passing through the
points (1, 1, 0), (1, 2, 1) and (−2, 2, –1).

20. vkys[k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft, % 6

vf/kdre % Z = 4x + y
O;ojks/kksa ds vUrxZr %
x + y ≤ 50,

3x + y ≤ 90,

x ≥ 0, y ≥ 0.

4331/(Set : B) P. T. O.

( 16 ) 4331/(Set : B)
Solve the following linear programming problem
by graphical method :

Maximize : Z = 4x + y

subject to the constraints :

x + y ≤ 50,

3x + y ≤ 90,

x ≥ 0, y ≥ 0.

s

4331/(Set : B)

CLASS : 12th (Sr. Secondary) Code No. 4331
Series : SS-M/2019
Roll No. SET : C

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh/Re-appear Candidates)
Time allowed : 3 hours ] [ Maximum Marks : 80
• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr i`"B 16 rFkk
iz'u 20 gSaA
Please make sure that the printed pages in this
question paper are 16 in number and it contains
20 questions.
• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks
Nk= mÙkj&iqfLrdk ds eq[;&i`"B ij fy[ksaA
The Code No. and Set on the right side of the
question paper should be written by the candidate
on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad
vo'; fy[ksaA
Before beginning to answer a question, its Serial
Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMsa+A
Don’t leave blank page/pages in your answer-book.

4331/(Set : C) P. T. O.

(2) 4331/(Set : C)
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr%
vko';drkuqlkj gh fy[ksa vkSj fy[kk mÙkj u dkVsaA
Except answer-book, no extra sheet will be given.
Write to the point and do not strike the written answer.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the
question paper.
• d`i;k iz'uksa dk mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i=
iw.kZ o lgh gS] ijh{kk ds mijkUr bl lEcU/k esa dksbZ Hkh nkok
Lohdkj ugha fd;k tk;sxkA
Before answering the question, ensure that you
have been supplied the correct and complete
question paper, no claim in this regard, will be
entertained after examination.
lkekU; funsZ'k %
(i) bl iz'u-i= esa 20 iz'u gSa] tks fd pkj [k.Mksa % v] c]
l vkSj n esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ,d ç'u gS tks 16 (i-xvi) Hkkxksa
esa gS] ftuesa 6 Hkkx cgqfodYih; gSaA izR;sd
Hkkx 1 vad dk gSA
[k.M ^c* % bl [k.M esa 2 ls 11 rd dqy nl ç'u
gSaA çR;sd ç'u 2 vadksa dk gSA
[k.M ^l* % bl [k.M esa 12 ls 16 rd dqy ik¡p ç'u
gSaA çR;sd ç'u 4 vadksa dk gSA
[k.M ^n* % bl [k.M esa 17 ls 20 rd dqy pkj ç'u
gSAa çR;sd ç'u 6 vadksa dk gSA
(ii) lHkh ç'u vfuok;Z gSaA
(iii) [k.M ^n* ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSa]
muesa ls ,d gh iz'u dks pquuk gSA
4331/(Set : C)

(3) 4331/(Set : C)
(iv) fn;s x;s xzkQ-isij dks viuh mÙkj-iqfLrdk ds lkFk vo';
uRFkh djsaA
(v) xzkQ-isij ij viuh mÙkj-iqfLrdk dk Øekad vo'; fy[ksaA
(vi) dSYD;qysVj ds ç;ksx dh vuqefr ugha gSA
General Instructions :
(i) This question paper consists of 20 questions
which are divided into four Sections : A, B,
C and D :
Section 'A' : This Section consists of one
question which is divided into
16 (i-xvi) parts of which 6 parts
of multiple choice type. Each
part carries 1 mark.
Section 'B' : This Section consists of ten
questions from 2 to 11. Each
question carries 2 marks.
Section 'C' : This Section consists of five
questions from 12 to 16. Each
question carries 4 marks.
Section 'D' : This Section consists of four
questions from 17 to 20. Each
question carries 6 marks.
(ii) All questions are compulsory.
(iii) Section 'D' contains some questions where
internal choice have been provided. Choose
one of them.
(iv) You must attach the given graph-paper along
with your answer-book.
(v) You must write your Answer-book Serial No.
on the graph-paper.
(vi) Use of Calculator is not permitted.

4331/(Set : C) P. T. O.

(4) 4331/(Set : C)
[k.M – v
SECTION – A
1
1. (i) fog Kkr dhft, ;fn f (x ) = 8x 3
vkSj g (x ) = x 3 . 1
1
3
Find fog if f (x ) = 8x and g (x ) = x 3 .

(ii) (
tan −1 − 3 ) dk eku gS % 1
π π
(A) (B) –
3 6
π
(C) – (D) buesa ls dksbZ ugha
3
The value of tan −1 − 3 is : ( )
π π
(A) (B) –
3 6
π
(C) – (D) None of these
3

(iii) ,d ,sls 2 × 2 vkO;wg dh jpuk dhft,] ftlds vo;o
i
aij = }kjk çnÙk gSaA 1
j

Construct a 2 × 2 matrix whose elements
i
are given by aij = .
j

4331/(Set : C)

(5) 4331/(Set : C)
2 4 2x 4
(iv) = ds fy, x dk eku gS % 1
5 1 6 x

(A) ± 3 (B) ± 6

(C) ± 5 (D) buesa ls dksbZ ugha

2 4 2x 4
The value of x for which =
5 1 6 x

is :

(A) ± 3 (B) ± 6

(C) ± 5 (D) None of these

(v) x ds lkis{k sin(ax + b ) dk vodyu Kkr dhft,A 1

Differentiate sin(ax + b ) w. r. t. x.

(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj bldh f=T;k r ds
lkis{k tcfd r = 5 cm] gS % 1

(A) 5π cm2 /sec (B) 15π cm2 /sec

(C) 10π cm2 /sec (D) buesa ls dksbZ ugha

4331/(Set : C) P. T. O.

(6) 4331/(Set : C)
The rate of change of the area of a circle
with respect to its radius r when r = 5 cm
is :
(A) 5π cm2 /sec (B) 15π cm2 /sec
(C) 10π cm2 /sec (D) None of these

(vii) x = 2 ij oØ y = x 3 − x + 1 dh Li'kZjs[kk dh ço.krk
Kkr dhft,A 1
Find the slope of tangent to the curve
y = x 3 − x + 1 at x = 2.

(viii) eku Kkr dhft, % 1

(log x )2
∫ x dx
Evaluate :

(log x )2
∫ x dx
π

∫ x .sin x dx dk eku gS %
10 7
(ix) 1
−π

(A) π (B) –π
(C) 1 (D) 0
π

∫ x .sin x dx is :
10 7
The value of
−π

(A) π (B) –π
(C) 1 (D) 0
4331/(Set : C)

(7) 4331/(Set : C)
3
2 
2 
d 3y  d y  = 0 ] vodyu lehdj.k dh dksfV
(x) + x .
dx 3  dx 2 
 
gS % 1

(A) 2 (B) 3

(C) 1 (D) buesa ls dksbZ ugha
The order of the differential equation
3
d 3y 2  d 2y 
+x . = 0 , is :
dx 3  dx 2 
 

(A) 2 (B) 3

(C) 1 (D) None of these

4
 ds  d 2s
(xi)   + 3s 2 = 0, vodyu lehdj.k dh ?kkr
 dt  dt
gS % 1

(A) 4 (B) 2

(C) 1 (D) buesa ls dksbZ ugha
The degree of the differential equation
4
 ds  d 2s
  + 3s = 0, is :
 dt  dt 2

(A) 4 (B) 2

(C) 1 (D) None of these
4331/(Set : C) P. T. O.

(8) 4331/(Set : C)
(xii) ;fn P (A ) = 0.8, P (B ) = 0.5 vkSj P(B/A ) = 0.4,

rks P (A ∩ B ) Kkr dhft,A 1

If P (A ) = 0.8, P (B ) = 0.5 and P(B/A ) = 0.4,
then find P (A ∩ B ).

(xiii) ;fn P (A ) = 0.3, P (B ) = 0.6 rFkk A vkSj B LorU=

?kVuk,¡ gSa] rks P (A ;k B ) dk eku Kkr dhft,A 1

If P (A ) = 0.3, P (B ) = 0.6 and A and B are

independent events, then find the value of

P (A or B ).

(xiv) ,d FkSys esa 4 lQsn vkSj 6 dkyh xsansa gSAa nks xsansa çfrLFkkiu

ds lkFk ;kn`fPNd fudkyh x;h gSaA ,d xsan lQsn vkSj
,d xsan dkyh gksus dh izkf;drk Kkr dhft,A 1

A bag contains 4 white and 6 black balls.

Two balls are drawn at random with

replacement. Find the probability that one

ball is white and one ball is black.

4331/(Set : C)

(9) 4331/(Set : C)
→
(xv) a = 2iˆ + 3 ˆj + kˆ lfn'k ds vuqfn'k ek=d lfn'k Kkr
dhft,A 1

Find a unit vector in the direction of a
→
vector a = 2iˆ + 3 ˆj + kˆ .

(xvi) x-v{k] y-v{k vkSj z-v{k ds fnd~-dksT;k Kkr dhft,A 1

Find the direction cosines of x-axis, y-axis,
and z-axis.

[k.M – c
SECTION – B

2. eku yhft, fd N ,d f}vk/kkjh lafØ;k ∗] a ∗ b = a rFkk b
dk L.C.M. }kjk ifjHkkf"kr gSA 16 ∗ 24 Kkr dhft,A 2

Let ∗ be the binary operation on N given by a ∗ b =
L. C. M. of a and b. Find 16 ∗ 24.

 7π 
3. tan−1 tan  dk eku Kkr dhft,A 2
 6 

 7π 
Find the value of tan−1 tan .
 6 

4331/(Set : C) P. T. O.

( 10 ) 4331/(Set : C)

ekuk A = 
2 4 − 2 5
4. , B = , 3A – B dk eku Kkr
3 2  3 4
dhft,A 2
2 4 − 2 5
Let A =  , B = , find 3A – B.
3 2  3 4

;fn A = 
4 1
5. , rks n'kkZb, |3A| = 9|A|. 2
3 2
4 1
If A =  , then show that |3A| = 9|A|.
3 2

6. lehdj.k xy + y2 = tan x + y ls dy Kkr dhft,A 2
dx
dy
Find from the equation xy + y2 = tan x + y.
dx

7. ;fn x = sin t, y = cos 2t, rks dy Kkr dhft,A 2
dx
dy
If x = sin t, y = cos 2t, then find .
dx

8. eku Kkr dhft, % 2

∫ x .e dx
2 x

Evaluate :

∫ x .e dx
2 x

4331/(Set : C)

( 11 ) 4331/(Set : C)
9. eku Kkr dhft, % 2
π /4
∫0 tan x dx

Evaluate :
π /4
∫0 tan x dx

10. a rFkk b dks foyqIr djrs gq, oØ y = e 2x . (a + bx ) dks
fu:fir djus okys vodyu lehdj.k Kkr dhft,A 2
Find the differential equation corresponding to
y = e 2x . (a + bx ) , by eliminating a and b.

11. ,d U;k¸; flDds dks 10 ckj mNkyk x;k gSA vf/kdre N% fpr
vkus ds fy, çkf;drk Kkr dhft,A 2
A fair coin is tossed 10 times. Find the probability
of at most 6 heads.

[k.M – l
SECTION – C

12. ljy dhft, % 4
 cos x − sin x 
tan−1 
 cos x + sin x 
Simplify :
 cos x − sin x 
tan−1 
 cos x + sin x 
4331/(Set : C) P. T. O.

( 12 ) 4331/(Set : C)
13. k dk eku Kkr dhft, rkfd Qyu %

 kx + 1,
f (x ) = 
;fn x ≤5
] x = 5 ij larr gSA 4
3x − 5, ;fn x >5

Find the value of k, so that the function :
 kx + 1, if x ≤5
f (x ) = 
3x − 5, if x >5

is continuous at x = 5 .

14. vUrjky Kkr dhft, ftuesa Qyu %

f (x ) = 4x 3 − 6x 2 − 72x + 30 ls çnÙk Qyu f fujUrj

o/kZeku gSA 4

Find the intervals in which the function

f (x ) = 4x 3 − 6x 2 − 72x + 30 is strictly increasing.

15. ,d flDds dh nks mNkyksa esa fprksa dh la[;k dk izkf;drk caVu
Kkr dhft,A 4

Find the probability distribution of the number
of heads in two tosses of a coin.

4331/(Set : C)

( 13 ) 4331/(Set : C)
16. lekUrj prqHkZqt dk {ks=Qy Kkr dhft, ftldh layXu Hkqtk,¡¡
→
a = iˆ − ˆj + 3kˆ vkSj b→ = 2iˆ − 7 ˆj + kˆ }kjk nh xbZ gSaA 4

Find the area of a parallelogram whose adjacent
→ →
sides are a = iˆ − ˆj + 3kˆ and b = 2iˆ − 7 ˆj + kˆ .

[k.M – n

SECTION – D

17. fuEu lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6

x – y + 2z = 7,

3x + 4y – 5z = –5,

2x – y + 3z = 12.

Solve the following equations by matrix method :

x – y + 2z = 7,

3x + 4y – 5z = –5,

2x – y + 3z = 12.
4331/(Set : C) P. T. O.

( 14 ) 4331/(Set : C)
2
y2
18. nh?kZo`Ùk x + = 1 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 6
4 9

x 2 y2
Find the area enclosed by the ellipse + = 1.
4 9

vFkok

OR

eku Kkr dhft, % 6

π
x sin x
∫ 1 + cos2 x dx
0

Evaluate :

π
x sin x
∫ 1 + cos2 x dx
0

x +1 y +1 z +1 x − 3 y − 5 z − 7
19. = = , = = js[kkvksa ds
7 −6 1 1 −2 1

chp dh U;wure nwjh Kkr dhft,A 6

Find the shortest distance between the lines :
x +1 y +1 z +1 x − 3 y − 5 z − 7
= = , = =
7 −6 1 1 −2 1

4331/(Set : C)

( 15 ) 4331/(Set : C)
vFkok
OR

lery dk lehdj.k Kkr dhft,] tks fcUnqvksa (0, 1, 1),

(1, 1, 2) vkSj (−1, 2, –2) ls xqtjrk gSA 6

Find the equation of plane passing through the

points (0, 1, 1), (1, 1, 2) and (−1, 2, –2).

20. vkys[k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft, % 6

U;wure % Z = 200x + 500y

O;ojks/kksa ds vUrxZr %

x + 2y ≥ 10,

3x + 4y ≤ 24,

x ≥ 0, y ≥ 0.

4331/(Set : C) P. T. O.

( 16 ) 4331/(Set : C)
Solve the following linear programming problem
by graphical method :

Minimize : Z = 200x + 500y

subject to the constraints :

x + 2y ≥ 10,

3x + 4y ≤ 24,

x ≥ 0, y ≥ 0.

s

4331/(Set : C)

CLASS : 12th (Sr. Secondary) Code No. 4331
Series : SS-M/2019
Roll No. SET : D

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh/Re-appear Candidates)
Time allowed : 3 hours ] [ Maximum Marks : 80
• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr i`"B 16 rFkk
iz'u 20 gSaA
Please make sure that the printed pages in this
question paper are 16 in number and it contains
20 questions.
• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks
Nk= mÙkj&iqfLrdk ds eq[;&i`"B ij fy[ksaA
The Code No. and Set on the right side of the
question paper should be written by the candidate
on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad
vo'; fy[ksaA
Before beginning to answer a question, its Serial
Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMsa+A
Don’t leave blank page/pages in your answer-book.

4331/(Set : D) P. T. O.

(2) 4331/(Set : D)
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr%
vko';drkuqlkj gh fy[ksa vkSj fy[kk mÙkj u dkVsaA
Except answer-book, no extra sheet will be given.
Write to the point and do not strike the written answer.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the
question paper.
• d`i;k iz'uksa dk mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i=
iw.kZ o lgh gS] ijh{kk ds mijkUr bl lEcU/k esa dksbZ Hkh nkok
Lohdkj ugha fd;k tk;sxkA
Before answering the question, ensure that you
have been supplied the correct and complete
question paper, no claim in this regard, will be
entertained after examination.
lkekU; funsZ'k %
(i) bl iz'u-i= esa 20 iz'u gSa] tks fd pkj [k.Mksa % v] c]
l vkSj n esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ,d ç'u gS tks 16 (i-xvi) Hkkxksa
esa gS] ftuesa 6 Hkkx cgqfodYih; gSaA izR;sd
Hkkx 1 vad dk gSA
[k.M ^c* % bl [k.M esa 2 ls 11 rd dqy nl ç'u
gSaA çR;sd ç'u 2 vadksa dk gSA
[k.M ^l* % bl [k.M esa 12 ls 16 rd dqy ik¡p ç'u
gSaA çR;sd ç'u 4 vadksa dk gSA
[k.M ^n* % bl [k.M esa 17 ls 20 rd dqy pkj ç'u
gSAa çR;sd ç'u 6 vadksa dk gSA
(ii) lHkh ç'u vfuok;Z gSaA
(iii) [k.M ^n* ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSa]
muesa ls ,d gh iz'u dks pquuk gSA
4331/(Set : D)

(3) 4331/(Set : D)
(iv) fn;s x;s xzkQ-isij dks viuh mÙkj-iqfLrdk ds lkFk vo';
uRFkh djsaA
(v) xzkQ-isij ij viuh mÙkj-iqfLrdk dk Øekad vo'; fy[ksaA
(vi) dSYD;qysVj ds ç;ksx dh vuqefr ugha gSA
General Instructions :
(i) This question paper consists of 20 questions
which are divided into four Sections : A, B,
C and D :
Section 'A' : This Section consists of one
question which is divided into
16 (i-xvi) parts of which 6 parts
of multiple choice type. Each
part carries 1 mark.
Section 'B' : This Section consists of ten
questions from 2 to 11. Each
question carries 2 marks.
Section 'C' : This Section consists of five
questions from 12 to 16. Each
question carries 4 marks.
Section 'D' : This Section consists of four
questions from 17 to 20. Each
question carries 6 marks.
(ii) All questions are compulsory.
(iii) Section 'D' contains some questions where
internal choice have been provided. Choose
one of them.
(iv) You must attach the given graph-paper along
with your answer-book.
(v) You must write your Answer-book Serial No.
on the graph-paper.
(vi) Use of Calculator is not permitted.
4331/(Set : D) P. T. O.

(4) 4331/(Set : D)
[k.M – v
SECTION – A
1
1. (i) gof Kkr dhft, ;fn f (x ) = 8x 3
vkSj g (x ) = x 3 . 1
1
3
Find gof if f (x ) = 8x and g (x ) = x 3 .

 1
(ii) cos −1 −  dk eku gS % 1
 2
2π π
(A) (B)
3 4
π
(C) (D) buesa ls dksbZ ugha
2
 1
The value of cos −1 −  is :
 2
2π π
(A) (B)
3 4
π
(C) (D) None of these
2

(iii) ,d ,sls 2 × 2 vkO;wg dh jpuk dhft,] ftlds vo;o
(i + 2 j )2
aij = }kjk çnÙk gSaA 1
2
Construct a 2 × 2 matrix whose elements
(i + 2 j )2
are given by a ij = .
2
4331/(Set : D)

(5) 4331/(Set : D)
2 3 x 3
(iv) = ds fy, x dk eku gS % 1
4 5 2x 5

(A) 1 (B) 2

(C) 3 (D) buesa ls dksbZ ugha

2 3 x 3
The value of x for which =
4 5 2x 5

is :

(A) 1 (B) 2

(C) 3 (D) None of these

(v) x ds lkis{k tan(2x + 3) dk vodyu Kkr dhft,A 1

Differentiate tan(2x + 3) w. r. t. x.

(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj bldh f=T;k r ds
lkis{k tcfd r = 6 cm] gS % 1

(A) 6π cm2 /sec (B) 8π cm2 /sec

(C) 10π cm2 /sec (D) 12π cm2 /sec

4331/(Set : D) P. T. O.

(6) 4331/(Set : D)
The rate of change of area of a circle with
respect to its radius r, when r = 6 cm is :
(A) 6π cm2 /sec (B) 8π cm2 /sec

(C) 10π cm2 /sec (D) 12π cm2 /sec

(vii) x = 3 ij oØ y = x 3 − 3x + 2 dh Li'kZjs[kk dh
ço.krk Kkr dhft,A 1

Find the slope of tangent to the curve
y = x 3 − 3x + 2 at x = 3.

(viii) eku Kkr dhft, % 1
1
∫ x + x log x dx
Evaluate :
1
∫ x + x log x dx
π

∫ x cos x dx dk eku gS %
3
(ix) 1
−π

(A) π (B) –π
(C) –1 (D) 0
π

∫ x cos x dx is :
3
The value of
−π

(A) π (B) –π
(C) –1 (D) 0
4331/(Set : D)

(7) 4331/(Set : D)
d 2y dy
(x) 2x 2 2
−3 + y = 0, vodyu lehdj.k dh
dx dx
dksfV gS % 1

(A) 0 (B) 2

(C) 1 (D) buesa ls dksbZ ugha
The order of the differential equation
2
d y dy
2x 2 2
−3 + y = 0, is :
dx dx

(A) 0 (B) 2

(C) 1 (D) None of these

3
 d 2y  2
(xi)   +  dy  + 3y = 0, vodyu lehdj.k dh
 dx 2   dx 
 
?kkr gS % 1

(A) 1 (B) 2

(C) 3 (D) buesa ls dksbZ ugha
The degree of the differential equation
3
 d 2y  2
  +  dy  + 3y = 0, is :
 dx 2   dx 
 

(A) 1 (B) 2

(C) 3 (D) None of these

4331/(Set : D) P. T. O.

(8) 4331/(Set : D)
7 9
(xii) ;fn P (A ) = , P (B ) = vkSj P (A ∩ B ) = 4 ,
13 13 13
rks P (A/B ) Kkr dhft,A 1

7 9 4
If P (A ) = , P (B ) = and P (A ∩ B ) = ,
13 13 13
then find P (A/B ).

(xiii) ;fn P (A ) = 0.3, P (B ) = 0.6 rFkk A vkSj B LorU=

?kVuk,¡ gSa] rks P (A vkSj B ) dk eku Kkr dhft,A 1

If P (A ) = 0.3, P (B ) = 0.6 and A and B are

independent events, then find the value of

P (A and B ).

(xiv) rk'k ds 52 iÙkksa dh ,d lqfefJr xM~Mh ls nks iÙks
izfrLFkkiuk ds lkFk ;kn`fPNd fudkys x, gSaA nksuksa iÙks
bDds gksus dh izkf;drk Kkr dhft,A 1

Two cards are drawn with replacement from

a well shuffled pack of 52 cards. Find the

probability that both cards are aces.

4331/(Set : D)

(9) 4331/(Set : D)
→
(xv) a = iˆ + ˆj + 2kˆ lfn'k ds vuqfn'k ek=d lfn'k Kkr
dhft,A 1

Find the unit vector in a direction of a
→
vector a = iˆ + ˆj + 2kˆ .

(xvi) ,d js[kk ds fnd~-dksT;k Kkr dhft, tks funsZ'kkadksa ds lkFk

leku dks.k cukrh gSA 1

Find the direction cosines of a line which
makes equal angles with co-ordinate axis.

[k.M – c
SECTION – B

2. eku yhft, fd N ,d f}vk/kkjh lafØ;k ∗] a ∗ b = a rFkk b
dk L.C.M. }kjk ifjHkkf"kr gSA 12 ∗ 20 Kkr dhft,A 2

Let ∗ be the binary operation on N given by a ∗ b =
L. C. M. of a and b. Find 12 ∗ 20.

 8π 
3. cos −1  cos  dk eku Kkr dhft,A 2
 5 

 8π 
Find the value of cos −1 cos .
 5 
4331/(Set : D) P. T. O.

( 10 ) 4331/(Set : D)

ekuk A = 
2 4 − 2 5
4. ,B = , 3A – 2B dk eku Kkr
3 2  3 4
dhft,A 2

2 4 − 2 5
Let A =  , B = , find 3A – 2B.
3 2  3 4

;fn A = 
3 1
5. , rks n'kkZb, |2A| = 4|A|. 2
2 3

3 1
If A =  , then show that |2A| = 4|A|.
2 3

dy
6. x 2 + xy + y 2 = 100 lehdj.k ls Kkr dhft,A 2
dx
dy
Find from the equation x 2 + xy + y 2 = 100 .
dx

7. ;fn x = 4t, y = 4 ] rks dy Kkr dhft,A 2
t dx
4 dy
If x = 4t, y = , then find .
t dx

8. eku Kkr dhft, % 2

∫ x log x dx
Evaluate :

∫ x log x dx
4331/(Set : D)

( 11 ) 4331/(Set : D)
9. eku Kkr dhft, % 2
π /4
∫0 tan x dx

Evaluate :
π /4
∫0 tan x dx

10. a rFkk b dks foyqIr djrs gq, oØ y = ae 3x + be −2x dks
fu:fir djus okys vodyu lehdj.k Kkr dhft,A 2

Find the differential equation corresponding to

y = ae 3x + be −2x by eliminating a and b.

11. ,d ikls dks 6 ckj mNkyk tkrk gSA ikls ij le la[;k izkIr
gksuk ,d lQyrk gSA Bhd 5 lQyrk,¡ izkIr gksus dh izkf;drk
Kkr dhft,A 2

A die is tossed 6 times. Getting an even number

is considered a success. Find the probability of

exactly 5 successes.

4331/(Set : D) P. T. O.

( 12 ) 4331/(Set : D)
[k.M – l
SECTION – C

12. fuEu dks ljyre :i esa O;Dr dhft, % 4

 cos x 
tan−1 
 1 − sin x 
Write the simplest form of the following :
 cos x 
tan−1 
 1 − sin x 

13. a vkSj b dk eku Kkr dhft, rkfd Qyu %
 5, ;fn x ≤2

f (x ) = ax + b ;fn 2 < x < 10 ] larr gSA 4
 21 ;fn x ≥ 10

Find the value of a and b so that the function :
 5, if x ≤2

f (x ) = ax + b if 2 < x < 10
 21 if x ≥ 10

is continuous function.

14. vUrjky Kkr dhft, ftuesa Qyu %
f (x ) = 5x 3 − 15x 2 − 120x + 3 çnÙk Qyu f fujUrj Ðkleku
gSA 4
Find the interval in which the function :
f (x ) = 5x 3 − 15x 2 − 120x + 3 is strictly decreasing.
4331/(Set : D)

( 13 ) 4331/(Set : D)
15. ,d flDds dh pkj mNkyksa esa fprksa dh la[;k dk izkf;drk caVu
Kkr dhft,A 4

Find the probability distribution of number of
heads in four tosses of coin.

16. f=Hkqt dk {ks=Qy Kkr dhft, ftlds 'kh"kZ A(1, 1, 2),
B(2, 3, 5) vkSj C(1, 5, 5) gSaA 4

Find the area of a triangle whose vertices are
A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

[k.M – n
SECTION – D

17. fuEu lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6

2x – 3y + 5z = 11,

3x + 2y – 4z = –5,

x + y – 2z = –3.

Solve the following equations by a matrix method :

2x – 3y + 5z = 11,

3x + 2y – 4z = –5,

x + y – 2z = –3.

4331/(Set : D) P. T. O.

( 14 ) 4331/(Set : D)
18. o`Ùk x 2 + y 2 = a 2 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 6

Find the area enclosed by the circle

x 2 + y2 = a2 .

vFkok
OR

eku Kkr dhft, % 6

π
x dx
∫ 1 + sin x
0

Evaluate :

π
x dx
∫ 1 + sin x
0

→ →
19. r = (iˆ + 2 ˆj + 3kˆ) + λ(iˆ − 3 ˆj + 2kˆ ) vkSj r = 4iˆ + 5 ˆj + 6kˆ +

µ(2iˆ + 3 ˆj + kˆ ) js[kkvksa ds chp dh U;wure nwjh Kkr dhft,A 6

Find the shortest distance between the lines :

→
r = (iˆ + 2 ˆj + 3kˆ ) + λ(iˆ − 3 ˆj + 2kˆ ) and

→
r = 4iˆ + 5 ˆj + 6kˆ + µ(2iˆ + 3 ˆj + kˆ )

4331/(Set : D)

( 15 ) 4331/(Set : D)
vFkok
OR

lery dk lehdj.k Kkr dhft,] tks fcUnqvksa (–2, 6, –6),

(–3, 10, –9) vkSj (−5, 0, –6) ls xqtjrk gSA 6

Find the equation of plane passing through the

points (–2, 6, –6), (–3, 10, –9) and (−5, 0, –6).

20. vkys[k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft, % 6

vf/kdre % Z = 3x + 9y

O;ojks/kksa ds vUrxZr %

x + 3y ≤ 60,

x + y ≥ 10,

x ≤ y,

x ≥ 0, y ≥ 0.

4331/(Set : D) P. T. O.

( 16 ) 4331/(Set : D)
Solve the following linear programming problem
by graphical method :

Maximize : Z = 3x + 9y

subject to the constraints :

x + 3y ≤ 60,

x + y ≥ 10,

x ≤ y,

x ≥ 0, y ≥ 0.

s

4331/(Set : D)