HBSE Class 12 Mathematics Question Paper 2020

CLASS : 12th (Sr. Secondary) Code No. 4931
Series : SS-M/2020
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.

4931/(Set : A) P. T. O.

(2) 4931/(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
4931/(Set : A)

(3) 4931/(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.

4931/(Set : A) P. T. O.

(4) 4931/(Set : A)
[k.M – v
SECTION – A

1. (i) ;fn Qyu f : R → R tks f (x) = x3 }kjk ifjHkkf"kr gS]
rks f gS % 1
(A) ,dSdh ij vkPNknd ugha
(B) ,dSdh vkSj vkPNknd
(C) ,dSdh ugha ij vkPNknd
(D) u ,dSdh] u vkPNknd
Let f : R → R is defined as f (x) = x3 then f is :
(A) One-one, into
(B) One-one, onto
(C) Many-one, onto
(D) Many-one, into

(ii) tan−1 x dk eq[; eku gS % 1
 π
(A) 0, 2  (B) [0, π]

 π π
(C) − 2 , 2  (D) buesa ls dksbZ ugha

The principal value of tan−1 x is :
 π
(A) 0, 2  (B) [0, π]

 π π
(C) − 2 , 2  (D) None of these

4931/(Set : A)

(5) 4931/(Set : A)

;fn X + Y = 
5 2
vkSj X − Y = 
3 6
(iii)  , rks
0 9 −2 1
vkO;wg X dk eku gS % 1

 4 4  8 8 
(A)  −1 5  (B)  − 2 10 
   

1 − 2 
(C) 1 4  (D) buesa ls dksbZ ugha
 

5 2  3 6
If X + Y =   and X − Y =   , then
0 9  −2 1
matrix X is :

 4 4  8 8 
(A)  −1 5  (B)  − 2 10 
   

1 − 2 
(C) 1 4  (D) None of these
 

2 4 2x 4
(iv) ;fn lkjf.kd = , rks x dk eku gS % 1
5 1 6 x

(A) 6 (B) ±6

(C) –6 (D) buesa ls dksbZ ugha
2 4 2x 4
If det. = , then the value of x is :
5 1 6 x

(A) 6 (B) ±6
(C) –6 (D) None of these
4931/(Set : A) P. T. O.

(6) 4931/(Set : A)
(v) sec(tan x ) dk x ds lkis{k vodyu dhft,A 1

Differentiate sec(tan x ) with respect to x.

(vi) Qyu f (x ) = x 3 − 3x + 4 dk mPpre gS] tgk¡ x dk
eku gS % 1

(A) –1 (B) 1

(C) 0 (D) buesa ls dksbZ ugha
f (x ) = x 3 − 3x + 4 has a maxima at x is
equal to :

(A) –1 (B) 1

(C) 0 (D) None of these

(vii) Qyu f (x ) = log(sin x ) vUrjky ftlesa fujarj
Ðkleku gS] og gS % 1

 π π 
(A)  0,  (B)  , π
 2 2 

(C) (0, π) (D) buesa ls dksbZ ughas
f (x ) = log(sin x ) is strictly decreasing in
interval :

 π π 
(A)  0,  (B)  , π
 2 2 

(C) (0, π) (D) None of these
4931/(Set : A)

(7) 4931/(Set : A)
tan−1 x
(viii) ∫ dx dk eku Kkr dhft,A 1
1+ x2

tan−1 x
Find the value of ∫ dx .
1+ x2
π /2
(ix) ∫ sin3 x cos 2 x dx dk eku Kkr dhft,A 1
− π /2

π /2
Evaluate ∫ sin3 x cos 2 x dx .
− π /2

3
3 d 2y   dy 
2
 dy
(x) vody lehdj.k x
 2   dx 
+ +x +y = 0
 dx   dx 

dh ?kkr vkSj dksfV Kkr dhft,A 1

Find the degree and order of the differential
3
3
y  2 2
d  dy  dy
equation x +  +x +y = 0.
 dx 
2
 dx  dx
 

dy
(xi) vody lehdj.k (1 + x 2 ) = (1 + y 2 ) dks gy
dx

dhft,A 1

Solve the differential equation :

dy
(1 + x 2 ) = (1 + y 2 )
dx

4931/(Set : A) P. T. O.

(8) 4931/(Set : A)
(xii) ,d FkSys esa 4 lQsn vkSj 6 dkyh xsansa gSaA nks xsansa
izfrLFkkiu ds lkFk ;kn`fPNd fudkyh tkrh gSaA nksuksa xsan
ds 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 both the
balls are black.

(xiii) A vkSj B nks Lora= ?kVuk,¡ gSaA ;fn P (A ) = 0.3 vkSj
P (B ) = 0.4] rks P (A/B ) dk eku Kkr dhft,A 1

A and B are independent event such that
P (A ) = 0.3 and P (B ) = 0.4, find the P (A/B ).

(xiv) ,d ;kn`PN;k pj X dk izkf;drk caVu fuEufyf[kr gS % 1

X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k2 2k2 7k2 + k

k dk eku Kkr dhft,A

A random variable X has the following
probability distribution :

X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k2 2k2 7k2 + k

Find k.

4931/(Set : A)

(9) 4931/(Set : A)

(xv) lfn'kksa a→ = 2iˆ + 2 ˆj − 5kˆ vkSj b→ = ˆj − kˆ ds ;ksx dh
fn'kk esa bdkbZ lfn'k (unit vector) Kkr dhft,A 1

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

(xvi) ml js[kk dk lfn'k lehdj.k Kkr dhft, tks fcUnq
iˆ + 2 ˆj + 3kˆ ls xqtjrh gS vkSj 3iˆ + 2 ˆj − 2kˆ lfn'k
dh fn'kk esa gksA 1

Write the equation of line passing through
the point with position vector iˆ + 2 ˆj + 3kˆ
and in the direction 3iˆ + 2 ˆj − 2kˆ in vector
form.

[k.M – c
SECTION – B

1
2. ;fn f : R → R] f (x ) = (3 − x ) 3 3
}kjk iznf'kZr gS] rks
fof (x) Kkr dhft,A 2
1
3 3
If f : R → R be given by f (x ) = (3 − x ) , find fof (x).

3. fl) dhft, fd cos −1 4 + cos −1 12 = cos −1 33 2
5 13 65
4 12 33
Prove that cos −1 + cos −1 = cos −1
5 13 65

4931/(Set : A) P. T. O.

( 10 ) 4931/(Set : A)
− 2
4. ;fn A =  4  vkSj B = [1 3 − 6], rks (AB )' Kkr
 5 
dhft,A 2

− 2
If A =  4  and B = [1 3 − 6], find (AB )' .
 5 

y +k y y
5. fl) dhft, y y +k y = (3y + k ) k 2 2
y y y +k

y +k y y
Prove that y y +k y = (3y + k ) k 2
y y y +k

6. Kkr dhft, fd fuEufyf[kr Qyu x = 2 ij lrr gS ;k ugha % 2
f (x ) = x 3 − 3, x ≤ 2

= x 2 + 1, x > 2

Find out whether the following function is
continuous or not at x = 2 :
f (x ) = x 3 − 3, x ≤ 2

= x 2 + 1, x > 2

4931/(Set : A)

( 11 ) 4931/(Set : A)
7. ;fn x = a(cos θ + θ sin θ) 2
y = a (sin θ − θ cos θ) ,

rks θ = π ij dy dk eku Kkr dhft,A
4 dx
If x = a (cos θ + θ sin θ)
y = a (sin θ − θ cos θ) ,
dy π
then find , at θ = .
dx 4

x1 1 
8. ∫ e  x − x 2  dx dk eku Kkr dhft,A 2

1 1 
Evaluate ∫ e x  − 2  dx .
x x 

π /2
sin3 x
9. ∫ sin3 x + cos3 x dx dk eku Kkr dhft,A 2
0
π /2
sin3 x
Evaluate ∫ sin3 x + cos3 x dx.
0

10. vody lehdj.k x dy + 2y = x 2 , x ≠ 0 dk lkekU; gy
dx
Kkr dhft,A 2

Find the general solution of the differential
dy
equation x + 2y = x 2 , x ≠ 0.
dx
4931/(Set : A) P. T. O.

( 12 ) 4931/(Set : A)
11. ,d ikls dks 6 ckj Qsadk tkrk gSA le la[;k vkuk lQyrk gSA
4 lQyrk vkus dh izkf;drk Kkr dhft,A 2

A dice is thrown 6 times. If getting an even
number is success, find probability of getting 4
successes.

[k.M – l
SECTION – C

12. lehdj.k tan−1 1 − x = 1 tan−1 x , x > 0 dks gy dhft,A 4
1+ x 2
1− x 1
Solve the equation tan−1 = tan−1 x , x > 0.
1+ x 2

13. ;fn y = (sin x )sin x , 0 < x < π, rks dy Kkr dhft,A 4
dx
dy
If y = (sin x )sin x , 0 < x < π, find .
dx

14. fcUnq t = π 4 ij oØ x = a sin3 t , y = a cos3 t dh Li'kZ
js[kk dk lehdj.k Kkr dhft,A 4

Find the equation of tangent to the curve

x = a sin3 t , y = a cos 3 t at point t = π .
4

4931/(Set : A)

( 13 ) 4931/(Set : A)
15. ,d f=Hkqt ABC ds 'kh"kksZa ds fLFkfr lfn'k (position vector)
A(iˆ + ˆj + 2kˆ ), B (2iˆ + 3 ˆj + 5kˆ ) vkSj C (iˆ + 5 ˆj + 5kˆ ) gS]
rks ∆ABC dk {ks=Qy Kkr dhft,A 4
The vertices of a triangle ABC are given by
position vector A(iˆ + ˆj + 2kˆ ), B (2iˆ + 3 ˆj + 5kˆ ) and
C (iˆ + 5 ˆj + 5kˆ ) . Find its area.

16. fdlh fof'k"V leL;k dks A, B vkSj C }kjk Lora= :i ls gy
djus dh izkf;drk,¡ Øe'k% 1 , 1 vkSj 1 gSaA ;fn rhuksa
2 3 4
Lora= :i ls gy djrs gSa] rks leL;k gy gksus dh izkf;drk
Kkr dhft,A 4

Probability of solving
specific problem
1 1 1
independently by A, B and C are , and . If
2 3 4
they all try the problem independently, find the
probability that problem is solved.

[k.M – n
SECTION – D

17. fuEufyf[kr lehdj.kksa dks vkO;wg fof/k ls gy dhft, % 6

2x + 3y + 3z = 5,

x – 2y + z = – 4,

3x – y – 2z = 3.

4931/(Set : A) P. T. O.

( 14 ) 4931/(Set : A)
Solve the following system of equation by Matrix
method :

2x + 3y + 3z = 5,

x – 2y + z = – 4,

3x – y – 2z = 3.

18. o`Ùk x 2 + y 2 = 4 ls js[kk x + y = 2 }kjk dkVs x;s y?kq {ks=
dk {ks=Qy Kkr dhft,A 6

Find the area of smaller part of the circle
x 2 + y 2 = 4 cut-off by the line x + y = 2 .

vFkok
OR

fl) dhft, fd oØ y 2 = 4x vkSj x 2 = 4y, x = 0, y = 0
x = 4 vkSj y = 4 }kjk cus oxZ dks rhu cjkcj Hkkxksa esa ck¡Vrs

gSaA 6

Prove that the curves y 2 = 4x and x 2 = 4y divide
the area of the square bounded by x = 0, y = 0,
x = 4 and y = 4 in three equal parts.
4931/(Set : A)

( 15 ) 4931/(Set : A)
19. ml lery dk lehdj.k Kkr dhft, tks leryksa

→ →
r .(2iˆ + 2 ˆj − 3kˆ ) = 7 vkSj r .(2iˆ + 5 ˆj + 3kˆ ) = 9 ds izfrPNsn

ls xqtjrk gS vkSj (2, 1, 3) fcUnq ls Hkh xqtjrk gSA 6

Find the equation of the plane passing through the

→
intersection of the planes r .(2iˆ + 2 ˆj − 3kˆ ) = 7 and

→
r .(2iˆ + 5 ˆj + 3kˆ ) = 9 and through the point (2, 1, 3).

vFkok
OR

js[kkvksa → →
r = (iˆ + 2 ˆj + kˆ ) + λ(iˆ − ˆj + kˆ ) vkSj r = (2iˆ − ˆj + kˆ )

+ µ(2iˆ + ˆj + 2kˆ ) ds chp dh fuEure nwjh (S.D.) 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 ) .

4931/(Set : A) P. T. O.

( 16 ) 4931/(Set : A)
20. fuEu jSf[kd izksxzkeu leL;k (L.P.P.) dks xzkQh; fof/k }kjk gy
dhft, % 6

U;wure % Z = 18x + 10y
O;ojks/kksa ds vUrxZr %
4x + y ≥ 20,

2x + 3y ≥ 30,

x, y ≥ 0.

Solve the linear programming problem by
graphic method

Minimize : Z = 18x + 10y under the constraints :

4x + y ≥ 20,

2x + 3y ≥ 30,

x, y ≥ 0.

s
4931/(Set : A)

CLASS : 12th (Sr. Secondary) Code No. 4931
Series : SS-M/2020
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.

4931/(Set : B) P. T. O.

(2) 4931/(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
4931/(Set : B)

(3) 4931/(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.

4931/(Set : B) P. T. O.

(4) 4931/(Set : B)
[k.M – v
SECTION – A

1. (i) ;fn Qyu f : R → R+ tks f (x ) = x 4 }kjk ifjHkkf"kr
gS] rks f gS % 1
(A) ,dSdh vkSj vkPNknd
(B) ,dSdh] vkPNknd ugha
(C) ,dSdh ugha ij vkPNknd
(D) u ,dSdh] u vkPNknd

Let f : R → R+ defined by f (x ) = x 4 then f is :
(A) One-one, onto
(B) One-one, into
(C) Many-one, onto
(D) Many-one, into

(ii) cos −1 x dk eq[; eku gS % 1
 π π
(A) [0, π] (B) − 2 , 2 

 π π
(C) − ,  (D) buesa ls dksbZ ugha
 2 2
The principal value of cos −1 x is :
 π π
(A) [0, π] (B) − , 
 2 2
 π π
(C) − ,  (D) None of these
 2 2
4931/(Set : B)

(5) 4931/(Set : B)

;fn 2X + Y = 
0
vkSj 2X − Y = 
1 3 4
(iii)  ,
−3 2  −1 2 
rks X dk eku gS % 1
 4 4  1 1
(A) −4 4 (B)  −1 1
   
 −1 − 2 
(C)  −1 0  (D) buesa ls dksbZ ugha
 
 1 0  3 4
If 2X + Y =   and 2X − Y =  ,
−3 2  −1 2 
then X is equal to :
 4 4  1 1
(A) −4 4 (B)  −1 1
   
 −1 − 2 
(C)  −1 0  (D) None of these
 

3x 5 2 0
(iv) ;fn = , rks x dk eku gS % 1
1 x −1 2

(A) +2 (B) 2
3
(C) ± 3 (D) 0
3x 5 2 0
If = , then the value of x is :
1 x −1 2

(A) +2 (B) 2
3
(C) ± 3 (D) 0

4931/(Set : B) P. T. O.

(6) 4931/(Set : B)
(v) log(sec x ) dk x ds lkis{k vodyu dhft,A 1

Differentiate log(sec x ) with respect to x.

(vi) Qyu f (x ) = sin x + cos x dk LFkkuh; mPpre gS]
tgk¡ x dk eku gS % 1
π
(A) 0 (B)
6
π π
(C) (D)
4 2
f (x ) = sin x + cos x has a local maxima at x
is equal to :
π
(A) 0 (B)
6
π π
(C) (D)
4 2

(vii) Qyu f (x ) = log(sin x ) tgk¡ fujarj o/kZeku gS og
varjky gS % 1

 π π 
(A)  0,  (B)  , π
 2 2 
(C) (0, π) (D) buesa ls dksbZ ughas
f (x ) = log(sin x ) is strictly increasing in the
interval :
 π π 
(A)  0,  (B)  , π
 2 2 
(C) (0, π) (D) None of these
4931/(Set : B)

(7) 4931/(Set : B)
sin−1 x
(viii) ∫ dx dk eku Kkr dhft,A 1
1− x2

sin−1 x
Evaluate ∫ dx .
1− x2

1 x3
(ix) ∫ dx dk eku Kkr dhft,A 1
−1 1 + x 2

1 x3
Evaluate ∫ dx .
−1 1 + x 2

2
d 4y  dy 
(x) vody lehdj.k 4
− 5  − 6y = log x dh
dx  dx 

dksfV vkSj ?kkr Kkr dhft,A 1

Find the degree and order of the differential
2
d 4y  dy 
equation 4
− 5  − 6y = log x.
dx  dx 

(xi) vody lehdj.k dy = y tan x dks gy dhft,A 1
dx

Solve the differential equation :

dy
= y tan x
dx

4931/(Set : B) P. T. O.

(8) 4931/(Set : B)
(xii) fcuk izfrLFkkfir fd, nks xsan ,d ds ckn ,d ml FkSys ls
fudkyh tkrh gSa ftlesa 4 lQsn vkSj 6 dkyh xsansa gSaA
nksuksa xsanksa ds dkyh gksus dh izkf;drk Kkr dhft,A 1

A bag contains 4 white and 6 black balls.
Two balls are drawn at random one after
the other without replacement. Find the
probability that both the balls are black.

3
(xiii) ;fn P ( A ) = ] P (B ) = 1 vkSj P (A ∩ B ) = 1 ,
5 5 10
rks P (B/A ) Kkr dhft,A 1
3 1 1
If P (A) = , P(B ) = and P (A ∩ B ) = , find
5 5 10
P (B/A ).

(xiv) ;fn ,d ;kn`PN;k pj X dk izkf;drk caVu fuEufyf[kr
gS % 1

X 0 1 2 3 4
P(X) 0.1 k 2k 2k k
rks k dk eku Kkr dhft,A
The probability distribution of X is given
below :

X 0 1 2 3 4
P(X) 0.1 k 2k 2k k
Find k.
4931/(Set : B)

(9) 4931/(Set : B)
(xv) PQ lfn'k dh fn'kk esa bdkbZ lfn'k Kkr dhft, tgk¡
fcUnq P vkSj Q Øe'k% (1, 2, 3) vkSj (4, 5, 6) gSaA 1
Find a unit vector in the direction of PQ ,
where points P and Q are (1, 2, 3) and (4, 5, 6)
respectively.

(xvi) fcUnq (–3, 5, –6) ls xqtjus okyh vkSj fn'kk 2iˆ + 4 ˆj + 2kˆ
ds lekarj js[kk dk lehdj.k Kkr dhft,A 1

Find the equation of line passing through
the point (–3, 5, –6) and parallel to the
direction 2iˆ + 4 ˆj + 2kˆ.

[k.M – c
SECTION – B

1
2. ;fn f : R → R] f (x ) = (3 − x ) }kjk ifjHkkf"kr gS] rks
5 5

fof (x) Kkr dhft,A 2
1
5 5
If f : R → R be given by f (x ) = (3 − x ) , then find
fof (x).

3. fl) dhft, fd tan−1 1 − cos x = x , tgk¡ (0 < x < π) 2
1 + cos x 2

1 − cos x x
Prove that tan−1 = , where (0 < x < π)
1 + cos x 2

4931/(Set : B) P. T. O.

( 10 ) 4931/(Set : B)
3
4. ;fn A =  4  vkSj B = [1 − 2 3], rks (AB )' dk eku
− 2

Kkr dhft,A 2

3
If A =  4  and B = [1 − 2 3], then find (AB )' .
− 2

1 1 1
5. fl) dhft, a b c = (a − b ) (b − c ) (c − a ) 2
a b2 c 2
2

1 1 1
Prove that det. a b c = (a − b ) (b − c ) (c − a )
a b2 c 2
2

;fn Qyu f (x ) = 
kx + 1, x ≤5
6. x = 5 ij lrr gS] rks
3x − 5, x > 5

k dk eku Kkr dhft,A 2

 kx + 1, x ≤ 5
If f (x ) =  is continuous at x = 5,
3x − 5, x > 5
then find k.
4931/(Set : B)

( 11 ) 4931/(Set : B)
7. ;fn x = a(sin θ − θ cos θ) vkSj y = a (cos θ + θ sin θ) , rks
dy
dk eku θ = π ij Kkr dhft,A 2
dx 4
If x = a (sin θ − θ cos θ) and y = a (cos θ + θ sin θ),
dy π
find , at θ = .
dx 4

x 1
8. ∫ e  log x + x  dx dk eku Kkr dhft,A 2

 1
Evaluate ∫ e x  log x +  dx .
 x

a x
9. ∫ 0 x + a − x dx dk eku Kkr dhft,A 2

a x
Evaluate ∫ dx .
0 x + a−x

10. a vkSj b dks foyqIr djds oØ xy = ae x + be − x dk
vody lehdj.k cukb,A 2

Find the differential equation of the family of
curves given by xy = ae x + be − x by eliminating a
and b.

11. ,d ikls dks 6 ckj Qsadus ij 5 ds nks ckj vkus dh izkf;drk
Kkr dhft,A 2
Find the probability of getting 5 twice in 6
throws of a die.
4931/(Set : B) P. T. O.

( 12 ) 4931/(Set : B)
[k.M – l
SECTION – C

12. lehdj.k tan−1 1 − x  = 1 tan−1 x dks gy dhft,A 4
1 + x  2

1 − x  1
Solve the equation tan−1 −1
 = tan x .
 1 + x  2

13. (log x )log x , x > 1 dk x ds lkis{k vodyu dhft,A 4

Differentiate (log x )log x , x > 1 with respect to x.

14. fcUnq t = π ij oØ x = a cos 4 t , y = a sin4 t dh Li'kZ
4
js[kk dk lehdj.k Kkr dhft,A 4

If x = a cos 4 t , y = a sin4 t . Find the equation of
π
tangent to this curve at t = .
4

15. fcUnqvksa A(1, 1, 2), B(2, 3, 5) vkSj C(1, 5, 5) dks feykus
ls cuus okys f=Hkqt dk {ks=Qy Kkr dhft,A 4

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

4931/(Set : B)

( 13 ) 4931/(Set : B)
16. ,d cDls esa 5 yky vkSj 3 dkyh xsansa gSaA nwljs cDls esa 3
yky vkSj 5 dkyh xsansa gSaA ,d cDlk ;kn`PN;k pqudj mlesa ls
,d xsan fudkyh tkrh gSA ;fn og yky xsan gS] rks mlds igys
cDls ls fudyus dh izkf;drk Kkr dhft,A 4

An urn contains 5 red and 3 black balls and
second urn contains 3 red and 5 black balls. An
urn is selected at random and a ball is drawn
from it. If the ball is black, find the probability
that it is from 1st urn.

[k.M – n
SECTION – D

17. fuEufyf[kr lehdj.kksa dks vkO;wg fof/k ls gy dhft, % 6

x – y + z = 4,

2x + y – 3z = 0,

x + y + z = 2.

Solve the following system of equation by Matrix
method :

x – y + z = 4,

2x + y – 3z = 0,

x + y + z = 2.

4931/(Set : B) P. T. O.

( 14 ) 4931/(Set : B)
18. o`Ùk x 2 + y 2 = 9 ls js[kk x + y = 3 }kjk dkVs x;s y?kq

Hkkx dk {ks=Qy Kkr dhft,A 6

Find the area of smaller part of the circle

x 2 + y 2 = 9 cut-off by the line x + y = 3 .

vFkok

OR

oØ y = cos x vkSj x = 0] x = 2π ls f?kjs {ks= dk {ks=Qy

Kkr dhft,A 6

Find the area bounded by the curve y = cos x

between x = 0 to x = 2π.

→ →
19. leryksa r .(3iˆ − ˆj + 2kˆ ) − 4 = 0 vkSj r .(iˆ + ˆj − kˆ ) − 2 = 0

ds izfrPNsn ls tkus okys vkSj fcUnq (2, 2, 1) ls xqtjus okys

lery dk lehdj.k Kkr dhft,A 6

4931/(Set : B)

( 15 ) 4931/(Set : B)
Find the equation of the plane through the

→
intersection of the planes r .(3iˆ − ˆj + 2kˆ ) − 4 = 0 and

→
r .(iˆ + ˆj − kˆ ) − 2 = 0 and the point (2, 2, 1).

vFkok
OR

js[kkvksa x + 1 = y + 1 = z + 1 vkSj x − 3 = y − 5 = z − 7
7 −6 1 1 −2 1

ds chp dh y?kqÙke nwjh (Shortest distance) Kkr dhft,A 6
Find the shortest distance between the lines
x +1 y +1 z +1 x −3 y −5 z −7
= = and = = .
7 −6 1 1 −2 1

20. fuEu jSf[kd izksxzkeu leL;k dks xzkfQd fof/k }kjk gy dhft, %
vojks/kksa 6

2x + y ≥ 8,

x + 2y ≥ 10,

x ≥ 0, y ≥ 0

ds vUrxZr Z = 5x + 7y dk U;wurehdj.k dhft,A

4931/(Set : B) P. T. O.

( 16 ) 4931/(Set : B)
Solve the following linear programming problem
graphically constraints :

2x + y ≥ 8,

x + 2y ≥ 10,

x ≥ 0, y ≥ 0

and minimize Z = 5 x + 7y

s

4931/(Set : B)

CLASS : 12th (Sr. Secondary) Code No. 4931
Series : SS-M/2020
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.

4931/(Set : C) P. T. O.

(2) 4931/(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
4931/(Set : C)

(3) 4931/(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.

4931/(Set : C) P. T. O.

(4) 4931/(Set : C)
[k.M – v
SECTION – A

1. (i) ;fn Qyu f : N → N tks f (x) = x3 }kjk ifjHkkf"kr gS]
rks f gS % 1

(A) ,dSdh vkSj vkPNknd
(B) ,dSdh ij vkPNknd ugha
(C) ,dSdh ugha ij vkPNknd
(D) u ,dSdh] u vkPNknd
Let f : N → N defined as f (x) = x3 then f is :
(A) One-one, onto
(B) One-one, into
(C) Many-one, onto
(D) Many-one, into

(ii) sin−1 x dk eq[; eku gS % 1

 π π
(A) [0, π] (B) − 2 , 2 

(C) [0, 2π] (D) buesa ls dksbZ ugha
The principal value of sin−1 x is :
 π π
(A) [0, π] (B) − 2 , 2 

(C) [0, 2π] (D) None of these
4931/(Set : C)

(5) 4931/(Set : C)

;fn 2X + 3Y = 
3 2
vkSj 2X − 3Y = 
1 0
(iii)  ,
1 4  −3 2
rks vkO;wg Y dk eku gS % 1

1 4 2 2 2
(A) (B)
5  − 2 6  4 2
 

12 2
(C) (D) buesa ls dksbZ ugha
6  4 2 

3 2  1 0
If 2X + 3Y =   and 2X − 3Y =  ,
1 4  −3 2
then matrix Y is :

1 4 2 2 2
(A) (B)
5  − 2 6  4 2
 

12 2
(C) (D) None of these
6  4 2 

x 2 6 2
(iv) ;fn = , rks x dk eku gS % 1
18 x 18 6

(A) 6 (B) –6
(C) ±6 (D) 0

x 2 6 2
If = , then the value of x is :
18 x 18 6

(A) 6 (B) –6
(C) ±6 (D) 0

4931/(Set : C) P. T. O.

(6) 4931/(Set : C)

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

Differentiate e sin x with respect to x.

(vi) Qyu f (x ) = sin x − cos x , 0 < x < 2π dk LFkkuh;
mPpre gS tgk¡ x dk eku gS % 1
π 3π
(A) (B)
4 4
5π
(C) (D) buesa ls dksbZ ugha
4
The function f (x ) = sin x − cos x , 0 < x < 2π
has a local maxima at x is equal to :
π 3π
(A) (B)
4 4
5π
(C) (D) None of these
4

(vii) Qyu f (x ) = log(sin x ) ftl varjky esa fujarj
Ðkleku gS] og gS % 1

 π π 
(A)  0,  (B)  , π
 2 2 
(C) (0, π) (D) buesa ls dksbZ ughas
f (x ) = log(sin x ) is strictly decreasing in
interval :
 π π 
(A)  0,  (B)  , π
 2 2 
(C) (0, π) (D) None of these
4931/(Set : C)

(7) 4931/(Set : C)
−1
e tan x
(viii) ∫ dx dk eku Kkr dhft,A 1
1+ x2

−1
e tan x
Evaluate ∫ dx .
1+ x2

π/4
(ix) ∫ tan3 x dx dk eku Kkr dhft,A 1
−π /4

π/4
Evaluate ∫ tan3 x dx .
−π /4

2
d 2y  dy 
(x) vody lehdj.k 2  dx 
+ + 2y = 0 dh dksfV
dx  

vkSj ?kkr Kkr dhft,A 1

Find the order and degree of the differential
2
d 2y
 dy 
2  dx 
equation + + 2y = 0 .
dx  

(xi) vody lehdj.k dy = − 4xy 2 dks gy dhft,A 1
dx

Solve the differential equation :

dy
= − 4xy 2
dx

4931/(Set : C) P. T. O.

(8) 4931/(Set : C)
(xii) ,d tksM+k ikls dks Qsadus ij le vHkkT; la[;k (Even
Prime) izR;sd ikls ij vkus dh izkf;drk Kkr dhft,A 1

Find the probability of getting an even prime
number on each die, when a pair of dice is
rolled.

(xiii) ;fn E vkSj nks ?kVuk,¡ bl izdkj gSa fd
F
P (E ) = 0.6, P (F ) = 0.3 vkSj P (E ∩ F ) = 0.2, rks
P (E/F ) Kkr dhft,A 1

If E and F be two events such that
P (E ) = 0.6, P (F ) = 0.3 and P (E ∩ F ) = 0.2,
then find P (E/F ).

(xiv) ;fn ,d ;kn`PN;k pj dk izkf;drk caVu bl izdkj gS % 1

X 0 1 2 3 4 5 6

P(X) 0.2 k 2k 2k 3k k 0.1

rks k dk eku Kkr dhft,A
A random variable X has the following
probability distribution :

X 0 1 2 3 4 5 6

P(X) 0.2 k 2k 2k 3k k 0.1

Determine the value of k.

4931/(Set : C)

(9) 4931/(Set : C)
r r
(xv) lfn'kksa a = 2iˆ + 2 ˆj − 5kˆ vkSj b = ˆj − kˆ ds varj dh
fn'kk esa bdkbZ lfn'k Kkr dhft,A 1

Find a unit vector in the direction of the
r
difference of vectors a = 2iˆ + 2 ˆj − 5kˆ and
r
b = ˆj − kˆ .

(xvi) fcUnq (5, 2, –4) ls xqtjus okyh vkSj lfn'k
3iˆ + 2 ˆj − 8kˆ ds lekarj js[kk dk lehdj.k Kkr
dhft,A 1

Find the equation of the line passing
through the point (5, 2, –4) and parallel to
the vector 3iˆ + 2 ˆj − 8kˆ .

[k.M – c
SECTION – B

2. ;fn f (x ) = 4x + 3 , x ≠ 2 , rks n'kkZb, fd fof (x) = x
6x − 4 3
izR;sd x ≠ 2 ds fy,A 2
3
4x + 3 2
If f (x ) = , x ≠ , show that fof (x) = x for all
6x − 4 3
2
x ≠ .
3

4931/(Set : C) P. T. O.

( 10 ) 4931/(Set : C)
 cos x − sin x  π
3. fl) dhft, fd tan−1  = − x, tgk¡
 cos x + sin x  4
0<x<π 2
 cos x − sin x  π
Prove that tan−1  = − x, for 0 < x < π.
 cos x + sin x  4
4
4. ;fn A = − 2 vkSj B = [3 1 − 6], rks (AB )' Kkr
 5 
dhft,A 2
4
If A = − 2 and B = [3 1 − 6], then find (AB )' .
 5 
b +c a a
5. fl) dhft, b c +a b = 4abc 2
c c a +b
b +c a a
Prove that b c +a b = 4abc
c c a +b
6. Kkr dhft, fd fuEufyf[kr Qyu x = 0 ij lrr gS ;k ugha
|x |
f (x ) = ,x ≠ 0
x 2
= 0, x = 0
Find out whether the following function is
continuous or not at x = 0 :
|x |
f (x ) = ,x ≠ 0
x
= 0, x = 0
4931/(Set : C)

( 11 ) 4931/(Set : C)
7. ;fn x = 2 cos θ − cos 2θ vkSj y = 2 sin θ − sin 2θ, rks

fl) dhft, dy = tan 3θ . 2
dx 2

If x = 2 cos θ − cos 2θ and y = 2 sin θ − sin 2θ,
dy 3θ
then prove that = tan .
dx 2

x 1 
8. ∫ e  tan
−1
x+ 2
dx dk eku Kkr dhft,A 2
1+ x 

 1 
Evaluate ∫ e x  tan−1 x +  dx .
 1+ x2 

π /2 sin x
9. ∫0 dx dk eku Kkr dhft,A 2
sin x + cos x

π /2 sin x
Evaluate ∫ dx .
0 sin x + cos x

10. vody lehdj.k dy + (sec x )y = tan x , (0 < x < π ) dks
dx 2

gy dhft,A 2

Solve the differential equation :
dy π
+ (sec x )y = tan x , (0 < x < )
dx 2
4931/(Set : C) P. T. O.

( 12 ) 4931/(Set : C)
11. 52 iÙkksa dh vPNh rjg QsaVh xbZ rk'k dh xìh ls 4 iÙks ,d
ds ckn ,d izfrLFkkfir djds fudkys x;s gSaA 3 gqdqe ds iÙks
vkus dh izkf;drk Kkr dhft,A 2
4 cards are drawn with replacement one by one
from a well shuffled pack of 52 cards. Find the
probability of getting 3 spades.

[k.M – l
SECTION – C

12. lehdj.k tan−1 2x + tan−1 3x = π dks gy dhft,A 4
4

π
Solve the equation tan−1 2x + tan−1 3x = .
4

13. ;fn y = (x cos x )x , rks dy Kkr dhft,A 4
dx

dy
If y = (x cos x )x , find .
dx

14. oØ ay 2 = x 3 ds fcUnq (am 2, am 3 ) ij Li'kZ js[kk dk
lehdj.k Kkr dhft,A 4

Find the equation of tangent to the curve

ay 2 = x 3 at the point (am 2 , am 3 ) .

4931/(Set : C)

( 13 ) 4931/(Set : C)

15. f=Hkqt ABC ds 'kh"kZ A(1, 2, 3), B(–1, 0, 0) vkSj C(0, 1, 2)
gSaA f=Hkqt dk {ks=Qy Kkr dhft,A 4

The vertices of ∆ABC are A(1, 2, 3), B(–1, 0, 0)

and C(0, 1, 2). Find its area.

16. ,d dkj[kkus esa nks e'khu A vkSj B gSaA A dqy mRiknu dk
60% vkSj B 40% mRiknu djrh gSA A ds mRiknu dk 1%

vkSj B dk 2% [kjkc gSA ;fn dqy mRiknu ls ,d oLrq pquh
tk;s vkSj og [kjkc gS] rks mlds A ds }kjk mRikfnr gksus dh
izkf;drk Kkr dhft,A 4

A factory has two machines A and B. Past record

shows that A produces 60% and B produces

40% of items. Further 1% of machine A and 2%

of machine B produces defective items. If from the

total production 1 item is selected and is found

defective, find the probability that it was

produced by machine A.

4931/(Set : C) P. T. O.

( 14 ) 4931/(Set : C)
[k.M – n
SECTION – D

17. fuEufyf[kr lehdj.k fudk; dks vkO;wg fof/k ls gy dhft, % 6
2x + y + z = 1,
2x – 4y – 2z = 3,
3y – 5z = 9.
Solve the following system of equations by Matrix
method :
2x + y + z = 1,
2x – 4y – 2z = 3,
3y – 5z = 9.

x 2 y2 x y
18. nh?kZo`Ùk + = 1 vkSj js[kk + = 1 ls f?kjs y?kq {ks=
9 4 3 2
dk {ks=Qy Kkr dhft,A 6
Find the area of smaller region bounded by the
x 2 y2 x y
ellipse + = 1 and line + = 1.
9 4 3 2
vFkok
OR

ijoy; 4y = 3x 2 vkSj js[kk 2y = 3x + 12 ls f?kjs {ks= dk
{ks=Qy Kkr dhft,A 6

Find the area enclosed by the parabola 4y = 3x 2
and line 2y = 3x + 12 .

4931/(Set : C)

( 15 ) 4931/(Set : C)

19. ml lery dk lehdj.k Kkr dhft, tks leryksa →
r .(iˆ + ˆj + kˆ ) = 1

vkSj →
r .(2iˆ + 3 ˆj − kˆ ) + 4 = 0 ds izfrPNsn ls xqtjrk gS vkSj

fcUnq (1, 1, 1) ls Hkh xqtjrk gSA 6

Find the equation of the plane through the line
→
of intersection of the planes r .(iˆ + ˆj + kˆ ) = 1 and

→
r .(2iˆ + 3 ˆj − kˆ ) + 4 = 0 and through the point (1, 1, 1).

vFkok
OR

js[kk,¡ x − 1 = y − 2 = z − 1 vkSj x − 2 = y + 1 = z + 1
1 −1 1 2 1 2
ds chp dh y?kqÙke nwjh Kkr dhft,A 6
Find the shortest distance between the lines
x −1 y − 2 z −1 x − 2 y +1 z +1
= = and = = .
1 −1 1 2 1 2

20. fuEufyf[kr jSf[kd izksxzkeu leL;k dks xzkfQd fof/k ls gy
dhft, % 6

mPprehdj.k dhft, Z = 5x + 3y
vojks/kksa ds vUrxZr 3x + 5y ≤ 15,

5x + 2y ≤ 10,

x ≥ 0, y ≥ 0.
4931/(Set : C) P. T. O.

( 16 ) 4931/(Set : C)
Solve the following linear programming problem
graphically :
Maximize Z = 5x + 3y
Subject to constraints
3x + 5y ≤ 15,
5x + 2y ≤ 10,
x ≥ 0, y ≥ 0.

s

4931/(Set : C)

CLASS : 12th (Sr. Secondary) Code No. 4931
Series : SS-M/2020
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.

4931/(Set : D) P. T. O.

(2) 4931/(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
4931/(Set : D)

(3) 4931/(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.

4931/(Set : D) P. T. O.

(4) 4931/(Set : D)
[k.M – v
SECTION – A

1. (i) eku yhft, f : R → R, f (x) = 3x }kjk ifjHkkf"kr gS]
rks lgh mÙkj dk p;u dhft, % 1
(A) f ,dSdh] vkPNknd gS
(B) f cgq,dh] vkPNknd gS
(C) f ,dSdh gS] vkPNknd ugha gS
(D) f u ,dSdh gS] u vkPNknd
Let f : R → R be defined as f (x) = 3x choose
the correct answer :
(A) f is one-one, onto
(B) f is many one, onto
(C) f is one-one, into
(D) f is many one, into

(ii) tan−1 x dk eq[; eku gS % 1
 π π  π π
(A) − 2 , 2  (B) − , 
 2 2
 π
(C) [0, π] (D)  0, 
 2
The principal value of tan−1 x is :
 π π  π π
(A) − ,  (B)  − , 
 2 2  2 2
 π
(C) [0, π] (D)  0, 
 2
4931/(Set : D)

(5) 4931/(Set : D)

;fn 2X + 3Y = 
2 3
vkSj Y = 
3 2
(iii)   , rks X
4 0 1 4
dk eku gS % 1

−4 −3  1  − 7 −1 
(A)  1 − 12  (B)
  2  2 − 8 

1−7 −3 
(C) (D) buesa ls dksbZ ugha
2  1 − 12 

2 3 3 2
If 2X + 3Y =   and Y =   , then
4 0 1 4
X is equal to :

−4 −3  1  − 7 −1 
(A)  1 − 12  (B)
  2  2 − 8 

1−7 −3 
(C) (D) None of these
2  1 − 12 

3 x 3 2
(iv) ;fn = , rks x dk eku gS % 1
x 1 4 1

(A) 2 (B) 4

(C) ±2 2 (D) buesa ls dksbZ ugha
3 x 3 2
If = , then the value of x is :
x 1 4 1

(A) 2 (B) 4

(C) ±2 2 (D) None of these
4931/(Set : D) P. T. O.

(6) 4931/(Set : D)

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

Differentiate e sin x with respect to x.

(vi) Qyu f (x ) = cos x − sin x dk LFkkuh; mPpre gS
tgk¡ x dk eku gS % 1
π 3π
(A) (B)
4 4
5π 7π
(C) (D)
4 4
f (x ) = cos x − sin x has a local maxima at x
is equal to :
π 3π
(A) (B)
4 4
5π 7π
(C) (D)
4 4

(vii) Qyu f (x ) = log(cos x ) tgk¡ fujarj o/kZeku gS] og
varjky gS % 1
 π
(A)  0,  (B) (0, π)
 2
 π π
(C)  − ,  (D) buesa ls dksbZ ughas
 2 2
f (x ) = log(cos x ) is strictly increasing in the
interval :
 π
(A)  0,  (B) (0, π)
 2
 π π
(C) − ,  (D) None of these
 2 2
4931/(Set : D)

(7) 4931/(Set : D)
cos x
(viii) ∫ dx dk eku Kkr dhft,A 1
1 + sin x

cos x
Evaluate ∫ dx .
1 + sin x

π /2
(ix) ∫
−π /2
sin5 x dx dk eku Kkr dhft,A 1

π /2
Evaluate ∫ sin5 x dx .
−π /2

4 2
(x) vody lehdj.k  dy  + 3y d y2 = 0 dh ?kkr vkSj
 dx  dx

dksfV Kkr dhft,A 1

Find the degree and order of the differential
4
 dy  d 2y
equation   + 3y 2 = 0 .
 dx  dx

2
(xi) vody lehdj.k dy = 1 + y 2 dks gy dhft,A 1
dx 1+ x

Solve the differential equation :

dy 1 + y 2
=
dx 1 + x 2

4931/(Set : D) P. T. O.

(8) 4931/(Set : D)
(xii) ,d FkSys esa 10 lQsn vkSj 15 dkyh xsansa gSaA nks xsan fcuk

izfrLFkkfir fd;s ,d-,d dj fudkyh tkrh gSA 1 lQsn
vkSj 1 dkyh xsan fudyus dh izkf;drk Kkr dhft,A 1

A bag contains 10 white and 15 black balls.
Two balls are drawn one by one without
replacement. Find the probability of one
white and one black ball.

5 5
(xiii) ;fn P (A) = , P (B ) = , A vkSj B Lora= ?kVuk,¡
26 13
gSa] rks P (A/B ) Kkr dhft,A 1

5 5
If P (A ) = , P (B ) = , A and B are
26 13
independent, find P (A/B ).

(xiv) nks U;k¸; flDdksa dks Qsadus ij fpr (Head) vkus dh

la[;k dk izkf;drk caVu Kkr dhft,A 1

Find the probability distribution of number

of heads in two tosses of a coin.

4931/(Set : D)

(9) 4931/(Set : D)
r r
(xv) ;fn lfn'k a vkSj b bl izdkj gSa fd ar = 5iˆ + 2ˆj − 4kˆ
r r r
vkSj b = 3iˆ + 2 ˆj − 4kˆ, rks a + b ds lekarj bdkbZ
lfn'k Kkr dhft,A 1
r r
If a and b are two vectors such that
r r
a = 5iˆ + 2 ˆj − 4kˆ and b = 3iˆ + 2 ˆj − 4kˆ . Find the
r r
unit vector parallel to a + b .

(xvi) fcUnq (1, 2, 3) ls xqtjus okyh vkSj lfn'k iˆ + 2 ˆj − kˆ
ds lekarj js[kk dk lehdj.k Kkr dhft,A 1
Find the equation of the line passing
through the point (1, 2, 3) and parallel to
the vector iˆ + 2 ˆj − kˆ .

[k.M – c
SECTION – B

2. ;fn f (x ) = x 2 + 4 vkSj f : R + → [4, ∞ ), rks f −1(x )

Kkr dhft,A 2

If f (x ) = x 2 + 4 and f : R + → [4, ∞ ), find f −1(x ) .

x x
3. ;fn |x| < a, rks fl) dhft, tan−1 = sin−1 2
2
a −x 2 a

x x
Prove that tan−1 = sin−1 , |x| < a.
a2 − x2 a

4931/(Set : D) P. T. O.

( 10 ) 4931/(Set : D)
1
4. ;fn A = 2 vkSj B = [2 − 3 1], rks (AB )' dk eku
1
Kkr dhft,A 2
1
If A = 2 and B = [2 − 3 1], then find (AB )' .
1

1 a bc
5. fl) dhft, 1 b ca = (a − b ) (b − c ) (c − a ) 2
1 c ab
1 a bc
Prove that 1 b ca = (a − b ) (b − c ) (c − a ) .
1 c ab
6. ;fn f (x ) = λ(x 2 − 2x ), x ≤ 1
= 4x + 1 , x > 1
λ ds fdl eku ds fy;s f (x) ,d lrr Qyu gSA 2
2
If f (x ) = λ(x − 2x ), x ≤ 1
= 4x + 1 , x > 1
for what value of λ the function f(x) is
continuous.

7. ;fn x = a(cos t + t sin t ) vkSj y = a(sin t − t cos t ), rks
dy
dk t = π ij eku Kkr dhft,A 2
dx 3
If x = a (cos t + t sin t ) and y = a (sin t − t cos t ), then
dy π
find at t = .
dx 3
4931/(Set : D)

( 11 ) 4931/(Set : D)

∫ e (sec x + sec x tan x )dx dk eku Kkr dhft,A
x
8. 2

Evaluate ∫ e x (sec x + sec x tan x )dx .

5 x
9. ∫ 0 x + 5 − x dx dk eku Kkr dhft,A 2

5 x
Evaluate ∫ dx .
0 x + 5−x

dy  2x  cot x
10. vody lehdj.k + y = dks gy
dx  1 + x 2  1+ x2

dhft,A 2

Solve the differential equation :

dy  2x  cot x
+ 2 y =
dx  1 + x  1+ x2

11. ;fn fdlh <sj esa 5% [kjkc phts gSaA 6 oLrqvksa ds U;kn'kZ
(sample) esa 3 [kjkc phtksa ds gksus dh izkf;drk Kkr
dhft,A 2

There are 5% defective items in a bulk of items.
Find the probability that in a sample of 6 items
3 are defective.

4931/(Set : D) P. T. O.

( 12 ) 4931/(Set : D)
[k.M – l
SECTION – C

12. lehdj.k 2 tan−1(cos x ) = tan−1 (2 cosec x) dks gy dhft,]
x≠0A 4

Solve the equation 2 tan−1(cos x ) = tan−1 (2 cosec x)

x ≠ 0.

13. (x )x cos x dk x ds lkis{k vodyu dhft,A 4

Differentiate (x )x cos x with respect to x.

14. oØ y = x 2 − 2x + 7 dh ml Li'kZ js[kk dk lehdj.k Kkr
dhft, tks 2x − y + 9 = 0 ds lekarj gSA 4
Find the equation of tangent to the curve
y = x 2 − 2x + 7, which is parallel to the line
2x − y + 9 = 0 .

15. ∆ABC dk {ks=Qy Kkr dhft, ;fn 'kh"kksZa ds fLFkfr lfn'k
(position vectors) Øe'k% A(2iˆ − ˆj + kˆ ), B (iˆ − 3 ˆj − 5kˆ )
vkSj C (3iˆ − 4 ˆj − 4kˆ ) gSA 4
Find the area of the triangle ∆ABC whose vertices
have the position vectors as A(2iˆ − ˆj + kˆ ),
B (iˆ − 3 ˆj − 5kˆ ) and C (3iˆ − 4 ˆj − 4kˆ ) .

4931/(Set : D)

( 13 ) 4931/(Set : D)
16. ,d FkSys esa 4 yky vkSj 4 dkyh xsansa gSaA nwljs FkSys esa 2 yky
vkSj 6 dkyh xsansa gSaA ;fn ;kn`PN;k ,d FkSyk pqudj mlesa ls
,d xsan fudkyh tk;s vkSj og yky gks] rks ml xsan ds igys
FkSys ls gksus dh izkf;drk Kkr dhft,A 4

A bag contains 4 red and 4 black balls, and
another contains 2 red and 6 black balls. One of
the two bags is selected at random and a ball is
drawn from it. The ball is found to be red. Find
the probability that ball is drawn from first bag.

[k.M – n
SECTION – D

17. fuEufyf[kr lehdj.k fudk; dks gy dhft, % 6

x – y + 2z = 7,

3x + 4y – 5z = –5,

2x – y + 3z = 12.

Solve the following system of linear equations :

x – y + 2z = 7,

3x + 4y – 5z = –5,

2x – y + 3z = 12.

4931/(Set : D) P. T. O.

( 14 ) 4931/(Set : D)
2
y2
18. nh?kZo`Ùk x + = 1 vkSj js[kk
x y
+ = 1 ls f?kjs y?kq {ks=
4 9 2 3

dk {ks=Qy Kkr dhft,A 6

Find the area of smaller part of the ellipse

x 2 y2 x y
+ = 1 and the line + = 1.
4 9 2 3
vFkok
OR

ijoy; x2 = y vkSj js[kk y = x +2 ls f?kjs {ks= dk
{ks=Qy Kkr dhft,A 6

Find the area bounded by the parabola x 2 = y
and line y = x + 2 .

19. ml lery dk lehdj.k Kkr dhft, tks leryksa
→ →
r .(2iˆ + 3 ˆj + 4kˆ ) = −5 vkSj r .(iˆ + ˆj + kˆ ) = 6 ds izfrPNsn
okyh js[kk ls xqtjrk gS vkSj fcUnq (1, 1, 1) ls Hkh xqtjrk gSA6
Find the equation of the plane through the line
→
of intersection of the planes r .(2iˆ + 3 ˆj + 4kˆ ) = −5

→
and r .(iˆ + ˆj + kˆ ) = 6 and the point (1, 1, 1).

4931/(Set : D)

( 15 ) 4931/(Set : D)
vFkok
OR

→ →
js[kkvksa r = iˆ + ˆj + λ(2iˆ − ˆj + kˆ ) vkSj r = 2iˆ + ˆj − kˆ

+ µ(3iˆ − 5 ˆj + 2kˆ ) ds chp dh fuEure nwjh (S.D.) 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 ) .

20. fuEufyf[kr jSf[kd izksxzkeu leL;k dks xzkfQd fof/k ls gy
dhft, % 6

mPprehdj.k dhft, Z = 4x + y

vojks/kksa ds vUrxZr x + y ≤ 50,

3x + y ≤ 90,

x ≥ 0, y ≥ 0

4931/(Set : D) P. T. O.

( 16 ) 4931/(Set : D)
Solve the following linear programming problem
graphically :

Maximize Z = 4x + y

Subject to constraints

x + y ≤ 50,

3x + y ≤ 90,

x ≥ 0, y ≥ 0

s

4931/(Set : D)