Rajasthan Board 12th Maths Sample Paper 2021

mPp ek/;fed ijh{kk ekWMy iz'u&i= 2021
SENIOR SECONDARY EXAMINATION, MODEL QUESTION PAPER-2021
xf.kr
MATHEMATICS

le;% 3¼ ?k.Vs iw.kkZd 80

ijh{kkfFkZ;ksa ds fy, lkekU; funZs'k %&
GENERAL INSTRUCTION TO THE EXAMINEES:

1- ijh{kkFkhZ loZizFke vius iz'u i= ij ukekad vfuok;Zr% fy[ksaA
Candidate mast write first his/her Roll No. on the question paper
compulsorily.

2- lHkh iz'u djus vfuok;Z gSA
All the questions are compulsory.

3- izR;sd iz'u dk mÙkj nh xbZ mÙkj iqfLrdk esa gh fy[ksAa
Write the answer to each question in the given answer book only.

4- ftu iz'uksa esa vkUrfjd [k.M gS mu lHkh ds mÙkj ,d lkFk gh fy[ksAa
For questions having more than one part the answers to those parts are
to be written together in continuity.

5- iz'u i= ds fgUnh ij vaxzsth :ikUrj.k esa fdlh izdkj dh =qfV@ vUrj@fojks/kkHkkl gksus ij
fgUnh Hkk"kk ds iz'u dks lgh ekusaA
If there is any error/difference/contradiction in Hindi & English version
of the question paper, the question of the Hindi version should be treated
valid.
6- iz'u dk mÙkj fy[kus ls iwoZ iz'u dk Øekad vo'; fy[ksaA
Write down the serial number of the question before attempting it.
7- iz'uksa dk vadHkkj fuEukuqlkj gSA
Weightage of marks for the question is as follows.

[k.M iz'uksa dh la[;k dqy vad Hkkj vad izR;sd iz'u
[k.M&v(A) 1 ( i to x ), 2 to 11=20 20 1
[k.M&c(B) 8 16 2
[k.M&l(C) 4 16 4
[k.M&n(D) 2 10 5
[k.M&;(E) 3 18 6

7- iz'u la[;k 20 ls 28 esa vkUrfjd fodYi fn;s x;s gSaA

[k.M & v

Section - A

1- (i) cot  tan 1   cot 1   dk eku gSaA

¼v½ 1 ¼c½  ¼l½ 0 ¼n½ buesa ls dksbZ ugha 1

cot  tan 1   cot 1   is

(A) 1 (B)  (C) 0 (D) None of above

(ii) ;fn vkO;wg A   aij 24 gks rks A esa vo;oksa dh la[;k gksxh \&

¼v½ 6 ¼c½ 8 ¼l½ 2 ¼n½ buesa ls dksbZ ugha 1

Number of element in a matrix A   aij  is
24

(A) 6 (B) 8 (C) 2 (D) None of above

K 8
(iii) ;fn lkjf.kd  4 gks rks K dk eku gksxk\
2 4

¼v½ 4 ¼c½ 5 ¼l½ 6 ¼n½ buesa ls dksbZ ugha 1
K 8
If  4 then the value of K is
2 4

(A) 4 (B) 5 (C) 6 (D) None of above

(iv) vkO;wg A vO;qRØe.kh; vkO;wg dgykrk gSa ;fn

¼v½ A  0 ¼c½ A  1 ¼l½ A  1 ¼n½ buesa ls dksbZ ugha 1

Matrix A is called Non-Singular if

(A) A  0 (B) A  1 (C) A  1 (D) None of above

(v) Qyu y  cos  x  dk x ds lkis{k vodyu gksxk\

sin x sin x
¼v½ ¼c½  ¼l½ 2 x .sin x ¼n½ 2 x .sin x 1
2 x 2 x

The Differentiation of y  cos  x  w.r. to x will be
sin x sin x
(A) (B)  (C) 2 x .sin x (D) 2 x .sin x
2 x 2 x

(vi) oØ y  x3  x ] fcUnq x  2 ij Li’kZ js[kk dh izo.krk gksxh\

¼v½ 6 ¼c½ 9 ¼l½ 11 ¼n½ 13 1

The gradient of a tangent x  2 to the curve y  x3  x will be

(A) 6 (B) 9 (C) 11 (D) 13

(vii)  log x dx cjkcj gSa &

¼v½ x log  xe   c ¼c½ x log x  c ¼l½ x log    c ¼n½ log    c
x x
1
e e

 log x dx is equal to -
 x x
(A) x log  xe   c (B) x log x  c (C) x log    c (D) log    c
e e
2
 d2y 
(viii) vody lehdj.k  2      0 dh dksfV rFkk ?kkr Øe’k% gksxh\
dy
 dx   dx 

¼v½ 1] 2 ¼c½ 2] 2 ¼l½ 2] 1 ¼n½ 1] 1 1
2
 d 2 y   dy 
The order and degree of differential equation  2      0 will be
 dx   dx 
1, 2 (B) 2, 2 (C) 2, 1 (D) 1,1
(ix) lfn’k a  i  j  k dk ifjek.k a gksxk\
¼v½ 3 ¼c½ 3 ¼l½ 1 ¼n½ buesa ls dksbZ ugha 1

Magnitude a of a vector a  i  j  k is

(A) 3 (B) 3 (C) 1 (D) None of above

(x) ;fn A vkSj B Lora= ?kVuk,a gSa tgka P  A  0.3, P  B   0.6 rc P  A  B  cjkcj
gksxk\
1
¼v½ ¼c½ 2 ¼l½ 1-8 ¼n½ 0-18 1
2

If A and B are independent events and P  A  0.3, P  B   0.6 then P  A  B
equal to

1
(A) (B) 2 (C) 1.8 (D) 0.18
2

2- ,d 4  4 Øe dk bdkbZ vkO;wg fyf[k,A 1
Write down the unit matrix of order 4  4 -

5 20
3- lkjf.kd esa vo;o 5 dk milkjf.kd fyf[k,A 1
3 1
5 20
Write the Minors of element 5 in the determinant -
3 1
4- ;fn fcUnq A  x,  2 , B  5, 2 , C 8, 8 lajs[k gks rks dk x eku fyf[k,A 1
If points A  x,  2  , B  5, 2  , C 8, 8 are collinear, then find the value of x .
dy
5- Qyu y  sin x0 dk x ds lkis{k vodyu djus ij dk eku fyf[k,A 1
dx

dy
Differentiate y  sin x0 w. r. of x and find the value of .
dx

1  cos 2 x
6-  . dx dk eku fyf[k,A 1
1  cos 2 x

1  cos 2 x
Evaluate :  1  cos 2 x . dx
7- o`r x2  y 2  a 2 dk {ks=Qy fyf[k,A 1

Find the area enclosed by circle x2  y 2  a 2 -

  
8- 2i  3 j  4k . 3i  4 j  4k dk eku fyf[k,A 1

 
Find the value : 2i  3 j  4k . 3i  4 j  4k 
9- ;fn ,d js[kk ds fnd~ vuqikr 2] &1] &2 gks rks bldh fnd~ dksT;k,a ---------------------- gksxh\ 1
If Direction Ratio of a line 2, -1, -2 . Then what will the Direction cosine ………… .

dy
10- vody lehdj.k  sin x  x dks gy djus ij y  ............ gksxk\ 1
dx

dy
Find the value of y  ......... if differential equation  sin x  x is given.
dx

1
11-   ...............  c gksxk\ 1
 a  x2 
2

1
  a  x   ............... .
2 2

[k.M & c

Section - B
12- 2 tan  tan 1 x  tan 1 x3  dk eku Kkr dhft,A 2

Find the value of 2 tan  tan 1 x  tan 1 x3 

 2  1  0 4
13- ;fn vkO;wg A    rFkk B    gks rks 3 A2  2B dk eku Kkr djksA 2
3 2   1 7 

 2  1  0 4
If A    and B    then find the value of 3 A2  2B .
3 2   1 7 

1 3 2
14- ;fn lkjf.kd A  4  1 2 gks rks izFke LrEHk ds vo;oksa dh milkjf.kd ,oa lg[k.M
3 5 2
fyf[k,A 2

1 3 2
Evaluate the determinant A  4  1 2 and write the minors and cofactors of
3 5 2
elements of first column.
15- Qyu f  x   x   x dh x  3 ij larrrk dk ijh{k.k dhft,A 2
Examine the continuity of f  x   x   x  at x  3 -
16-  cos x. dx Kkr dhft,A 2
3

Evaluate  cos3 x. dx .

17- vody lehdj.k 1  x2  dy  1  y 2  dx dks gy dhft,A 2

Solve the differential equation 1  x 2  dy  1  y 2  dx

18- fcUnq ¼5] 2] &4½ ls tkus okyh rFkk lfn’k 3i  2 j  8k ds lekUrj js[kk dk lfn’k
lehdj.k Kkr dhft,A 2
Find the vector equation of the line passing through the point (5, 2, -4) and parallel to
the vector 3i  2 j  8k -

6 5 7
19- ;fn P  A  ] P  B  vkSj P  A  B   gks rks Kkr dhft,A
11 11 11

(ii) P  
A
(i) P  A  B  2
B

6 5 7
If P  A  ] P  B  and P  A  B   then find
11 11 11

(ii) P  
A
(i) P  A  B 
B

[k.M & l

Section - C

20- fuEu f=dks.kferh; lehdj.k dks gy dhft, &

sin 1 x  sin 1 2 x  4
3

;k

;fn tan 1 x  tan 1 y  tan 1 z  gks rks
2

fl) dhft, fd xy  yz  zx  1

Solve the following equation :


sin 1 x  sin 1 2 x 
3

or


If tan 1 x  tan 1 y  tan 1 z  , then prove that xy  yz  zx  1
2

21- og vUrjky Kkr dhft,] ftlesa Qyu f  x   2 x3  3x2  36 x  7

(i) o/kZeku gaSA (ii) gkzleku gaSA
;k

oØ y  x2  2 x  7 dh Li’kZ js[kk dk lehdj.k Kkr dhft, tks js[kk 2 x  y  9  0 ds
lekUrj gSaA

Find the intervals in which the function f  x   2 x3  3x 2  36 x  7

(i) Increasing (ii) Decreasing
or

Find the equation of the tangent to curve y  x 2  2 x  7 . Which is parallel to the
line 2 x  y  9  0 .

22- lfn’kksa 4i  j  3k rFkk 2i  j  2k ds yEcor 9 bdkbZ ifjek.k okyk lfn’k Kkr
dhft,A 4
;k

   
fcUnqvksa P i  2 j  k vkSj Q i  j  k dks feykus okyh js[kk dks 2 % 1 ds vuqikr esa
vUr foHkkftr djus okys fcUnq R dk fLFkfr lfn’k Kkr dhft,A
Find the vector with magnitude 9 units which is perpendicular to the vector
4i  j  3k and 2i  j  2k .

or

Find the position vector of a point R which divides the line joining two points P and

Q whose position vectors are P i  2 j  k  and Q  i  j  k  respectively in the
ratio 2 : 1 internally.

23- js[kk 2 x  y  4, x  v{k ,oa dksfV;ksa x  0 ,oa x  3 ls f?kjs {ks= dk {ks=Qy Kkr
dhft,A 4
;k
dy
;fn y  log x  log x  log x  ..........  gks rks Kkr dhft,A
dx

Find the area bounded by line 2 x  y  4, x  axis and ordinates x  0 and x  3 .
or
dy
Find :
dx

y  log x  log x  log x  .......... 

[k.M & n

Section - D

24- vkO;wg fl)kar dk iz;ksx dj fuEufyf[kr jSf[kd lehdj.k fudk; dks gy dhft,A
3x  y  z  3
2x  y  z  2 5
x  y  z  1

;k

2 3 
;fn vkO;wg A    gks rks fl) dhft, fd A  4 A  I  O
2

1 2 

0 0  1 0
tgka O    ,oa I    rFkk Hkh Kkr dhft,A
0 0  0 1

Solve the following system of equations using the matrix method.

3x  y  z  3
2x  y  z  2
x  y  z  1

or

2 3 
If matrix A    then prove that A  4 A  I  O
2

1 2 

1
25- ,d fo’ks"k leL;k dks A vkSj B }kjk Lora= :i ls gy djus dh izkf;drk,a Øe’k% o
2
1
gSaA ;fn nksuksa Lora= :i ls leL;k dks gy djus dk iz;kl djrs gSa rks izkf;drk Kkr dhft,
3
fd
(i) leL;k gy gks tkrh gaSA

(ii) muesa ls rF;r% dksbZ ,d leL;k gy dj ysrk gSA 5
;k
,d flDds dks bl izdkj vfHkur fd;k x;k gS fd flDds ij fpr vkus dh laHkkouk iV
vkus dh vis{kk rhu xquk gSA ;fn flDds dks nks ckj mNkyk tkrk gks rks iVksa dh la[;k ds fy,
izkf;drk caVu Kkr dhft,A
1 1
Probability of solving specific problem independently by A and B are and
2 3
respectively. If both try to solve the problem independently. Find the probability that

(i) The problem is solved

(ii) Exactly one the them solves the problem.
or

A coin is biased, so that the head is 3 times as likely to occur as tail. If the coin is
tosses twice, find the probability distribution of number of tails.

[k.M & ;

Section - E

 /4

26- fl) dhft, &  log 1  tan x  dx  8 log 2
0
e e 6

;k

 log  x  x  a  dx dk eku Kkr dhft,A
2 2

;k

p
fl) dhft, fd Qyu y  sin p  .cosq  dk eku tan   ij mfPp"B gSaA
q

 /4

Prove that :  log 1  tan x  dx  8 log 2
0
e e

or
Evaluate :  log  x  x 2  a 2  dx .
 
or

p
Prove that function y  sin p  .cosq  has maximum value at tan  
q
dy
  4 x  y  1
2
27- gy dhft, 6
dx
;k

dy y  y
gy dhft,   tan  
dx x x

;k

d2y cos x
;fn y  sec x  tan x , rc fl) dhft, fd 
1  sin x 
2 2
dx

dy
  4 x  y  1
2
Solve :
dx
or
dy y  y
Solve :   tan  
dx x x
or
2
d y cos x
If y  sec x  tan x , then prove that 
1  sin x 
2 2
dx

28-       
js[kkvksa r  i  2 j  k   i  j  k vkSj r  2i  j  k   2i  j  2k ds e/; dh
U;wure nwjh Kkr dhft,A 6
;k
x  4 y  3 z 1 x  1 y  1 z  10
fl) dhft, fd js[kk,a   vkSj   izfrPNsn
1 4 7 2 3 8
djrh gSA buds izfrPNsn fcUnq ds funsZ’kkad Hkh Kkr dhft,A
;k
x6 y 7 z 7
fcUnq ¼1] 2] 3½ ls js[kk   ij Mkys x, yEc dh yEckbZ Kkr dhft,A
3 2 2

Find the shortest distance between the lines whose vector equations are
  
r  i2jk  i jk  and r   2i  j  k     2i  j  2k  .
or
x  4 y  3 z 1 x  1 y  1 z  10
Prove that the lines   and   intersect each
1 4 7 2 3 8
other. Find the Co-ordinates of their intersecting points.
or

Find the length of perpendicular drawn from point the (1, 2, 3) on the line
x6 y 7 z 7
 
3 2 2