Rajasthan Board Class 12 Question Paper 2023 Maths

Zm_m§H$ Roll No.

Tear Here
Sl.No. :

No. of Questions – 23 SS–15–Mathematics (D&D)
No. of Printed Pages – 15

Cƒ _mÜ`{_H$ (‘yH$-~{Ya) narjm, 2023
SENIOR SECONDARY (DEAF & DUMB)

TEAR HERE TO OPEN THE QUESTION PAPER
EXAMINATION, 2023
J{UV

àíZ nÌ H$mo ImobZo Ho$ {bE `hm± \$m‹S>|
MATHEMATICS
g_` : 4 KÊQ>o 15 {_{ZQ>
nyUmªH$ : 80

narjm{W©`m| Ho$ {bE gm_mÝ` {ZX}e …
GENERAL INSTRUCTIONS TO THE EXAMINEES :

1) narjmWu gd©àW_ AnZo àíZ nÌ na Zm_m§H$ A{Zdm`©V… {bI|&
Candidate must write first his/her Roll No. on the question paper
compulsorily.
2) g^r àíZ H$aZo A{Zdm`© h¢&
All the questions are compulsory.
3) àË`oH$ àíZ H$m CÎma Xr JB© CÎma-nwpñVH$m _| hr {bI|&
Write the answer to each question in the given answer-book only.
4) {OZ àíZmo§ _| AmÝV[aH$ IÊS> h¡§, CZ g^r Ho$ CÎma EH$ gmW hr {bI|&
`hm± go H$m{Q>E

For questions having more than one part, the answers to those parts are to
be written together in continuity.

SS–15–Mathematics (D&D) 3116 [ Turn Over

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5) àíZ nÌ Ho$ {hÝXr d A§J«oOr ê$nmÝVa _o| {H$gr àH$ma H$s Ìw{Q> / AÝVa / {damoYm^mg hmoZo na {hÝXr ^mfm
Ho$ àíZ H$mo hr ghr _mZ|&
If there is any error / difference / contradiction in Hindi & English versions
of the question paper, the question of Hindi version should be treated
valid.

6) àíZ H$m CÎma {bIZo go nyd© àíZ H$m H«$_m§H$ Adí` {bI|&
Write down the serial number of the question before attempting it.

7) àíZ g§»`m 17 go 23 _| AmÝV[aH$ {dH$ën {X`o JE h¡&
Q. Nos. 17 to 23 having internal choices.

8) àíZ g§»`m 23 J«m’$ nona na hb H$aZm h¡&
Solve Question number 23 on graph paper.

SS–15–Mathematics (D&D) 3116

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IÊS> - A
SECTION - A

1) ~hþ{dH$ënr` àíZ …
Multiple Choice Questions :

`{X f : R  R, f  x   sin x VWm g : R  R, g  x   x V~  f  g  ( x) ~am~a h¡& [1]
2
i)
A) sin x2 ~) sin x
g) sin2 x2 X) sin2 x
If f : R  R, f  x   sin x and g : R  R, g  x   x then  f  g  ( x) is equal to :
2

A) sin x2 B) sin x
C) sin2 x2 D) sin2 x
ii) ¶{X {H$gr Amì¶yh H$s H$mo{Q> m × n h¢, Vmo Bg‘| Ad¶dm| H$s g§»¶m h¢ - [1]
A) m ~) n
g) mn X) m – n
If the order of a matrix is m × n, then the number of elements in it are -
A) m B) n
C) mn D) m – n
d2y
iii) ¶{X y  x  log e x , Vmo 2 H$m ‘mZ hmoJm - [1]
dx
1 1
A) ~)
1 x x
g) log e (1  x ) X) 1  log e x

d2y
If y  x  log e x , then the value of 2 will be -
dx
1 1
A) B)
1 x x
C) log e (1  x ) D) 1  log e x

SS–15–Mathematics (D&D) 3116 [ Turn Over

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 1 
iv)  x   H$m x Ho$ gmnoj à{V AdH$bO h¡ - [1]
 x
1 13 1 2 23 1 2
A) x  2x 2  C ~) x  x C
3 3 2

2 32 1 3 32 1 12
g) x  2x 2  C X) x  x C
3 2 2

 1 
The anti derivative of  x   with respect to x -
 x

1 13 1 2 23 1 2
A) x  2x 2  C B) x  x C
3 3 2

2 32 1 3 32 1 12
C) x  2x 2  C D) x  x C
3 2 2

 cos x dx H$m _mZ h¡ -
2
v) [1]

x 1 1
A)  sin 2 x  C ~) x 2  sin 2 x  C
2 4 4

x 1 x2 1 2
g)  sin x  C X)  sin x  C
4 2 2 2

The value of  cos 2 x dx is -

x 1 1
A)  sin 2 x  C B) x 2  sin 2 x  C
2 4 4

x 1 x2 1 2
C)  sin x  C D)  sin x  C
4 2 2 2

SS–15–Mathematics (D&D) 3116

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vi) dH«$ y = x2 Ed§ aoIm y = 4 go {Kao joÌ H$m joÌ’$b h¡ - [1]
33 8
A) ~)
2 3
32 4
g) X)
3 3
The area of the region bounded by the curve y = x2 and the line y = 4 is -
33 8
A) B)
2 3
32 4
C) D)
3 3
vii) iˆ  ( ˆj  kˆ )  ˆj  (iˆ  kˆ )  kˆ  (iˆ  ˆj ) H$m ‘mZ h¡ - [1]
A) 0 ~) –1
g) 1 X) 3
The value of iˆ  ( ˆj  kˆ)  ˆj  (iˆ  kˆ)  kˆ  (iˆ  ˆj ) is -
A) 0 B) –1
C) 1 D) 3
     
viii) `{X Xmo g{Xem| a VWm b Ho$ n[a‘mU H«$‘e: 3 d 2 h¢ Am¡a a  b  6 hmo, Vmo a VWm b Ho$
~rM H$m H$moU h¡ - [1]
 
A) ~)
2 3
 
g) X)
6 4
 
If the magnitude of two vectors a and b are 3 and 2 respectively and
   
a  b  6 , then the angle between a and b is -
 
A) B)
2 3
 
C) D)
6 4

SS–15–Mathematics (D&D) 3116 [ Turn Over

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ix) x, y Am¡a z-Ajm| na H«$‘e: 2, 3 Am¡a 4 A§V: I§S> H$mQ>Zo dmbo g‘Vb H$m g‘rH$aU h¡ - [1]

A) 4x + 6y + 3z = 12 ~) 6x + 4y + 3z = 12

g) 3x + 4y + 6z = 12 X) 5x + 4y + 3z = 0

The equation of the plane with intercepts of 2, 3 and 4 on the x, y and z-axes
respectively is -

A) 4x + 6y + 3z = 12 B) 6x + 4y + 3z = 12

C) 3x + 4y + 6z = 12 D) 5x + 4y + 3z = 0

x) `{X P(A) 
7
13
,P(B) 
9
13
Am¡a P(A  B) 
4
13
 
hmo, Vmo P A B H$m ‘mZ h¡ - [1]

4 7
A) ~)
9 9

5 5
g) X)
9 13

If P(A) 
7
13
, P(B) 
9
13
4
and P(A B)  , then the value of P A B is -
13
 
4 7
A) B)
9 9

5 5
C) D)
9 13

SS–15–Mathematics (D&D) 3116

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xi) ¶{X nmgm| H$m EH$ Omo‹S>m CN>mbm OmVm h¡, Vmo à˶oH$ nmgo na g‘ A^mÁ¶ g§»¶m àmá hmoZo H$s
àm{¶H$Vm h¡ - [1]

1
A) 0 ~)
3

1 1
g) X)
12 36

If a pair of dice is thrown, then the probability of getting an even prime
number on each die is -

1
A) 0 B)
3

1 1
C) D)
12 36

xii) ¶{X EH$ {g¸o$ H$mo VrZ ~ma CN>mbm J¶m h¡, Ohm± E: Vrgar CN>mb na {MV, F: nhbr XmoZm| CN>mbm| na
{MV hmo, Vmo P(E/F) H$m ‘mZ h¡ - [1]

1 1
A) ~)
8 2

1 1
g) X)
4 3

If a coin is tossed three times, where E: head on third toss; F: heads on first
two tosses, then the value of P(E/F) is -

1 1
A) B)
8 2

1 1
C) D)
4 3

SS–15–Mathematics (D&D) 3116 [ Turn Over

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2) [aº$ ñWmZm| H$s ny{V© H$s{OE :
Fill in the blanks :

2  1 10 
i) `{X x    y      hmo, Vmo (x + y) = .............. hmoJm& [1]
3  1 5    

2  1 10 
If x    y      , then (x + y) = ________.
3 1 5

ii) cos  x  H$m x Ho$ gmnoj AdH$bZ ............. h¡& [1]

The derivative of cos  x  with respect to x is _______.
iii) dH«$ y = 3x4 – 4x Ho$ x = 4 na ñne© aoIm H$s àdUVm H$m ‘mZ ................ hmoJm& [1]
The slope of the tangent line at x = 4 to the curve y = 3x4 – 4x will be ____ .

 1 
 x 1  x  dx H$m ‘mZ ................. hmoJm&
2
iv) 2
[1]

 1 
The value of  x 2 1  2  dx will be _________.
 x 
v) ¶{X {~ÝXþAm| A, B, C Am¡a D Ho$ {ZX}em§H$ H«$‘e: (1, 2, 3), (4, 5, 7), (–4, 3, –6) Am¡a (2, 9, 2)
h¡, Vmo AB Am¡a CD aoImAm| Ho$ ~rM H$m ݶyZ H$moU .............. hmoJm& [1]

If the coordinates of the points A, B, C and D are then (1, 2, 3), (4, 5, 7),
(–4, 3, –6) and (2, 9, 2) respectively, the acute angle between the lines AB
and CD will be __________.

vi) ¶{X Xmo {Zînj nmgm| H$s EH$ OmoS‹ >r H$mo EH$ ~ma CN>mbm OmVm h¡, Vmo XmoZm| nmgm| na A§H$m| H$m ¶moJ 5 hmoZo H$s
àm{¶H$Vm H$m ‘mZ .............. hmoJm& [1]

If a pair of two unbiased dice is thrown once, then the probability that the
sum of the numbers on both the dice is 5 will be _______.

SS–15–Mathematics (D&D) 3116

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3) A{V bKwÎmamË_H$ àíZ :
Very short answer type questions :

 1
i) sin 1    H$m ‘w»¶ ‘mZ kmV H$s{OE& [1]
 2
 1
Find the principal value of sin 1    .
 2
ii) {ZåZ{b{IV g‘rH$aU go x VWm y Ho$ ‘mZm| H$mo kmV H$s{OE :
x 5  3 4   7 6 
2    [1]
 7 y  3 1 2  15 14 
Find the values of x and y from the following equation :
x 5  3 4   7 6 
2  
7 y  3 1 2  15 14 

102 18 36
iii) gma{UH$ 1 3 4 H$m ‘mZ kmZ H$s{OE& [1]
17 3 6

102 18 36
Evaluate 1 3 4
17 3 6

iv) x = 3 na ’$bZ f (x) = 2x2 – 1 Ho$ gm§V˶ H$s Om±M H$s{OE& [1]
Examine the continuity of the function f (x) = 2x2 – 1 at x = 3.

v) {H$gr CËnmX H$s x BH$mB¶m| Ho$ {dH«$¶ go àmá Hw$b Am¶ R(x) ê$n¶m| ‘| R(x) = 13x2 + 26x + 15
go àXV h¡& gr‘m§V Am¶ kmV H$s{OE, O~ x = 7 h¡& [1]
The total revenue in Rupees received from the sale of x units of a product is
given by R(x) = 13x2 + 26x + 15. Find the marginal revenue, when x = 7.

vi) àW‘ MVwWmªe ‘| dH«$ y2 = 9x; x = 2, x = 4 Ed§ x-Aj go {Kao joÌ H$m joÌ’$b kmV H$s{OE&[1]
Find the area of the region bounded by y2 = 9x; x = 2, x = 4 and the x-axis in
the first quadrant.

SS–15–Mathematics (D&D) 3116 [ Turn Over

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vii) {~ÝXþAm| P(iˆ  2 ˆj  kˆ ) Am¡a Q( iˆ  ˆj  kˆ) H$mo {‘bmZo dmbr aoIm H$mo 2 : 1 Ho$ AZwnmV ‘| AÝV:
{d^m{OV H$aZo dmbo {~ÝXþ R H$m pñW{V g{Xe kmV H$s{OE& [1]

Find the position vector of a point R which internally divides the line joining
two points P and Q whose position vectors are (iˆ  2 ˆj  kˆ) and ( iˆ  ˆj  kˆ )
respectively in the ratio 2 : 1.

viii) g{Xem| iˆ  2 ˆj  3kˆ Am¡a 3iˆ  2 ˆj  kˆ Ho$ ~rM H$m H$moU kmV H$s{OE& [1]

Find the angle between the vectors iˆ  2 ˆj  3kˆ and 3iˆ  2 ˆj  kˆ .

ix) Xem©BE {H$ {~ÝXþAm| (1, –1, 2) Am¡a (3, 4, –2) go hmoH$a OmZo dmbr aoIm, {~ÝXþAm| (0, 3, 2) Am¡a
(3, 5, 6) go OmZo dmbr aoIm na b§~ h¡& [1]

Show that the line through the points (1, –1, 2) and (3, 4, –2) is perpendicular
to the line through the points (0, 3, 2) and (3, 5, 6).

x 5 y  4 z 6
x) EH$ aoIm H$m H$mVu¶ g‘rH$aU   h¡& BgH$m g{Xe g‘rH$aU kmV H$s{OE&
3 7 2
[1]

x 5 y  4 z 6
The cartesian equation of a line is   . Write its vector form.
3 7 2

xi) g‘Vb 2x + y – z = 5 Ûmam {ZX}er Ajmo na H$mQ>o JE A§V: I§S>m| H$mo kmV H$s{OE& [1]

Find the intercepts cut off by the plane 2x + y – z = 5 on co-ordinate axes.

xii) EH$ AZ{^ZV (unbiased) nmgo H$mo Xmo ~ma CN>mbm J¶m& ‘mZ bo A KQ>Zm "nhbr CN>mb na {df‘
g§»¶m àmá hmoZm' Am¡a B KQ>Zm "{ÛVr¶ CN>mb na {df‘ g§»¶m àmá hmoZm' Xem©Vo h¡& KQ>ZmAm| A Am¡a B
Ho$ ñdmV§Í¶ H$m narjU H$s{OE& [1]

An unbiased die is thrown twice. Let the event A be 'odd number on the first throw'
and B the event 'odd number on the second throw'. Check the independence
of the events A and B.

SS–15–Mathematics (D&D) 3116

11

IÊS> - ~
SECTION - B
bKwCÎmar¶ àíZ :
Short answer type questions :
(4 x  3) 2 2
4) `{X f ( x)  , x  , Vmo {gÕ H$s{OE H$s g^r x  Ho$ {bE  f  f  ( x)  x h¡& [2]
(6 x  4) 3 3

(4 x  3) 2 2
If f ( x)  , x  , show that  f  f  ( x)  x for all x  .
(6 x  4) 3 3

 sin  cos  
5) `{X A=  hmo, Vmo g˶m{nV H$s{OE A' A = I [2]
  cos  sin  

 sin  cos  
If A=  , then verify that A' A = I.
  cos  sin  

a 2 ab ac
6) {gÕ H$s{OE H$s ba b 2 bc  4a 2b 2c 2 . [2]
ca cb c 2

a 2 ab ac
Prove that ba b 2 bc  4a 2b 2c 2 .
ca cb c 2

7) Xem©BE H$s {~ÝXþ A(a, b + c), B(b, c + a) Am¡a C(c, a + b) g§aoI h¢& [2]
Show that the points A(a, b + c), B(b, c + a) and C(c, a + b) are collinear.

SS–15–Mathematics (D&D) 3116 [ Turn Over

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 x 2  3, ¶{X x  0
8) {gÕ H$s{OE H$s ’$bZ f ( x)   x = 0 na g§VV Zht h¡& [2]
 1 , ¶{X x  0

 x 2  3, if x  0
Prove that the function f given by f ( x)   is not continuous at x = 0.
 1 , if x  0

9) A§Vamb kmV H$s{OE {Og‘| f(x) = x2 – 4x + 6 go àXÎm ’$bZ f [2]
i) dY©‘mZ h¡
ii) ömg‘mZ h¡&
Find the intervals in which the function f given by f(x) = x2 – 4x + 6 is
i) Increasing
ii) Decreasing

10) x ‘rQ>a ^wOm dmbo KZ H$s ^wOm ‘| 2% H$s d¥{Õ Ho$ H$maU go KZ Ho$ Am¶VZ ‘| g{ÞH$Q> n[adV©Z kmV H$s{OE&
[2]
Find the approximate change in the volume of a cube of side x meters caused by
increasing the side by 2%.

sec2 x
11)  tan x  4 dx H$m ‘mZ kmV H$s{OE&
2
[2]

sec2 x
Evaluate  dx .
tan x  4
2

12) nadb¶ y2 = 4ax Am¡a CgHo$ Zm{^b§~ go n[a~Õ joÌ H$m joÌ’$b kmV H$s{OE& [2]
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum.

SS–15–Mathematics (D&D) 3116

13
13) y-Aj H$mo ‘yb q~Xÿ na ñne© H$aZo dmbo d¥Îmm§o Ho$ Hw$b H$m AdH$b g‘rH$aU kmV H$s{OE& [2]
Form the differential equation of the family of circles touching the y-axis at origin.

   
14) {XE hþE g{Xem| a  2iˆ  ˆj  2kˆ Am¡a b  iˆ  ˆj  kˆ Ho$ {bE g{Xe a  b Ho$ AZw{Xe ‘mÌH$ g{Xe
kmV H$s{OE& [2]
 
For given vectors, a  2iˆ  ˆj  2kˆ and b  iˆ  ˆj  kˆ , find the unit vector in the
 
direction of the vector a  b .

 
15) g‘Vbm|, {OZHo$ g{Xe g‘rH$aU r  (2iˆ  2 ˆj  3kˆ)  5 Am¡a r  (3iˆ  3 ˆj  5kˆ)  3 h¡, Ho$ ~rM H$m
H$moU kmV H$s{OE& [2]

Find the angle between the planes whose vector equations are r  (2iˆ  2 ˆj  3kˆ)  5

and r  (3iˆ  3 ˆj  5kˆ)  3 .

16) ¶{X EH$ ݶm¶ {g¸o$ H$mo 10 ~ma CN>mbm J¶m, Vmo R>rH$ N>:{MV AmZo H$s àm{¶H$Vm kmV H$s{OE& [2]
If a fair coin is tossed 10 times, find the probability of exactly six heads.

IÊS> - g
SECTION - C

 1 
17) ¶{X sin  sin 1  cos 1 x   1 , Vmo x H$m ‘mZ kmV H$s{OE& [3]
 5 

 1 
If sin  sin 1  cos 1 x   1 , then find the value of x.
 5 
AWdm/OR

1 3 8 84
Xem©BE {H$ sin  sin 1  cos 1 . [3]
5 17 85
1 3 8 84
Show that sin  sin 1  cos 1 .
5 17 85

SS–15–Mathematics (D&D) 3116 [ Turn Over

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18) x Ho$ gmnoj (log x)cos x H$m AdH$bZ H$s{OE& [3]
Differentiate (log x)cos x with respect to x.
AWdm/OR
d2y
¶{X y = 500e + 600e h¡, Vmo Xem©BE H$s 2  49 y .
7x –7x
[3]
dx
d2y
If y = 500e + 600e , show that
7x –7x  49 y .
dx 2

1
19)  ( x  1)( x  2) dx H$m ‘mZ kmV H$s{OE& [3]

1
Evaluate  dx
( x  1)( x  2) .
AWdm/OR
x2  1
 x2  5x  6 dx H$m ‘mZ kmV H$s{OE& [3]

x2  1
Evaluate  2 dx .
x  5x  6

20) Xem©BE H$s g{Xe 2iˆ  ˆj  kˆ, iˆ  3 ˆj  5kˆ Am¡a 3iˆ  4 ˆj  4kˆ EH$ g‘H$moU {Ì^wO Ho$ erfm| H$s
aMZm H$aVo h¢& [3]

Show that the vectors 2iˆ  ˆj  kˆ, iˆ  3 ˆj  5kˆ and 3iˆ  4 ˆj  4kˆ form the vertices of
a right angled triangle.
AWdm/OR
EH$ {Ì^wO H$m joÌ’$b kmV H$s{OE, {OgHo$ erf© A (1, 1, 1), B(1, 2, 3) Am¡a C(2, 3, 1) h¢& [3]
Find the area of a triangle having the points A (1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as
its vertices.

SS–15–Mathematics (D&D) 3116

15

IÊS> - X
SECTION - D

1
21)  5 x 4 x5  1 dx H$m ‘mZ kmV H$s{OE& [4]
1

1
Evaluate  5 x 4 x5  1 dx .
1

AWdm/OR


 sin x dx H$m ‘mZ kmV H$s{OE&
2
4
[4]
 4



Evaluate  4 sin 2 x dx .
4

22) {H$gr ~¢H$ ‘| ‘ybYZ H$s d¥{Õ 5% dm{f©H$ H$s Xa go hmoVr h¡& Bg ~¢H$ ‘| Rs. 1,000 O‘m H$am¶o OmVo h¡& kmV
H$s{OE H$s 10 df© ~mX ¶h am{e {H$VZr hmo Om¶oJr (e0.5 = 1.648). [4]
In a bank, principal increases continuously at the rate of 5% per year. An amount
of Rs. 1,000 is deposited with this bank. How much will it worth after 10 years
(e0.5 = 1.648).
AWdm/OR
AdH$b g_rH$aU y dx – (x + 2y ) dy = 0 H$m ì¶mnH$ hb kmV H$s{O¶o&
2
[4]
Find the general solution of the differential equation y dx – (x + 2y2) dy = 0.

23) {ZåZ{b{IV ì¶damoYm| Ho$ A§VJ©V Z = 5x + 3y H$m AmboIr¶ {d{Y go A{YH$V‘rH$aU H$s{OE& [4]
3x + 5y < 15, 5x + 2y < 10, x > 0, y > 0
Maximize Z = 5x + 3y subject to constraints 3x + 5y < 15, 5x + 2y < 10, x > 0, y > 0
by using graphical method.
AWdm/OR
{ZåZ{b{IV ì¶damoYm| Ho$ AÝVJ©V Z = 200x + 500y H$m AmboIr¶ {d{Y go ݶyZV‘rH$aU H$s{OE& [4]
x + 2y > 10, 3x + 4y < 24, x > 0, y > 0.
Minimize Z = 200x + 500y subject to constraints x + 2y > 10, 3x + 4y < 24, x > 0, y > 0 by
using graphical method.

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SS–15–Mathematics (D&D) 3116

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