Rajasthan Board Class 12 Question Paper 2019 Mathematics

Zm_m§ H $ Roll No.

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Sl.No. :

No. of Questions – 30 SS–15–Mathematics
No. of Printed Pages – 11

Cƒ _mÜ`{_H$ narjm, 2019
SENIOR SECONDARY EXAMINATION, 2019

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àíZ nÌ H$mo ImobZo Ho$ {bE `hm± \$m‹S>|
MATHEMATICS

g_` : 3¼ KÊQ>o
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.
`hm± go H$m{Q>E

4) {OZ àíZmo§ _| AmÝV[aH$ IÊS> h¡§, CZ g^r Ho$ CÎma EH$ gmW hr {bI|&
For questions having more than one part, the answers to those parts are to
be written together in continuity.

SS–15–Mathematics 1310 [ 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) IÊS> àíZ g§»`m A§H$ àË`oH$ àíZ
A 1 - 10 1
~ 11 - 15 2
g 16 - 25 3
X 26 - 30 6
Section Q. Nos. Marks per question
A 1 - 10 1
B 11 - 15 2
C 16 - 25 3
D 26 - 30 6

7) àíZ g§»`m 16, 21, 24, 28 Am¡a 30 _| AmÝV[aH$ {dH$ën h¢& BZ àíZm| _o| go AmnH$mo EH$
hr {dH$ën H$aZm h¡&
There are internal choices in Q. Nos. 16, 21, 24, 28 and 30. You have to
attempt only one of the alternatives in these questions.

8) àíZ g§»`m 25 H$m boIm{MÌ J«m\$$ nona na ~ZmZm h¡&
Draw the graph of Q. No. 25 on the graph paper.

SS–15–Mathematics 1310

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IÊS> - A
SECTION - A

1) ¶{X f : R  R, f(x) = sin x VWm g : R  R, g(x) = x2 Vmo gof(x) kmV H$s{OE&
If f : R  R, f(x) = sin x and g : R  R, g(x) = x2 then find gof(x).

  1 
2) sin  tan 1 1  cos 1    H$m ‘mZ kmV H$s{OE&
  2 

  1 
Find the value of sin  tan 1 1  cos 1   .
  2 

a  b 4   6 4
3) ¶{X  3  hmo, Vmo a d b Ho$ ‘mZ kmV H$s{OE&
 ab   3 8 

a  b 4   6 4
If  3  , then find the value of a and b.
 ab   3 8 

 cos  sin  
4) ¶{X Amì¶yh A    hmo, Vmo A–1 kmV H$s{OE&
  sin  cos  

 cos  sin  
If matrix A    , then find A–1.
  sin  cos  

1  cos 2 x
5)  1  cos 2 x dx kmV H$s{OE&
1  cos 2 x
Find  dx .
1  cos 2 x

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6) g{Xe 2iˆ  ˆj VWm iˆ  2 ˆj Ho$ ‘ܶ H$m H$moU kmV H$s{OE&

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

    
7) ¶{X a  10, b  2 VWm a .b  12 hmo, Vmo sinH$m ‘mZ kmV H$s{OE& Ohm± , g{Xe a d b Ho$ ‘ܶ
H$m H$moU h¡&
   
If a  10, b  2 and a .b  12 , then find the value of sin, where  is the angle
 
between vectors a and b .

8) {~ÝXþAm| (1, 0, 0) VWm (0, 1, 1) go JwOaZo dmbr aoIm H$s {XH²$-H$mogmBZ kmV H$s{OE&
Find the direction cosines of the line passing through the points (1, 0, 0) and
(0, 1, 1).

9) {ZåZ ì`damoYm| Ho$ A§VJ©V gwg§JV joÌ CÎma nwpñVH$m _| Xem©BE&
2x  3y  6 ; x0 ; y0

Show the region of feasible solution under the following constraints

2x  3y  6 ; x0 ; y0 .

 
10) ¶{X P(A) = 0.6, P(B) = 0.3 Am¡a P  A  B   0.2 hmo, Vmo P A B kmV H$s{OE&

If P(A) = 0.6, P(B) = 0.3 and P  A  B   0.2 , then find P A B .  

SS–15–Mathematics 1310

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IÊS> - ~
SECTION - B

x3
11) `{X f ( x)  hmo, Vmo f[f{f(x)}] kmV H$s{OE&
x 1

x3
If f ( x)  , then find f[f{f(x)}].
x 1

 2 3  1 1
12) ¶{X A    VWm B    hmo, Vmo {gÕ H$s{OE {H$ (AB)T = BTAT.
 1 4  2 5 

 2 3  1 1
If A    and B    , then prove that (AB)T = BTAT.
 1 4  2 5 

 sin x
  cos x; x  0
13) ¶{X ’$bZ f  x    x {~ÝXþ x = 0 na g§VV h¡, Vmo K H$m ‘mZ kmV H$s{OE&
 K ; x0

 sin x
  cos x; x  0
If function f  x    x is continuous at point x = 0, then find the
 K ; x0
value of K.

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1
14)  cos  3x  2 .dx kmV H$s{OE&
2

1
Find  .dx .
cos  3x  2 
2

15) ¶{X {H$gr {Ì^wO H$s Xmo ^wOmE± g{Xe iˆ  2 ˆj  2kˆ VWm 3iˆ  2 ˆj  kˆ go {Zê${nV hmo, Vmo {Ì^wO H$m
joÌ’$b kmV H$s{OE&

If two sides of a triangle are represented by vectors iˆ  2 ˆj  2kˆ and 3iˆ  2 ˆj  kˆ ,
then find the area of the triangle.

IÊS> - g
SECTION - C

1 63 1 3
16) {gÕ H$s{OE cos  2 tan 1  sin 1
65 5 5

AWdm


g‘rH$aU tan 1 3 x  tan 1 2 x  H$mo hb H$s{OE&
4

1 63 1 3
Prove that cos  2 tan 1  sin 1 .
65 5 5

OR


Solve the equation tan 1 3 x  tan 1 2 x  .
4

SS–15–Mathematics 1310

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1 a b c
17) {gÕ H$s{OE {H$ a 1 b c  1  a  b  c  .
a b 1 c

1 a b c
Prove that a 1 b c  1  a  b  c  .
a b 1 c

18) a¡{IH$ g‘rH$aU {ZH$m¶ x + y + 2z = 0, x + 2y – z = 9, x – 3y + 3z = –14 H$mo Amì¶yh {gÕmÝV Ûmam
hb H$s{OE&

Solve the system of linear equations x + y + 2z = 0, x + 2y – z = 9, x – 3y + 3z = –14 by
using matrix method.

19) dH«$ y = x2 – 2x + 3 H$s ñne© aoIm H$m g‘rH$aU kmV H$s{OE, Omo aoIm 2x – y + 9 = 0 Ho$ g‘mÝVa h¡&

Find the equation of tangent of a curve y = x2 – 2x + 3 which is parallel to the line
2x – y + 9 = 0.

20) EH$ Jmobo H$s {ÌÁ¶m 7 go‘r ‘mnr OmVr h¡ {Og‘| 0.01 go‘r H$s Ìw{Q> h¡& Bg Ìw{Q> Ho$ H$maU BgHo$ Am¶VZ H$s
JUZm ‘| g{ÝZH$Q>Z Ìw{Q> kmV H$s{OE&

If the radius of a sphere is measured as 7 cm with an error of 0.01 cm, then find the
approximate error in calculating its volume.

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cos x
21)  4  sin x . dx kmV H$s{OE&
2

AWdm

 x tan x . dx kmV H$s{OE&
1

cos x
Find  . dx .
4  sin 2 x

OR

Find  x tan 1 x . dx .

22) nadb¶ x2 = 4y VWm aoIm y = 3 go n[a~Õ joÌ H$m joÌ’$b kmV H$s{OE& (CÎma nwpñVH$m ‘| {MÌ ~ZmBE)
Find the area bounded by the parabola x2 = 4y and line y = 3. (Draw the figure in
answer-book)

23) {ZåZ{b{IV joÌ H$m joÌ’$b kmV H$s{OE :

 x2 y2 
 x , y    1 VWm x 2
 y 2
 9 
 9 4 
(CÎma nwpñVH$m ‘| {MÌ ~ZmBE)
Find the area of the region given by :

 x2 y 2 
 x , y   1 and x 2  y 2  9  .
 9 4 

(Draw the figure in answer-book)

SS–15–Mathematics 1310

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           
24) ¶{X a  b  c  d VWm a  c  b  d , Vmo {gÕ H$s{OE {H$ a  d Ed§ b  c g‘mÝVa h¡&

AWdm

EH$ MVwî’$bH$ Ho$ Mmam| erf© H«$‘e… O(0, 0, 0), A(1, 2, 1), B(2, 1, 3) VWm C(1, 1, 2) h¡&
MVwî’$bH$ H$m Am¶VZ kmV H$s{OE&
           
If a  b  c  d and a  c  b  d , then prove that a  d is parallel to b  c .

OR

The four vertices of a tetrahedron are respectively O(0, 0, 0), A(1, 2, 1),
B(2, 1, 3) and C(1, 1, 2). Find the volume of the tetrahedron.

25) {ZåZ a¡{IH$ àmoJ«m‘Z g‘ñ¶m H$mo Ambo{I¶ {d{Y Ûmam hb H$s{OE&

A{YH$V‘ z = 20x + 30y

ì¶damoY x + 2y  20

3x + 2y  30

x  0, y  0.

By the graphical method, solve the following linear programming problem for

Maximize z = 20x + 30y

Constraints x + 2y  20

3x + 2y  30

x  0, y  0.

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IÊS> - X
SECTION - D

1 1 d2y dy
26) ¶{X x  y  t      2  0.
2 2 4 4 2
VWm x y t 2 , V~ {gÕ H$s{OE
x 2
t t dx dx

1 1 d2y dy
If x  y  t      2  0.
2 2 4 4 2
and x y t 2 , then prove that
x 2
t t dx dx


x sin x
27)  1  cos x . dx H$m ‘mZ kmV H$s{OE&
0
2


x sin x
Find the value of  . dx .
0
1  cos 2
x

28) AdH$b g_rH$aU x(x – y)dy = y(x + y)dx H$m hb kmV H$s{OE&

AWdm

dy
AdH$b g‘rH$aU cos 2 x  y  tan x H$m hb kmV H$s{OE&
dx

Solve the differential equation x(x – y)dy = y(x + y)dx.

OR

dy
Solve the differential equation cos x
2
 y  tan x .
dx

SS–15–Mathematics 1310

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x4 y z2
29) {~ÝXþ P(1, 1, 3) go aoIm   na S>mbo J¶o bå~ H$m nmX kmV H$s{OE gmW hr {X¶o J¶o {~ÝXþ
2 1 1
go aoIm H$s bå~dV² Xÿar kmV ^r H$s{OE&

Find the foot of the perpendicular drawn from the point P(1, 1, 3) to the line
x4 y z2
  . Also find the perpendicular distance of the line from the given
2 1 1
point.

30) EH$ ì¶{³V Ho$ ~mao ‘| kmV h¡ {H$ dh 3 ‘| go 2 ~ma g˶ ~mobVm h¡& dh EH$ nmgo H$mo CN>mbVm h¡ Am¡a ~VbmVm h¡
{H$ Cg na AmZo dmbr g§»¶m 6 h¡& BgH$s àm{¶H$Vm kmV H$s{OE {H$ nmgo na AmZo dmbr g§»¶m dmñVd ‘| 6 h¡&

AWdm

EH$ H$be ‘| 4 g’o$X VWm 2 bmb J|X| h¢& Xmo J|Xm| Ho$ ¶mÑÀN>¶m {ZH$mb ‘| bmb J|Xm| H$s g§»¶m H$m àm{¶H$Vm
~§Q>Z VWm BgH$m ‘mܶ ^r kmV H$s{OE&

A man is known to speak truth 2 out of 3 times. He throws a die and reports that it
is a six. Find the probability that it is actually a six.

OR

An urn contains 4 white and 2 red balls. Find the probability distribution and its
mean of the number of red balls, if 2 balls are drawn at random.



SS–15–Mathematics 1310

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