Rajasthan Board Class 12 Question Paper 2020 Mathematics

 Roll No.

No. of Questions – 30
No. of Printed Pages – 8 SS-15-Mathematics

 (MATHEMATICS)
  , 2020
 3¼ 
 80

     

GENERAL INSTRUCTIONS TO THE EXAMINEES :

(1)    -     

Candidate must write first his / her Roll No. on the question paper compulsorily.

(2)      

All the questions are compulsory.

(3)       -    

Write the answer to each question in the given answer-book only.

SS-15-Mathematics [ Turn over

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(4)      ,         
For questions having more than one part, the answers to those parts are to be written
together in continuity.

(5) -    
       //      
    

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)      

 1 – 10 1

 11 – 15 2

 16 – 25 3

 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)   25        
Draw the graph of Q. No. 25 on the graph paper.

3
 – 
SECTION – A

1.  f : R  R, f(x) = x2 + 5x + 9 ,  f –1 (8)  f –1(9)     
If f : R  R, f(x) = x2 + 5x + 9, then find the value of f –1 (8) and f –1(9).

2. 2 tan (tan–1 x + tan–1 x3)     

Find the value of 2 tan (tan–1 x + tan–1 x3 ).

 k+4 –1   a –1 
3.    
=
 

 ,  a     
 3 k–6   3 –4 
 k+4 –1   a –1 
If   
=
 

, then find the value of a.
 3 k–6   3 –4 

4.        
Define singular and Non-singular matrix.

5.     (–1, 1)   f(x) = x2 – x + 1          
Prove that in interval (–1, 1) function f(x) = x2 – x + 1 is neither increasing nor decreasing.

. 1
6.  dx   
. 1 + sin x

. 1
Find  . dx.
 1 + sin x

7. (2^i – 3^j + 4k^)  (3^i + 4^j – 4k^)     

Find the value of (2^i – 3^j + 4k^)  (3^i + 4^j – 4k^).

SS-15-Mathematics [ Turn over

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8.    A(2, 3, 4), B(–1, 2, –3)  C(– 4, 1, –10)    
Show that the points A(2, 3, 4), B(–1, 2, –3) and C (– 4, 1, –10) are Collinear.

9.          
Define the feasible solution of the Linear programming problem.

6 5 7
10.  P(A) = , P(B) =  P(A  B) = ,  P(A  B)   
11 11 11
If P(A) = 6/11, P(B) = 5/11 and P(A  B) = 7/11, then find P(A  B).

 - 
SECTION – B

11.   f  g            (gof)  (gof)–1 
    
(gof)–1 = f –1og–1.
If f and g are one-one onto function such that composite function (gof) and (gof)–1 are
defined, then show that (gof)–1 = f –1og–1.

 –1 –2 3 
 
12.  A – 2I = 2 1 –1 ,  AAT  ,  I, 3  3      
 
 –3 1 0 
 –1 –2 3 
 
If A – 2I = 2 1 –1 , then find AAT, where I is unit matrix of order 3  3.
 
 –3 1 0 

2x + 5
13.  x     
(x2 + 3x + 1)
2x + 5
Integrate with respect to x.
(x2 + 3x + 1)

5
dy
14.  y = log x + log x + log x + ....  ,  dx   
dy .
If y = log x + log x + log x + .... , then find
dx

15.  2x2 – y2 = 14    x + 3y = 6        
Find the equation of the normals to the curve 2x2 – y2 = 14 which are parallel to the line
x + 3y = 6.

 – 
SECTION – C

16.       

sin–1 x + sin–1 2x =
3
Solve the following trigonometrical equation :

sin–1 x + sin–1 2x = .
3

 1+a 1 1

17.   
 1 1+b 1 = abc 1 + + + 
 1 1 1
   a b c
 1 1 1+c 
 1+a 1 1

Prove that
 1 1+b 1 = abc 1 + + + .
 1 1 1
   a b c
 1 1 1+c 

18.       
 3 0 3   x   8   2y 
 2 1 0  y = 1 + z 
      
 4 0 2   z   4   3y 
Solve the following system of equations :
3 0 3 x 8 2y
      
 2 1 0  y = 1 + z 
      
 4 0 2   z   4   3y 
SS-15-Mathematics [ Turn over

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 x 
19. cos– 1    x     
a + x
 x 
Integrate cos– 1   with respect to x.
a + x

20.  y2 = 4ax  x2 = 4by        
Find the area of the region enclosed between the two Parabolas y2 = 4ax and x2 = 4by.

21.  x2 + y2 = 32   y = x  x-          
 
Find the area of the region in the First quadrant enclosed by the x-axis, the line y = x and
the circle x2 + y2 = 32.

dy
22.    dx + (2x tan–1y – x3) (1 + y2) = 0
dy
Solve : + (2x tan–1y – x3) ( 1 + y2) = 0.
dx

dy
23.    (1 + y2) + (x – etan–1 y) dx = 0.
–1 y dy
Solve : (1 + y2) + (x – etan ) = 0.
dx

24.                 tan–1 2   
Show that the semi vertical angle of a cone of maximum volume and given slant height is
tan–1 2.

25.           
 z = 2x + 3y
 4x + 6y  60
2x + y  20
 x  0, y  0
Solve the following Linear Programming problem by graphical method :
Max z = 2x + 3y
Constraints 4x + 6y  60
2x + y  20
and x  0, y  0

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 – 
SECTION – D

26.  f(x) = | x – 1 | + 2| x – 2 | + 3| x – 3 |   x = 1, 2, 3     
  
Examine the continuity and differentiability of the function f(x) = | x – 1 | + 2 | x – 2 | + 3 | x – 3 |
at point x = 1, 2, 3.


. 1
27.    I = 
 log (1 + cos x) dx =  loge  
. 2
0
Prove that :

. 1
I =  .log (1 + cos x) dx =  loge  .
 2
0

28.   
      
(i) [ a + b b + c c + a ] = 2[ a b c ]
      
(ii) [( a  b ) ( b  c ) ( c  a )] = [ a b c ]2
Prove that :
      
(i) [ a + b b + c c + a ] = 2[ a b c ]
      
(ii) [( a  b ) ( b  c ) ( c  a )] = [ a b c ]2

29.          :
x–3 y–4 z+1 x–1 y–3 z–1
(i)
2
=
1
=
–3
 –1
=
3
=
2
.
 ^  
(ii) r = ^i + 2^j – 4k^ + (2^i + 3^j + 6k) r = 3^i + 3^j – 5k^ + (2^i + 3^j + 6k)
^

Find the shortest distance between the following pair of lines :
x–3 y–4 z+1 x–1 y–3 z–1
(i) = = and = = .
2 1 –3 –1 3 2
 ^ and 
(ii) r = ^i + 2^j – 4k^ + (2^i + 3^j + 6k) r = 3^i + 3^j – 5k^ + (2^i + 3^j + 6k)
^

SS-15-Mathematics [ Turn over

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30.          5   3            
        1            
  1  
A man is known to speak the truth 3 out of 5 times. He throw a die and reports that it is
‘1’. Find the probability that it is actually 1.
____________