CBSE Class 12 Mathematics Question Paper 2020 Set 65-5-3

SET – 3
Series : HMJ/5
 .
Code No. 65/5/3
 .
   -  - 
Roll No.   
Candidates must write the Code on
the title page of the answer-book.

 NOTE
(I)       -   (I) Please check that this question
 15   paper contains 15 printed pages.
(II) -          (II) Code number given on the right
  -  -    hand side of the question paper
should be written on the title page
of the answer-book by the
candidate.
(III)      -  36  (III) Please check that this question
  paper contains 36 questions.
(IV)         (IV) Please write down the Serial
,       Number of the question in the
answer-book before attempting it.
(V)  -     15   (V) 15 minute time has been allotted
     -    to read this question paper. The
 10.15     10.15   question paper will be distributed
10.30     -   at 10.15 a.m. From 10.15 a.m. to
      -  10.30 a.m., the students will read
     the question paper only and will
not write any answer on the
answer-book during this period.


MATHEMATICS
{ZYm©[aV g‘¶ : 3 KÊQ>o A{YH$V‘ A§H$ : 80
Time allowed : 3 hours Maximum Marks : 80

. 65/5/3. 334C 1 P.T.O.

  :

           

(i) -        – , ,        36    
   

(ii) -    1  20  20          

(iii) -    21  26  6          

(iv) -    27  32  6          

(v) -    33  36  4      :    

(vi) -          -     , - 
   , -       -      
               

(vii)  , ,            
(viii)        

 – 

 1  10           :

1.  A   3     ,  |A|   
(a) 3 (b) 0 (c) 9 (d) 27

2.  ^i, ^j, k^      , 
(a) ^i  ^j = 1 (b) ^i  ^j = 1 (c) ^i  k^ = 0 (d) ^i  k^ = 0
.65/5/3. 2

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper comprises four sections – A, B, C and D.
This question paper carries 36 questions. All questions are compulsory.
(ii) Section A – Question no. 1 to 20 comprises of 20 questions of one mark
each.
(iii) Section B – Question no. 21 to 26 comprises of 6 questions of two
marks each.
(iv) Section C – Question no. 27 to 32 comprises of 6 questions of four
marks each.
(v) Section D – Question no. 33 to 36 comprises of 4 questions of six
marks each.
(vi) There is no overall choice in the question paper. However, an internal
choice has been provided in 3 questions of one mark, 2 questions of two
marks, 2 questions of four marks and 2 questions of six marks. Only
one of the choices in such questions have to be attempted.
(vii) In addition to this, separate instructions are given with each section
and question, wherever necessary.
(viii) Use of calculators is not permitted.

Section – A
Question numbers 1 to 10 are multiple choice questions. Select the correct
option :
1. If A is a skew symmetric matrix of order 3, then the value of |A| is
(a) 3 (b) 0 (c) 9 (d) 27

2. If ^i, ^j, k^ are unit vectors along three mutually perpendicular directions, then

(a) ^i  ^j = 1 (b) ^i  ^j = 1 (c) ^i  k^ = 0 (d) ^i  k^ = 0

.65/5/3. 3 P.T.O.

3.   52       ,          , 
       
1 4 1 1
(a) (b) (c) (d)
3 13 4 2

4.  A  3  3    |A| = 8 ,  |3A|  
(a) 8 (b) 24 (c) 72 (d) 216

5. . x2 ex3 dx  
.
1 x3 1 x4 1 x3 1 x2
(a) e +C (b) e +C (c) e +C (d) e +C
3 3 2 2

x2 d2y
6.  y = loge e2 ,  dx2   
 
1 1 2 2
(a) – (b) – (c) (d) –
x x2 x2 x2

7.              3   ,  A  
    5   ,  B        P(AB) 
2 3
(a) (b) (c) 0 (d) 1
5 5

8. ABCD      E     
   
 EA + EB + EC + ED   
   
(a) 0 (b) AD (c) 2BC (d) 2AD

9.   (0, 0, 0)   –2x + 6y – 3z = – 7   
(a) 1  (b) 2  (c) 2 2  (d) 3 

.65/5/3. 4

3. A card is picked at random from a pack of 52 playing cards. Given that the
picked card is a queen, the probability of this card to be a card of spade is
1 4 1 1
(a) (b) (c) (d)
3 13 4 2

4. If A is a 3  3 matrix such that |A| = 8, then |3A| equals.
(a) 8 (b) 24 (c) 72 (d) 216

5. . x2 ex3 dx equals
.
1 x3 1 x4 1 x3 1 x2
(a) e +C (b) e +C (c) e +C (d) e +C
3 3 2 2

x2 d2y
6. If y = loge  2 , then equals
e  dx2
1 1 2 2
(a) – (b) – (c) (d) –
x x2 x2 x2

7. A die is thrown once. Let A be the event that the number obtained is greater
than 3. Let B be the event that the number obtained is less than 5. Then
P(AB) is
2 3
(a) (b) (c) 0 (d) 1
5 5

  
8. ABCD is a rhombus whose diagonals intersect at E. Then EA + EB + EC +

ED equals
   
(a) 0 (b) AD (c) 2BC (d) 2AD

9. The distance of the origin (0, 0, 0) from the plane –2x + 6y – 3z = – 7 is
(a) 1 unit (b) 2 units (c) 2 2 units (d) 3 units

.65/5/3. 5 P.T.O.

10.  2x + 3y > 6     :
(a)       
(b)       ,   2x + 3y = 6       
(c)  XOY-,   2x + 3y = 6      
(d)  XOY-

 11  15 ,    
11.  A  B  3      |A| = 5, |B| = 3 ,  |3 AB|  
 ______

b
12.  f(x) = ax + x (a > 0, b > 0, x > 0)     _____.

13.  (3, 4, –7)  (1, – 1, 6)         _______.

       _______   

dy
14.   x dx + 2y = x2     _____.

dy 2
  1 + dx = x   ______  

15.  A     _______   ,  A     
  

 16  20        
 1 –2 
16.   4 3          

.65/5/3. 6

10. The graph of the inequality 2x + 3y > 6 is
(a) half plane that contains the origin.
(b) half plane that neither contains the origin nor the points of the line
2x + 3y = 6.
(c) whole XOY – plane excluding the points on the line 2x + 3y = 6.
(d) entire XOY plane.

Fill in the blanks in Questions from 11 to 15.
11. If A and B are square matrices each of order 3 and |A| = 5, |B| = 3, then
the value of |3 AB| is ______

b
12. The least value of the function f(x) = ax + (a > 0, b > 0, x > 0) is _____.
x

13. The vector equation of a line which passes through the points (3, 4, –7)
and (1, – 1, 6) is _______.
OR
The line of shortest distance between two skew lines is _______ to both the
lines.

dy
14. The integrating factor of the differential equation x + 2y = x2 is _____.
dx
OR
2
dy
The degree of the differential equation 1 +   = x is ______ .
dx

15. A relation in a set A is called _______ relation, if each element of A is
related to itself.

Q. Nos. 16 to 20 are of very short answer type questions.
 1 –2 
16. Find the cofactors of all the elements of  .
 4 3 

.65/5/3. 7 P.T.O.

17.  f(x) = x|x|, x  R,  x = 0      

  17
18.    sin–1 sin – 8  .

4
.
19.    
.
|x – 5|dx
1

20.  f(x) = x4 – 10 ,  f(2.1)      


 y = 2 sin2 (3x)   x = 6        

 – 
  21  26     2   

. x+1
21.   : 
.
 (x + 2) (x + 3)
dx

4x + 3 2 2
22.  f(x) = 6x – 4 , x  3 ,     x  3   (fof) (x) = x, f  
  

    ℝ  R = {(a, b) : a < b}    (i)  ,
(ii)   

23. A  B    ,  P(A) = 0.3  P(B) = 0.6, P(A  B)   

2
 –. 1 1  2x
24.    
. x  e dx
 2x2
1
.65/5/3. 8

17. Let f(x) = x|x|, for all x  R check its differentiability at x = 0.

  17
18. Find the value of sin–1 sin –  .
  8 

4
.
19. Find the value of . |x – 5|dx.

1

20. If f(x) = x4 – 10, then find the approximate value of f(2.1).
OR

Find the slope of the tangent to the curve y = 2 sin2 (3x) at x = .
6

Section – B

Q. Nos. 21 to 26 carry 2 marks each.

. x+1
21. Find  . dx.
 (x + 2) (x + 3)

4x + 3 2 2
22. If f(x) = , x  , then show that (fof) (x) = x, for all x  . Also, write
6x – 4 3 3
inverse of f.
OR
Check if the relation R in the set of real numbers defined as
R = {(a, b) : a < b} is (i) symmetric, (ii) transitive

23. Given two independent events A and B such that P(A) = 0.3 and
P(B) = 0.6, find P(A  B)

2
. 1 1 
24. Evaluate .  – 2 e2x dx.
  x 2x 
1
.65/5/3. 9 P.T.O.

d2y
25.  x = a cos ; y = b sin  ,  dx2   


ecosx   sin2 x     

1
.  1 – 2x 
26.    :  .tan–1   dx
 1 + x – x2
0

 – 

  27  32     4   
5 12 
27. sin–1   + sin–1   = (x  0)  x     
x x 2

28.   yex/y dx = (xex/y + y2) dy, y  0      

dy
29.  y = (log x)x + xlogx ,  dx   

30.                    
                  
   

 X     30  A      40  B      
       Y ,    50  A      60  B
                     
   B              Y   
.65/5/3. 10

d2y
25. If x = a cos ; y = b sin , then find .
dx2
OR
Find the differential of sin2 x w.r.t. ecosx.

1
.  1 – 2x 
26. Find the value of  .tan–1   dx.
 1 + x – x2
0

Section – C

Q. Nos. 27 to 32 carry 4 marks each.
5 12 
27. Solve the equation x : sin–1   + sin–1   = (x  0)
x x 2

28. Find the general solution of the differential equation
yex/y dx = (xex/y + y2) dy, y  0

dy .
29. If y = (log x)x + xlogx, then find
dx

30. Three rotten apples are mixed with seven fresh apples. Find the
probability distribution of the number of rotten apples, if three apples are
drawn one by one with replacement. Find the mean of the number of rotten
apples.
OR
In a shop X, 30 tins of ghee of type A and 40 tins of ghee of type B which
look alike, are kept for sale. While in shop Y, similar 50 tins of ghee of type
A and 60 tins of ghee of type B are there. One tin of ghee is purchased from
one of the randomly selected shop and is found to be of type B. Find the
probability that it is purchased from shop Y.

.65/5/3. 11 P.T.O.

31.                A     
   5    10       B        8
   8                3  20 
    4      A      ` 100   B 
    ` 120              
  -            
      

  a = ^i + 2^j + 3^k  b = 2^i + 4^j – 5^k       
 
32.

   ,           



   ,  ABC   A (1, 2, 3), B(2, –1, 4)  C (4, 5, – 1),
    

 – 

  33  36     6   

33.  P(3, 4, 4)           A(3, – 4, – 5)  B(2, – 3, 1)
   ,  2x + y + z = 7    

34. (ax + by)     ,  x y = c2

.65/5/3. 12

31. A company manufactures two types of novelty souvenirs made of plywood.
Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each
for assembling. Souvenirs of type B require 8 minutes each for cutting and 8
minutes each for assembling. Given that total time for cutting is 3 hours 20
minutes and for assembling 4 hours. The profit for type A souvenir is ` 100
each and for type B souvenir, profit is ` 120 each. How many souvenirs of
each type should the company manufacture in order to maximize the profit ?
Formulate the problem as an LPP and solve it graphically.

32. If a = ^i + 2^j + 3^
k and b = 2^i + 4^j – 5^
 
k represent two adjacent sides of a
parallelogram, find unit vectors parallel to the diagonals of the
parallelogram.

OR

Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3),
B(2, –1, 4) and C (4, 5, – 1).

Section – D

Q. 33 to 36, carry 6 marks each.

33. Find the distance of the point P(3, 4, 4) from the point, where the line
joining the points A(3, – 4, – 5) and B(2, – 3, 1) intersects the plane
2x + y + z = 7.

34. Find the minimum value of (ax + by), where xy = c2.

.65/5/3. 13 P.T.O.

35.  a, b, c     p, q, r   ,    
 log a p 1 
 log b q 1 
 =0
 log c r 1 

 2 –3 5 
 A =  3 2 –4  ,  A–1   
 1 1 –2 

A–1   ,        

2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = – 3

36.       x2 + y2 = 9  (x – 3)2 + y2 = 9     
  

            :
4
. 2
 .(x – x) dx

1

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.65/5/3. 14

35. If a, b, c are pth, qth and rth terms respectively of a G.P, then prove that
 log a p 1 
 log b q 1 
 =0
 log c r 1 
OR
 2 –3 5 
 
If A =  3 2 –4  , then find A–1.
 1 1 –2 
Using A–1, solve the following system of equations :
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = – 3

36. Using integration find the area of the region bounded between the two
circles x2 + y2 = 9 and (x – 3)2 + y2 = 9.
OR
4
.
Evaluate the following integral as the limit of sums  .(x2 – x) dx.

1

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.65/5/3. 15 P.T.O.

.65/5/3. 16