CBSE Class 12 Mathematics Question Paper 2020 Set 65-4-3

SET – 3
Series : HMJ/4
 .
Code No. 65/4/3
 .
   -  - 
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

.65/4/3. 333C 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    
1.   x = ay + b, z = cy + d  x = ay + b, z = cy + d   , 
a c a c
(a) + =1 (b) + = –1 (c) aa + cc = 1 (d) aa + cc = –1
a c a c

2 3 2
2.   x x x  + 3 = 0 ,  x    
4 9 1
(a) 3 (b) 0 (c) –1 (d) 1
.65/4/3. 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 of 1 mark each.
You have to select the correct choice :

1. The two lines x = ay + b, z = cy + d; and x = ay + b, z = cy + d are
perpendicular to each other, if
a c a c
(a) + =1 (b) + = –1 (c) aa + cc = 1 (d) aa + cc = –1
a c a c

2 3 2
 
2. If  x x x  + 3 = 0, then the value of x is
4 9 1
(a) 3 (b) 0 (c) –1 (d) 1

.65/4/3. 3 P.T.O.

3.     ,    z = ax + by       
    ,     ,       
 :
(a) 0 (b) 2 (c)  (d) 

a
4.  { 1,2,3,4,5 }    a  b (a ≠ b)      b   
   :
1 1 1 3
(a) (b) (c) (d)
3 4 2 5

/8
5. . tan2 (2x)   
.
0

4– 4+ 4– 4–
(a) (b) (c) (d)
8 8 4 2

  1    
6.  a . b = 2 | a | | b| ,  a  b      
(a) 0° (b) 30° (c) 60° (d) 90°

7.    3 , 4   2      2      
 ,           :
1 1 1 1
(a) (b) (c) (d)
18 36 12 24

1  5 
8. tan–1  cos–1      
2  3 
3+ 5 3– 5 –3 + 5 –3 – 5
(a) (b) (c) (d)
2 2 2 2

.65/4/3. 4

3. In an LPP, if the objective function z = ax + by has the same maximum
value on two corner points of the feasible region, then the number of
points at which zmax occurs is
(a) 0 (b) 2 (c) finite (d) infinite

4. From the set { 1,2,3,4,5 }, two numbers a and b (a ≠ b) are chosen at
a
random. The probability that is an integer is :
b
1 1 1 3
(a) (b) (c) (d)
3 4 2 5

/8
5. . tan2 (2x) is equal to
.
0

4– 4+ 4– 4–
(a) (b) (c) (d)
8 8 4 2

  1    
6. If a . b = | a | | b|, then the angle between a and b is
2
(a) 0° (b) 30° (c) 60° (d) 90°

7. A bag contains 3 white, 4 black and 2 red balls. If 2 balls are drawn at
random (without replacement), then the probability that both the balls
are white is
1 1 1 1
(a) (b) (c) (d)
18 36 12 24

1  5
8. The value of tan–1  cos–1   is
2  3 
3+ 5 3– 5 –3 + 5 –3 – 5
(a) (b) (c) (d)
2 2 2 2

.65/4/3. 5 P.T.O.

a 0 0
9.  A = 0 a 0 ,  det(adj A)   

0 0 a
(a) a 27 (b) a9 (c) a6 (d) a2

x–2 y–3 z–4
10.    3 = 4 = 5   ,  
(a) 2x + 3y + 4z = 0
(b) 3x + 4y – 5z = 7
(c) 2x + y – 2z = 0
(d) x – y + z = 2

 11  15         /  :
11.  y = x3 – x   (2, 6)       _______.

      ,   r  ,  r = 3  ,  _________.

12.  f : R → R, f(x) = (3 – x3)1/3   ,  fof (x) = _________

   
13.  a    ,  ( a .^i) ^i + ( a .^j) ^j + ( a .k)
^ k^   _________.


 ^i + ^j   ^i – ^j    _________.

x+y 7  2 7
14.   9 =
x – y  9

4 ,  x y = _________
.

15.  f(x) = x|x|,  f  (x) = _________.

 16  20       
dy 2 dy
16.    y = ax + 2a2,   2 dx + x dx – y = 0     

2 –1
17.  A = 4 
3  ,  adj A   

.65/4/3. 6

a 0 0
 0 , then det (adj A) equals
9. If A = 0 a

0 0 a
(a) a 27 (b) a9 (c) a6 (d) a2

x–2 y–3 z–4
10. The line = = is parallel to the plane
3 4 5
(a) 2x + 3y + 4z = 0
(b) 3x + 4y – 5z = 7
(c) 2x + y – 2z = 0
(d) x – y + z = 2

In Q. Nos. 11 to 15, fill in the blanks with correct word / sentence :
11. The slope of the tangent to the curve y = x3 – x at the point (2, 6) is
_______.
OR
The rate of change of the area of a circle with respect to its radius r, when
r = 3 cm, is _________.

12. If f : R → R be given by f(x) = (3 – x3)1/3, then fof (x) = ________

    ^ ^
13. If a is a non-zero vector, then ( a .^i) ^i + ( a .^j) ^j + ( a .k) k equals _________.
OR
The projection of the vector ^i – ^j on the vector ^i + ^j is _________.

x + y 7  2 7
14. If  = 
4 , then x y = ________
.
 9 x – y  9

15. If f(x) = x | x |, then f ʹ (x) = _________.

Q. Nos. 16 to 20 are very short answer type questions.
16. Show that the function y = ax + 2a2 is a solution of the differential equation
dy2 dy
2   + x   – y = 0.
dx dx

2 –1
17. Find adj A, if A = 4 
 3 

.65/4/3. 7 P.T.O.

a
. dx 
18.  
. = ,  ‘a’     
 1 + 4x2 8
0

. dx
  : 
. x+x

. 1
19.    
.
 x(1 + x2)
dx

3/2
.
20.  [x]    ,  
. [x2] dx   

0

 – 
  21  26    2    
       
21.  | a| = 2| b|  ( a + b)  ( a – b) = 12 ,  | a|  | b|   

 
 a = 4^i + 3^j + k^  b = 2^i – ^j + 2k^          

 dy
22.  =     ,  x = cos  – cos 2, y = sin  – sin 2.
3 dx

23.        y-  3     xz –  
  

24.      A = { 1, 2, 3, 4, 5, 6 }  R = { (x, y) : y, x    } 
  R (i)   (ii)   

   :
9 9 1 9 2 2
– sin–1   = sin–1  
8 4 3 4  3 

x 3
25.    f(x) = 3 + x ,  (–3, 0) (0, 3)    

.65/4/3. 8

a
. dx 
18. If . = , then find the value of a.
 1 + 4x 2 8
0
OR
. dx
Find .
 x+x

. 1
19. Find . dx
 x(1 + x2)

3/2
.
20. If [x] denotes the greatest integer function, then find . [x2] dx

0

Section – B
Q. Nos. 21 to 26 carry 2 marks each.
       
21. Find | a| and | b|, if | a| = 2| b| and ( a + b) . ( a – b) = 12.
OR

Find the unit vector perpendicular to each of the vectors a = 4^i + 3^j + k^

and b = 2^i – ^j + 2k.
^

dy 
22. Find the value of at  = , if x = cos  – cos 2, y = sin  – sin 2.
dx 3

23. Find the equation of the plane with intercept 3 on the y-axis and parallel
to xz – plane.

24. Check if the relation R on the set A = { 1, 2, 3, 4, 5, 6 } defined as
R = { (x, y) : y is divisible by x } is (i) symmetric (ii) transitive
OR
Prove that :
9 9 1 9 2 2
– sin–1   = sin–1  
8 4 3 4  3 

x 3
25. Show that the function f(x) = + decreases in the intervals (–3, 0) (0, 3).
3 x

.65/4/3. 9 P.T.O.

26.  50              
    2  3     

 – 
  27  32    4    
27.        I, II  III     I  II  12 
          III      5    
      M  N    ,     
       M  N          
    ( )      
    ( )

I II III
M 1 2 1
N 2 1 1.25
  M  ` 600     N  ` 400          
       ,     ,      
     ,      ?     ?

28.     Z  R = {(x, y) : (x – y)   5  }    R, 
   

29.                   
                    
        

 100   5  1000   25           
     ,           
.65/4/3. 10

26. Three distinct numbers are chosen randomly from the first 50 natural
numbers. Find the probability that all the three numbers are divisible by
both 2 and 3.

Section – C

Q. Nos. 27 to 32 carry 4 marks each.
27. A manufacturer has three machines I, II and III installed in his factory.
Machine I and II are capable of being operated for atmost 12 hours
whereas machine III must be operated for atleast 5 hours a day. He
produces only two items M and N each requiring the use of all the three
machines.
The number of hours required for producing 1 unit of M and N on three
machines are given in the following table :
Number of hours required on machines
Items
I II III
M 1 2 1
N 2 1 1.25
He makes a profit of ` 600 and ` 400 on one unit of items M and N
respectively. How many units of each item should he produce so as to
maximize his profit assuming that he can sell all the items that he
produced. What will be the maximum profit ?

28. Prove that the relation R on Z, defined by R {(x, y) : (x – y) is divisible by
5} is an equivalence relation.

29. A coin is biased so that the head is three times as likely to occur as tail. If
the coin is tossed twice, find the probability distribution of number of tails.
Hence find the mean of the number of tails.
OR
Suppose that 5 men out of 100 and 25 women out of 1000 are good orators.
Assuming that there are equal number of men and women, find the
probability of choosing a good orator.

.65/4/3. 11 P.T.O.

 1+x+ 1–x dy –1
30.  y = sin–1  2
 ,   
 dx
=
2 1 – x2



  
 – 2 , 2    f(x) = ex cos x        

1
.
31.    : 
. 3 – 2x – x2 dx

0

dy 1 ey
32.   dx + x = x      

 – 

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

33.       ,   ,  ,  x – ,  y = x
  x2 + y2 = 32    

34.          :

a–b b+c a
b–c c+a b  = a3 + b3 + c3 – 3 abc.
 
c–a a+b c



1 3 2 
 A =  2 0 –1  ,    A3 – 4A2 – 3A + 11 I = O  A–1   

1 2 3 

.65/4/3. 12

 1+x+ 1–x dy –1
30. If y = sin–1   , then show that =
 2  dx 2 1 – x2

OR

  
Verify the Rolle’s Theorem for the function f(x) = ex cos x in – , 
 2 2

1
.
31. Evaluate . 3 – 2x – x2 dx

0

dy 1 ey
32. Find the general solution of the differential equation + = .
dx x x

Section – D

Q. Nos. 33 to 36 carry 6 marks each.

33. Find the area of the region lying in the first quadrant and enclosed by the
x – axis, the line y = x and the circle x2 + y2 = 32.

34. Using properties of determinates prove that :

a–b b+c a
b–c c+a b  = a3 + b3 + c3 – 3 abc.
 
c–a a+b c

OR

1 3 2 
 –1  , then show that A3 – 4A2 – 3A + 11 I = O. Hence find A–1.
If A =  2 0

1 2 3 

.65/4/3. 13 P.T.O.

35.       f(x) = (x – 1)3 (x – 2)2 ()    () 
  



36         ,        

            

36.  x – 2y = 0   (–1, 3, 4)     

____________

.65/4/3. 14

35. Find the intervals on which the function f(x) = (x – 1)3 (x – 2)2 is (a) strictly
increasing (b) strictly decreasing.

OR

Find the dimensions of the rectangle of perimeter 36 cm which will sweep
out a volume as large as possible, when revolved about one of its side.
Also, find the maximum volume.

36. Find the image of the point (–1, 3, 4) in the plane x – 2y = 0.

____________

.65/4/3. 15 P.T.O.

.65/4/3. 16