CBSE Class 12 Mathematics Question Paper 2020 Set 65-4-2

SET – 2
Series : HMJ/4
 .
Code No. 65/4/2
 .
   -  - 
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

.65/4/2. 333B 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 – 2y + 4z = 10  18x + 17y + kz = 50  ,  k    
(a) –4 (b) 4 (c) 2 (d) –2

3 2
2.  A = [2 –3 4], B =   , X = [1 2 3]  Y =  3  ,  AB + XY   
2
2 4
(a) [28] (b) [24] (c) 28 (d) 24

.65/4/2. 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.
Select the correct option :

1. The two planes x – 2y + 4z = 10 and 18x + 17y + kz = 50 are
perpendicular, if k is equal to
(a) –4 (b) 4 (c) 2 (d) –2

3 2
2  
2. If A = [2 –3 4], B =   , X = [1 2 3] and Y =  3  , then AB + XY equals
2 4
(a) [28] (b) [24] (c) 28 (d) 24

.65/4/2. 3 P.T.O.

 3
3. sin–1 cos     
 5
 3 – –3
(a) (b) (c) (d)
10 5 10 5

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

6.   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

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

200 50 50 40
8.  A =  10  
2   B =  2

3  ,  |AB|  
(a) 460 (b) 2000 (c) 3000 (d) –7000

       
9.  a = ^i – 2^j + 3k^    b   a . b = | b|2  | a – b|= 7 , | b|
  
(a) 7 (b) 14 (c) 7 (d) 21

.65/4/2. 4

 3
3. The value of sin–1 cos  is
 5
 3 – –3
(a) (b) (c) (d)
10 5 10 5

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

6. 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

7. 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

200 50 50 40
8. Let A =  10  
2  and B =  2

3  , then |AB| is equal to

(a) 460 (b) 2000 (c) 3000 (d) –7000

9.
 ^ If 
Let a = ^i – 2^j + 3k.
    
b is a vector such that a . b = | b|2 and | a – b|= 7,

then | b| equals
(a) 7 (b) 14 (c) 7 (d) 21
.65/4/2. 5 P.T.O.

10.            5     :
5 1 1 1
(a) (b) (c) (d)
216 6 36 49

 11  15         /  :
   
11.  a    ,  ( a .^i) ^i + ( a .^j) ^j + ( a .k)
^ k^   _________.

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

x+y 7  2 7
12.   9 =
x – y  9

4 ,  x y = _________
.

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

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

15.  f(x) = 2|x|+ 3|sin x|+ 6 ,  x = 0  f(x)      
_______ 

 16  20       
2
.
16.    : 
. |sin x|dx

0

a
. dx 
17.  .
 1 + 4x 2 = ,  ‘a’     
8
0


. dx
  : 
. x+x

.65/4/2. 6

10. Three dice are thrown simultaneously. The probability of obtaining a total
score of 5 is
5 1 1 1
(a) (b) (c) (d)
216 6 36 49

In Q. Nos. 11 to 15, fill in the blanks with correct word/sentence :
    ^ ^
11. 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
12. If  = 
4 , then x y = ________
.
 9 x – y  9

13. 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 _________.

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

15. If f(x) = 2|x|+ 3|sin x| + 6, then the right hand derivative of f(x) at x = 0
is _______.

Q Nos. 16 to 20 are very short answer type questions.
2
.
16. Evaluate . |sin x|dx

0

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

OR
. dx
Find .
 x+x
.65/4/2. 7 P.T.O.

dy2 dy
18.    y = ax + 2a2,   2 dx + x dx – y = 0     

.  x  x
19.   : 
.
sin5   . cos   dx
 2  2

1 0
20.  A =  1 
1  ,  A   
3

 – 
  21  26    2    
21.      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 

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

3 2 3
23.  P(A) = 10 , P(B) = 5  P(AB) = 5 ,  [P(B/A) + P(A/B)]   

24.    f  f(x) = (x – 1) ex + 1,      x > 0   
  
.65/4/2. 8

18. Show that the function y = ax + 2a2 is a solution of the differential equation
dy2 dy
2   + x   – y = 0.
dx dx

.  x  x
19. Find . sin5   . cos   dx
  2  2

1 0
20. If A =  1 
1  , then find A .
3


Section – B

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

21. 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 

       
22. 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.
^

3 2 3
23. Find [P(B/A) + P(A/B)], if P(A) = , P(B) = and P(AB) = .
10 5 5

24. Show that the function f defined by f(x) = (x – 1) ex + 1 is an increasing
function for all x > 0.

.65/4/2. 9 P.T.O.

25. log x   xlog x     

26.   2x + y + 2z = 8  4x + 2y + 4z + 5 = 0       

 – 
  27  32    4    

27.                   
                    
        

 100   5  1000   25           
     ,           

28.        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          
       ,     ,      
     ,      ?     ?

.65/4/2. 10

25. Find the derivative of xlog x w.r.t. log x.

26. Find the distance between the parallel planes 2x + y + 2z = 8 and
4x + 2y + 4z + 5 = 0

Section – C

Q. Nos. 27 to 32 carry 4 marks each.

27. 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.

28. 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 ?

.65/4/2. 11 P.T.O.

 1+x+ 1–x dy –1
29.  y = sin–1  2
 ,   
 dx
=
2 1 – x2

  
 – 2 , 2    f(x) = ex cos x        

30.                
dy
(x + 1) = 2e–y + 1; y = 0  x = 0
dx

x
31.   f(x) = x2 + 1 ,  xR    f : R → R       
  

.
32.    : . |x3 – x| dx

–1

 – 
  33  36    6    
     
33.    r = a +  b  r = b +  a          
  
      r . ( a × b)= 0  

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 

.65/4/2. 12

 1+x+ 1–x dy –1
29. 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

30. For the differential equation given below, find a particular solution
satisfying the given condition
dy
(x + 1) = 2e–y + 1 ; y = 0 when x = 0.
dx

x
31. Show that the function f : R → R defined by f(x) = ,  xR is neither
x2 + 1
one-one nor onto.


.
32. Evaluate : . |x3 – x| dx

–1

Section – D
Q. Nos. 33 to 36 carry 6 marks each.
     
33. Show that the lines r = a +  b and r = b +  a are coplanar and the plane
  
containing them is given by r . ( a × b)= 0.

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 

.65/4/2. 13 P.T.O.

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



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

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

36.    ,  {(x, y) : 0 < y < x2, 0 < y < x, 0 < x < 2}   
 

____________

.65/4/2. 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. Using integration find the area of the region :
{(x, y) : 0 < y < x2, 0 < y < x, 0 < x < 2}
____________

.65/4/2. 15 P.T.O.

.65/4/2. 16