MBOSE Class 12 Question Paper 2020 for Mathematics Old Course

Total No. of Printed Pages—8
HS/XII/A.Sc.Com/M/OC/20

2020

MATHEMATICS

( Old Course )

Full Marks : 100

Time : 3 hours

General Instructions :
(i) Write all the answers in the Answer Script.
(ii) The question paper consists of three Sections—A, B
and C.
(iii) Section—A consists of 15 questions, carrying 2 marks
each.
(iv) Section—B consists of 10 questions, carrying 4 marks
each, out of which 3 questions have internal choices.
(v) Section—C has 5 questions, carrying 6 marks each,
out of which 2 questions have internal choices.

SECTION—A

1. Let S be the set of all real numbers and let R be a
relation in S defined by
R  {(a , b ) : (1  ab )  0}
Show that R is reflexive and symmetric. 2

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2. Let
 1  
R   a ,  : a  N and 1  a  5 
 a  
List the elements of the above relation. Find the domain
and range. 2

3. Show that the function f : r  r : f (x ) | x | is neither
one-one nor onto. 2

4. Show that the operation  on z , defined by
a  b  a  b  1  a, b  z

satisfies (a) closure property and (b) commutative law. 2

5. Construct a 3×2 matrix whose elements are given by
aij  i  2 j . 2

6. Let
2 3 1
A 
0 5 7 

Verify that ( A )  A . 2

7. Show that f (x )  x 3 is continuous at the point x  2. 2

8. Differentiate y  tan x with respect to x. 2

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9. If x 3  y 3  3axy , find dy . 2
dx

dy
10. Find , when x  at 2 , y  2at . 2
dx

11. A stone is dropped into a quiet lake and the waves move
in circles. If the radius of a circular wave increases at the
rate of 4 cm/s, find the rate of increase in its area at the
instant when its radius is 10 cm. 2

12. Evaluate : 2

2 x
 x e dx

13. Let

a  3iˆ  2 ˆj and b  2iˆ  3 ˆj

Is |a |  |b |? Is a  b ? 2

14. If
a  3iˆ  ˆj  4kˆ and b  6iˆ  5 ˆj  2kˆ

find a  b and |a  b |. 2

15. If E1 and E2 are two independent events such that
P (E1 )  0·35 and P (E1  E 2 )  0·60 , find P (E2 ). 2

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SECTION—B

16. Express the matrix
 3 4 
A 
1 1
as the sum of a symmetric and a skew-symmetric
matrix. 4
Or
If
3 2 
A 
2 1 
1
verify that A 2  4A  I  0 and hence find A . 4

17. Show that the semivertical angle of a right circular cone
of given surface area and maximum volume is
1
sin1   4
3

18. Verify Rolle’s theorem for the function f (x )  e x cos x in

   
 2 , 2  4

Or
Using Lagrange’s mean value theorem, find a point on
the curve y  x  2 , defined in the interval [2, 3] , where
the tangent is parallel to the chord joining the end points
of the curve. 4

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19. (a) The side of a square is increasing at the rate of
0·2 cm/s. Find the rate of increase of the perimeter
of the square. 2

(b) Find the approximate value of the cube root of 127. 2

20. Water is leaking from a conical funnel at the rate of
5 cm3/s. If the radius of the base of the funnel is 5 cm
and its altitude is 10 cm, find the rate at which the water
level is dropping when it is 2·5 cm from the top. 4

21. (a) If y  e x (sin x  cos x ) , prove that

d 2y dy
2  2y  0 2
2 dx
dx

(b) Solve the differential equation

dy 1  y2
 0 2
dx 1  x2

22. (a) Evaluate : 2
a
dx

0 ax  x 2

(b) Evaluate : 2

2
 2  x 2 dx
0

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Or

Evaluate : 4

x
 (a 2 cos2 x  b2 sin2 x ) dx
0

23. Find the intervals in which the function

f (x )  2x 3  9x 2  12x  1

is (a) strictly increasing and (b) strictly decreasing. 4

24. A line passes through the point (3, 4, 5) and is parallel
to the vector (2iˆ  2 ˆj  3kˆ ) . Find the equations of the line
in the vector as well as Cartesian forms. 4

25. Find the equations of the normals to the curve
3x 2  y 2  8 parallel to the line x  3y  4 . 4

SECTION—C

26. Using matrices, solve the following system of equations : 6

2 3 10
   4
x y z
4 6 5
   1
x y z
6 9 20
   2
x y z

HS/XII/A.Sc.Com/M/OC/20/32 [ Contd.

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Or
(a) Without expanding the determinants, prove that

a a 2 bc 1 a2 a3
b b2 ca  1 b 2 b 3
3
c c 2 ab 1 c2 c3

(b) Solve for x : 3

a x a x a x
a x a x a x 0
a x a x a x

27. Evaluate
b
 sin x dx
a

from the first principle. 6
Or

Find the area of the smaller region bounded by the
x2 y2 x y
ellipse   1 and the straight line   1. 6
2 2 a b
a b

28. (a) Find the angle between the lines whose direction
ratios are 2, 3, 6 and 1, 2, 2. 2
(b) Show that the lines

x 5 y 7 z 3 x 8 y 4 z 5
  and  
4 4 5 7 1 3
intersect each other. Also, find the point of their
intersection. 4

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29. A factory has three machines X, Y and Z, producing 1000
bolts, 2000 bolts and 3000 bolts per day respectively.
The machine X produces 1% defective bolts, Y produces
1·5% defective bolts and Z produces 2% defective bolts.
At the end of the day, a bolt is drawn at random and it
is found to be defective. What is the probability that this
defective bolt has been produced by machine X ? 6

30. A firm manufactures two types of products A and B and
sells them at a profit of R 5 per unit of type A and R 3
per unit of type B. Each product is processed on two
machines M1 and M2. One unit of type A requires one
minute of processing time on M1 and two minutes of
processing time on M2, whereas one unit of type B
requires one minute of processing time on M1 and one
minute of processing time on M2. Machines M1 and M2
are respectively available for at most 5 hours and 6
hours in a day. Find out how many units of each type of
products should the firm produce a day in order to
maximize the profit. Solve the problem graphically. 6

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HS/XII/A.Sc.Com/M/OC/20/32 20K—