MBOSE Class 12 Question Paper 2018 for Mathemtics

Total No. of Printed Pages—7
HS/XII/A. Sc. Com/M/18

2018

MATHEMATICS

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 2 questions have internal choices.
(v) Section—C has 5 questions, carrying 6 marks each, out of
which 2 questions have internal choices.

SECTION—A
4x + 3 2
1. If f (x ) = , x ¹ , show that ( f o f )(x ) = x.
6x - 4 3

2. For what value of k the function
ì x2 - 9
ï
f (x ) = í x - 3 , when x ¹ 3
ïî k , when x = 3

is continuous at x = 3?

/45

( 2 )

3. Find the domain and range of the function f : r ® r such
that f (x ) = x 2 + 1.

4. Show that ‘*’ on Q defined by a * b = ab + 1 is commutative
but not associative.

-1
5. Find the principal value of tan (- 3 ).

6. Find the matrix X such that 2A - B + X = 0, where
é3 1ù é -2 1 ù
A=ê ú and B = ê
ë0 2û ë 0 3 úû

7. Evaluate :

ò x log x dx

5
8. If A and B be two events such that 2P (A) = P (B ) = and
13
2
P (A|B ) = , find P (A È B ).
5

9. Prove that the points A (2 , 0 , 3), B (3 , 2 , - 1) and
C (1, - 2 , - 5) are collinear.

( 3 )

dy
10. Find , if x 2 + y 2 - 3xy = 1.
dx

11. If y = 2 sin x + 3 cos x, show that

d 2y
y+ =0
dx 2

r r
12. If a = 5i$ - $j + 7k$ and b = i$ - $j - lk$ , find the value of l for
r r r r
which a + b and a - b are perpendicular to each other.

13. Find the direction cosines of a line segment joining the
points A (2 , 5 , 7) and B (3 , 2 , 9).

14. Verify that y = A cos 2x + B sin 2x is a solution of the
differential equation

d 2y
+ 4y = 0
dx 2

15. Using the properties of determinants, prove that

a -b b -c c -a
b -c c -a a -b = 0
c -a a -b b -c

( 4 )

SECTION—B

16. Express the matrix
é -1 5 1 ù
A =ê 2 3 4ú
ê ú
ë 7 0 9û
as the sum of a symmetric and a skew-symmetric matrix.

17. Using vectors, find the area of DABC whose vertices are
A (1, 2 , 3), B (2 , 5 , - 1) and C (-1, 1, 2).

18. Find the equation of the plane passing through the
intersection of the planes 2x + 3y - z + 1 = 0 and
x + y - 2z + 3 = 0 and perpendicular to the plane
3x - y - 2z - 4 = 0.

Or

Find the image of the point (1, 2, 3) in the plane
x + 2y + 4z = 38.

19. Show that the function f (x ) = |x - 5|is continuous but not
differentiable at x = 5.

20. If y = x+ x+ x + ... ¥ , prove that

dy 1
=
dx 2y - 1

( 5 )

21. Evaluate :
2x + 9
ò (x + 2)(x - 3)2 dx

22. Verify Rolle’s theorem for the function
3 2
f (x ) = x - 7x + 16x - 12 in [ 2 , 3] .

23. Using the properties of definite integrals, prove that
1 1
ò0 x (1 - x) dx = 42
5

24. The volume of a spherical balloon is increasing at the rate
of 25 cubic centimeter per second. Find the rate of change
of its surface at the instant when its radius is 5 cm.

Or

Show that æç x + ö÷ has a maximum and a minimum, but
1
è xø
the maximum value is less than the minimum value.

25. Solve :
dy
(1 + x 2 ) + 2xy = cos x
dx

( 6 )

SECTION—C

26. Solve the following system of equations using matrix
method :
2x - 3y + 5z = 16
3x + 2y - 4z = - 4
x + y - 2z = - 3

27. Sketch the region common to the circle x 2 + y 2 = 16 and
the parabola x 2 = 6y. Also find the area of the region
using integration.

28. An insurance company insured 2000 scooters and
3000 motorcycles. The probability of an accident
involving a scooter is 0·01 and that of a motorcycle is
0·02. An insured vehicle met with an accident. Find the
probability that the accidented vehicle was a motorcycle.

29. A square piece of tin of side 18 cm is to be made into a
box without the top, by cutting a square piece from
each corner and fold up the flaps. What should be the
side of a square to be cut off so that the volume of the
box is maximum? Also find the maximum volume of the
box.

Or

A wire of length 36 cm is cut into two pieces; one of
the pieces is turned in the form of a square and the other
in the form of an equilateral triangle. Find the length of
each piece so that the sum of the areas of the two be
minimum.

( 7 )

30. A company makes two types of toys, A and B. Type A
requires 5 minutes each for cutting and 10 minutes each
for assembling. Type B requires 8 minutes each for
cutting and 8 minutes each for assembling. There are
3 hours available for cutting and 4 hours available for
assembling in a day. The profit is R 50 each on type A and
R 60 each on type B. How many toys of each type should
the company make in a day to maximize the profit?

Or

A small firm manufactures gold rings and chains. The
combined number of rings and chains manufactured per
day is at most 24. It takes one hour to make a ring and
half an hour for a chain. The maximum number of hours
available per day is 16. If the profit on a ring is R 300 and
that on a chain is R 190, how many of each should be
manufactured daily so as to maximize the profit?

HHH

8K—6170/45 HS/XII/A. Sc. Com/M/18