MBOSE Class 12 Question Paper 2019 for Mathematics

Total No. of Printed Pages—7
HS/XII/A. Sc. Com/M/19

2019

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

1. Show that the function f : r ® r defined by f (x ) = x 3 is
one-one and onto.

2. Evaluate :
ìp æ 1 öü
sin í - sin -1 ç - ÷ý
î3 è 2 øþ

/60

( 2 )

é 1 3ù é - 2 5ù
3. If A = ê ú and B = ê ú, then find A - 4B .
ë - 2 5û ë 3 4û

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

5. For what value of k the function
k cos x p
f (x ) = , when x ¹
p - 2x 2
p
=3 , when x =
2
p
is continuous at x = ?
2

6. Show that
d
(sin 2x sin 4x ) = 3 sin 6x - sin 2x
dx

7. The side of a square sheet of metal is increasing at 3 cm
per minute. At what rate the area is increasing when the
side is 10 cm long?

8. Evaluate :
p /2
ò- p/2 |sin x|dx

9. Solve the equation
æ dy ö
log ç ÷ = ax + by
è dx ø

( 3 )

10. Evaluate :
x æ1 + x log x ö
ò e çè x
÷ dx
ø

11. Find the value of the integral
p /2 sin x
ò0 1 + cos2 x
dx

r r
12. Find the unit vector perpendicular to both a and b where
r r
$ $ $
a = 3i + j - 2k and b = 2i$ + 3 $j - k$

13. Find the value of k so that the lines
1 - x 7y - 14 z - 3 7 - 7x 5 - y 6 - z
= = and = =
3 2k 2 3k 1 5
are at right angles.

r r r r r r
14. If a , b , c are unit vectors such that a + b + c = 0, find
the value of
r r r r r r
a ×b + b ×c + c ×a

15. The probability that a student selected at random from a
4
class will pass in Hindi is and the probability that he
5
1
passes in Hindi and English is . What is the probability
2
that he will pass in English if it is known that the student
has passed in Hindi?

( 4 )

SECTION—B

16. Let n be the set of natural numbers and let R be a relation
on n ´ n defined by
(a , b ) R (c , d ) Û ad = bc
Prove that R is an equivalence relation.

17. Using the properties of determinants, show that

1 a a 2 - bc
1 b b 2 - ca = 0
1 c c 2 - ab

1 + log t 3 + 2 log t dy
18. If x = and y = , then show that = t.
2 t dx
t

19. Evaluate :
-1
ò (sin x )2 dx

20. Using differential, find the approximate value of 3127.

21. Find the interval in which the function
2 3
f (x ) = 5 + 36x + 3x - 2x is (a) strictly increasing and
(b) strictly decreasing.

( 5 )

Or
Find the equation of the tangent to the curve x 2 + 3y = 3
which is parallel to the line y - 4x + 5 = 0.

22. Using the properties of definite integral, show that
p /3 1 p
òp/6 1 + tan x dx = 12

23. Evaluate the following integral as the limit of a sum :
4
ò1 (3x + 2x) dx
2

24. Find the equation of the plane passing through the point
(1, 0 , - 2) and perpendicular to each of the planes
2x + y - z - 2 = 0 and x - y - z - 3 = 0.

Or

Find the length and foot of the perpendicular from the
point (7, 14, 5) to the plane 2x + 4y - z = 2.

25. Find the shortest distance between the lines
r
r = (6i$ + 3k$ ) + l (2i$ - $j + 4k$ ) and
r
r = (- 9i$ + $j - 10k$ ) + m (4i$ + $j + 6k$ )

( 6 )

SECTION—C

26. Using integration, find the area of the region in the first
quadrant enclosed by the X-axis, the line y = x and the
circle x 2 + y 2 = 32.

27. Using matrices, solve the following system of equations :
2 3 3
- + = 10
x y z
1 1 1
+ + = 10
x y z
3 1 2
- + = 13
x y z

28. Show that the maximum volume of the cylinder which can
be inscribed in a sphere of radius 5 3 cm is (500 p) cm 3 .

Or

A window is in the form of a rectangle, surmounted by a
semicircular opening. The total perimeter of the window is
10 metres. Find the dimensions of the window to admit
maximum light through it.

29. A box contains 16 bulbs, out of which 4 bulbs are
defective. 3 bulbs are drawn one by one from the box
without replacement. Let X be the number of defective
bulbs drawn. Find the mean and variance of X.

( 7 )

30. A manufacturer produces two types of steel trunks. He
has two machines A and B. The first type of trunk requires
3 hours on machine A and 3 hours on machine B. The
second type requires 3 hours on machine A and 2 hours
on machine B. Machines A and B can work at most
18 hours and 15 hours per day respectively. He earns a
profit of R 30 and R 25 per trunk of first and second type
respectively. How many trunks of each type must he
make each day to make maximum profit?

Or

Two tailors A and B, earn R 300 and R 400 per day
respectively. A can stitch 6 shirts and 4 pairs of trousers
per day while B can stitch 10 shirts and 4 pairs of
trousers per day. How many days should each of them
work if it is desired to produce at least 60 shirts and
32 pairs of trousers at a minimum labour cost?

HHH

HS/XII/A. Sc. Com/M/19/60 K9—5980