Time : 3 Hrs.
I.
II.
III.
IV.
V.
MM : 100
GENERAL INSTRUCTIONS
All question are compulsory.
The question paper consist of 29 questions divided into three sections A, B
and C. Section A comprises of 10 questions of one mark each, section B
comprises of 12 questions of four marks each and section C comprises of 07
questions of six marks each.
All questions in Section A are to be answered in one word, one sentences or
as per the exact requirement of the question.
There is no overall choice. However, internal choice has been provided in
04 questions of four marks each and 02 questions of six marks each. You
have to attempt only one of the alternatives in all such questions.
Use of the calculator is not permitted. You may ask for logarithmic tables, if
required.
SECTION - A
1.
Evaluate : sin (3 sin–1 0.4)
2.
If sin–1 x =
3.
4.
p
for some x Î [–1, 1], then find the value of cos–1 x.
5
é a 0ù
é1 0ù
2
If A = ê
ú and B = ê
ú , find the values of a for which A = B.
1
1
ë
û
ë5 1û
Given : 2x – y – 4z = 2, x – 2y – z = – 4, x + y + lz = 4, then calculate the value
of l such that the given system of equations has no solution.
2
6
5.
For the determinant
6.
If f(2) = 4 and f ¢(2) = 1, then find lim
7.
Find the integrating factor of the differential equation x
8.
Show that the vector 2iˆ - 3jˆ + 4kˆ and -4iˆ + 6jˆ - 8kˆ are collinear..
-1 4
, Find M12, A21.
xf (2) - 2f (x)
.
x-2
x ®2
dy
- y = 2 x2
dx
1
2
9.
10.
Find the area of a parallelogram whose adjacent sides are i – 2j + 3k and
2i + 1j – 4k.
Find the angle between the planes 2x – y + z = 11 and x + y + 2z = 3.
SECTION - B
11.
Find the value of k for which
ìï log ∋1 ∗ ax ( , log ∋1, bx (
f ∋ x ( < ïí
x
ïï
î
k
OR
, if x ¹ 0 is continuous at x = 0
, if x < 0
æ dy ö
b
< .
If x = a sin 2t(1 + cos 2t) and y = b cos 2t(1 – cos 2t), show that ççç ÷÷÷
è dx øat t < p a
4
12.
Let f : R ® R : f(x) = 4x + 3 for all x Î R. Show that f is invertible and find
f –1. Does the truthfulness and honesty may have any relation ?
13.
Prove that 2 sin–1
3
07 p
, tan ,1
<
5
31 4
3
14.
-2
sin 3q
1
Show that if the determinant D = -7 8 cos 2q = 0, then sin q = 0 or .
2
-11 14
2
OR
b-a c-a
0
0
If D = a - b
a-c b-c
15.
Evaluate :
c - b , then show that D is equal to zero.
0
x
ò x + 1 dx
OR
1
Evaluate:
ò 5 x - 3 dx
0
16.
17.
2
Verify mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].
A truck driver is driving a truck on the dangerous path given by
ì
x2 -1
, x ¹1
ï 2
ï
f (x) = í x - 2 | x - 1| -1
1
ï
, x =1
ïî
2
Find the dangerous point (point of discontinuity) on the path. Whether the
driver should pass that point or not ? Justify your answers.
3
p/2
18.
Evaluate:
ò
0
x
dx
sin x + cos x
p
19.
x
Evaluate : ò 1 + sin x dx
0
20.
21.
dy y
+ = 0, where x denotes the percentage population living in a city & y
dx x
denotes the area for living a healthy life of population. Find the particular
solution when x = 100, y = 1. Is higher density of population is harmful ?
Justify your answer.
Find the values of p so that the lines 1 - x = 7y - 14 = z - 3 and
7 - 7x y - 5 6 - z
=
=
are at right angles.
3p
1
5
22.
3
2p
2
Find the probability distribution of the number of kings drawn when two
cards are drawn one by one, without replacement, from a pack of 52 playing
cards.
OR
A die is thrown twice and the sum of the numbers appearing is observed to
be 7. What is the conditional probability that the number 2 has appeared at
least once?
SECTION - C
23.
Let the two matrices A and B be given by
é 1 ,1 0ù
é 2
2 , 4ù
ê
ú
ê
ú
A < ê 2 3 4ú and B < ê, 4 2 , 4ú
ê
ú
ê
ú
ê 0 1 2ú
ê 2 ,1 5 ú
ë
û
ë
û
24.
Verify that AB = BA = 6I, where I is the unit matrix of order 3 and hence solve
the system of equations.
x – y = 3, 2x + 3y – 4z = 17 and y – 2z = 7
A dietician wishes to mix two types of foods in such a way that vitamin
contents of the mixture contains atleast 8 units of Vitamin A and 10 units of
Vitamin C. Food 'I' contains 2 units/kg of Vitamin A and 1 unit/kg of Vitamin
C. Food 'II' contains 1 unit/kg of Vitamin A and 2 units/kg of Vitamin C. It
costs ` 50 per kg to purchase Food 'I' and ` 70 per kg to purchase Food 'II'.
Formulate this problem as a linear programming problem to minimise the
cost of such a mixture and solve it graphically.
3
4
OR
An oil company has two depots A and B with capacities of 7000 L and 4000
L respectively. The company is to supply oil to three petrol pumps, D, E and
F whose requirements are 4500L, 3000L and 3500L respectively. The distances
(in km) between the depots and the petrol pumps is given in the following
table:
Distance in (km.)
From/To
A
B
D
7
3
E
6
4
F
3
2
25.
Assuming that the transportation cost of 10 litres of oil is ` 1 per km, how
should the delivery be scheduled in order that the transportation cost is
minimum? What is the minimum cost?
Using the method of integration, find the area of the region bounded by the
lines 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
OR
4
Evaluate
ò ∋2 x
1
26.
27.
28.
29.
4
2
, x( dx as limit of a sum.
A window is in the form of a rectangle surmounted by a semicircle. If the
perimeter be 30 metres, then find the ratio of the base to the height of the
window so that the greatest possible amount of light may be admitted
through the window. Write one importance of sunlight in our life.
Suppose the growth of population is proportional to the number present. If
the population of a town doubles in 6 years. Prove that the population
becomes 8 times at the end of 18 years. In context of conservation of energy
for energy requirements for future generation, which three things you do to
conserve energy ?
Find the foot of the perpendicular drawn from the point whose position
r
ˆ
vector is 2iˆ - ˆj + 5kˆ to the line r = (11 + 10l )iˆ + (-2 - 4l )ˆj + (-8 - 11l)k.
Also, find the length of the perpendicular.
A manufacturer has three machine operators A (skilled) B (semi-skilled) and
C (non- skilled). The first operator A produces 1% defective items whereas
the other two operators B and C produces 5% and 7% defective items
respectively. A is on the job for 50% of time B in the job for 30% of the time
and C is on the job for 20% of the time. A defective item is produced what is
the probability that it was produced by B ? What is the importance of a
skilled person ?