2013
MATHEMATICS
Full Marks: 100
Pass Marks: 30
Time: 3 hours
The figures in the margin indicate full marks for the questions
NEW COURSE
1. Answer the following questions: 1x10=10
- Given that
is an equivalence relation in the set of integers
. What is the number of partitions of
?
- Write down the domain of the function
- If A is a square matrix of order 3 such that
than what is the value of
?
- If
then what is the value of
- If
is the cube root of unity, what is the value of the one root of the equation
- What is the equation of the normal at the point
if
at this point does not exist?
- What is the value of
, if
, where
denotes the greatest integer
- What is the projection vector of
along
?
- What is the distance of the point
from z-axis?
- Can a vector have direction angles as
2. Show that the intersection of two equivalence relations in a set is again an equivalence relation in the set. 4
3. Show that: 4
Or
Find x, if 
4. Using the properties of determinant, prove that 4
5. Show that 
is a continuous function but it is not differentiable at
. 4
is a continuous function but it is not differentiable at. 4
Or
If 
find 
using parametric co-ordinates.
find using parametric co-ordinates.
6. Find
, if 2+2=4
, if 2+2=4
7. Integrate: 
4
4
8. Evaluate any one of the following: 4
9. Answer any two of the following: 4x2=8
- Solve:
- Find the particular solution of the differential equation
given that
when
.
- Solve the differential equation:
10. Give the geometrical interpretation of 
and find the area of a parallelogram having diagonals given by the vectors
. 4
and find the area of a parallelogram having diagonals given by the vectors. 4
Or
Find the value of 
if the scalar product of the vector 
with a unit vector along the sum of the vectors 
is equal to unity.
if the scalar product of the vector with a unit vector along the sum of the vectors is equal to unity.
11. Find the shortest distance between the lines given by 4
12. Find the probability distribution of the number of heads from the tossing of a fair coin thrice. 4
Or
Let 
denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of
.
denote the sum of the numbers obtained when two fair dice are rolled. Find the variance of.
13. If 
then prove that 
6
then prove that 6
Or
Let 
Find 
and use this to solve the following system of equations:
and use this to solve the following system of equations: 
14. Answer (a) or [(b) and (c)] 6
- Prove that the curves
and
cut at right angles if
.
- Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum 3
- Show that
is an increasing function of
throughout its domain. 3
15. Evaluate by expressing 
as the limit of a sum. 6
as the limit of a sum. 6
16. Find the Cartesian equation of the plane passing through the intersection of the planes 
and 
and also passing through the point
. 6
and and also passing through the point. 6
Or
Show that the lines 
are coplanar. Find also the equation of the plane.
are coplanar. Find also the equation of the plane.
17. Using integration, find the area of the region bounded by the triangle whose vertices are 
6
6
Or
Find the area lying above the
and enclosed by the circle 
and the parabola
.
and enclosed by the circle and the parabola.
18. A manufacturer of furniture makes two products: Chairs and tables. Processing of the products is done on two machines A and B. A chair requires 2 hours on machine A and 6 hours on machine B. A table requires 5 hours on machine A and 2 hours on machine B. There are 16 hours of time available on machine A and 22 hours on machine B. If the profit gained by the manufacturer from a chair and a table are Rs. 3 and Rs. 5 respectively, how many pieces of each of chairs and tables must be produced in order that the profit gained becomes maximum? 6
19. Suppose there are four boxes A, B, C and D containing coloured marbles as given below: 6
Box
|
Marble Colour
| ||
Red
|
White
|
Black
| |
A
B
C
D
|
1
6
8
0
|
6
2
1
6
|
3
2
1
4
|
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from (i) Box A? (ii) Box B? (iii) Box C?
Or
From a lot of 30 bulbs which include 6 defective, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.