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 zaxis?
 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
Or
If find using parametric coordinates.
6. Find, if 2+2=4
7. Integrate: 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
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.
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.
13. If then prove that 6
Or
Let
Find 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
16. Find the Cartesian equation of the plane passing through the intersection of the planes and and also passing through the point. 6
Or
Show that the lines 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
Or
Find the area lying above theand 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.