2016

MATHEMATICS

Full Marks: 100

Pass Marks: 30

Time: 3 hours

The figures in the margin indicate full marks for the questions

1. Answer the following questions: 1x10=10

- Let. For, let if and only if. Write down as a subset of.
- What is the domain of the function?
- If, find.
- For what value of, the matrix is not invertible?
- If A is a square matrix of order 3 such that, find.
- A function is defined as follows

Which one of the following is true?

- is continuous at 0 and 1.
- is continuous at 1 and 2.
- is continuous at 0 and 2.
- is continuous at 0, 1 and 2.

- Which one of the following is true? For the function

- is strictly decreasing in
- is strictly increasing in
- is neither increasing nor decreasing in

- What is the unit vector along the vector?
- What are the direction cosines of the normal to plane?
- What is the equation of the?

2. Show that the relation R in the set Z of integers given by, 3+1

is an equivalence relation. Find the set of all elements related to 0.

Or

A function is defined by, . Examine if is one-one and onto. 2+2

3. If, prove that. 4

Or

Prove that,

4. If, show that . Hence find. 2+2=4

Or

Using elementary row operations, find the inverse of the matrix

5. Show that is continuous but not differentiable at. 2+2=4

Or

If, find. 4

6. Find the point at which the tangent to the curve has its slope. Also find the equation of the tangent at that point. 2+2=4

Or

State Rolle’s Theorem and verify it for the following function. 1+3=4

7. Evaluate the following integrals: 4x2=8

Or

Or

8. Solve: 4

Or

From a differential equation by eliminating the arbitrary constants a and b from 4

9. Show that the family of curves for which the slope of the tangent at any point on its is, is given by. 4

10. Find the area of the parallelogram whose diagonals are given by the vectors and. 4

Or

If, show that and are perpendicular to each other. 4

11. Find the shortest distance between the lines whose vector equations are given by

and 4

Or

Find the acute angle between the planes whose vector equations are

And 4

12. Two cards are drawn successively, without replacement from a well-shuffled pack of 52 cards. Find the probability distribution of the number of aces. 4

Or

A box contains 2 told and 3 silver coins. Another box contains 3 gold and 3 silver coins. A box is chosen at random and a coin is drawn from it. If the selected coin is a gold coin, find the probability that it was drawn from the second box. 4

13. Prove that 6

Or

Show that 6

14. Answer any one:

- Find the maximum and minimum values, if any, of the following functions. 3+3=6

- Find the intervals in which the function is

- Increasing.
- Decreasing.

15. Find the area of the region in the first quadrant enclosed by the, the lineand the circle. 6

Or

Find the area of the region common to the two circles and. 6

16. Evaluate as a limit of a sum. 6

17. Find the vector equation of the plane passing through the intersection of the planes and and perpendicular to the vector. 6

Or

Find the equation of the plane that makes intercepts a, b and c on x, y and z-axes respectively. Also, if p is the length of the normal from the origin to this plane, prove that:- 3+3=6

18. Solve the following linear programming problem graphically. 6

Maximize and minimize

Subject to constraints

Or

Maximize the minimize 6

19. The sum and the product of the mean and variance of a binomial distribution are 24 and 128 respectively. Find the distribution. 6

Or

If a fair coin is tossed 10 times, find the probability of getting 2+2+2=6

- Exactly six heads.
- At least six heads.
- At most six heads.