LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

 

AB 03

THIRD SEMESTER – November 2008

MT 3102/MT 3100 – MATHEMATICS FOR PHYSICS

 

 

 

Date : 11-11-08                       Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

Section A

Answer ALL questions:                                                                                                  (10 x 2 = 20)

1. If   then show that .
2. Prove that the subtangent to the curve  is of constant length.

 

1. Show that

 

4. Find the rank of the matrix .
5. Find .

1. Find .
2. Write the expansion of tan nq in terms of tanq.
3. Prove that cosh²x – sinh²x = 1.
4. A bag contains 3 red, 6 white, 7 blue balls. What is probability that two balls drawn are white and blue balls?
5. A Poisson variate X is such that 2 P(X = 1) = 2P(X =2). Find the mean.

 

Section B

Answer any FIVE questions:                                                                                       (5 x 8 = 40)

11. Find the derivative of .
12. Find the maxima and minima of .
13. Prove that .
14. Find .
15. Ifprove that .
16. Expand in terms of cosq.
17. Find the mean and standard deviation for the following data:

 

Years under	10	20	30	40	50	60
No. of people	15	32	51	78	97	109

18. X is normally distributed with mean 12 and standard deviation 4. Find the probability of the following:

(i)            X ³ 20             (ii)  X £ 20             (iii) 0 £ X £ 12.

given that z_2.0 = 0. 4772, z_3.0 = 0. 4987, z_4.0 = 0. 4999.

 

Section C

Answer any TWO questions:                                                                             (2 x 20 = 40)

 

19. (a) Find the sum of the series to infinity:

(b) If   then prove that  and hence prove .                                          (10 +10) 

 

20. (a) Find the characteristic roots and characteristic vectors of the matrix

.

(b)  Verify Cayley Hamilton Theorem for matrix .

(12+8)

21. (a) Find the Laplace transform of

 

(b) Using Laplace transform, solve the equation y¢¢ + 2y¢ – 3y = sin t, given that y = y¢ = 0 when t = 0.

(8+ 12)

22. (a) Expand sin³qcos⁵q in a series of sines of multiplies of q.

 

(b) In the long run 3 vessels out of every 100 are sunk. If 10 vessels are out, what is

the probability that (i) exactly 6 will arrive safely.  (ii) at least 6 will arrive safely.

(10 +10)

 

 

 

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