LOYOLA COLLEGE (Autonomous), chennai – 600 034

B.Sc.  degree examination – physics

third semester -april 2003

 Mt  3100/ MAT 100 mathematics for physics

28.04.2003                                                                                     Max.: 100 Marks

9.00 – 12.00

 

PART – A                                       (10 ´ 2 = 20 Marks)

1. Define Laplace transform of f(t) and prove that L(e^–at) = .
2. Find .
3. Prove that the mean of the Poisson distribution P_r =, r = 0, 1, 2, 3 ….. is equal to m.
4. Mention any two significance of the normal distribution.
5. Find the .
6. Find L (1+ t)² .
7. Find L^-1.
8. Write down the real part of sin .
9. Prove that in the R.H. xy = c², the subnormal varies as the cube of the ordinate.
10. If y = log (ax +b), find y

 

PART – B                                          (5 ´ 8 = 40 Marks)

Answer any FIVE questions.  Each question carries EIGHT marks

11. (a) Find L^-1

           

• Find L .

12. Find L  .
13. Define orthoganal matrix and prove that the matrix is orthoganal.
14. Verify cayley-Hamilton theorem and hence find the inverse of
15. (i)  Prove that

• Find the sum to infinity of series

 

 

 

 

 

16. (i) Find q approximately to the nearest minute if cos q =

(ii)   Determine a, b, c such that     .

 

17. If cos (x + iy) = cos q + i sinq,  Show that cos 2x + cosh 2y =2.

 

18. What is the rank of .

 

PART – C                                      (2 ´ 20 = 40 Marks)

Answer any TWO questions. Each question carries twenty marks.

 

19. (a) If y = .

 

• Find the angle of intersection of the cardioids r = a(1+cosq) and r = b(1-cosq).

20. (a) Certain mass -produced articles of which 0.5 percent are defective, are packed

in  cartons each containing 130 article.  What Proportion of cartons are free

from defective articles, and what proportion contain 2 or more defectives

(given e^-2.2 = 0.6065)

 

• Of a large group of men 5 percent are under 60 inches in height and 40 percent are between 60 and 65 inches. Assuming a normal distribution find the mean height and standard deviation.

 

21. (a) Find the sum to infinity of the series

 

• From a solid sphere, matter is scooped out so as to form a conical cup, with vertex of the cup on the surface of the sphere, Find when the volume of the cup is maximum.

 

22. a) Prove that sin⁵q =
23. b) Prove that sin⁴q cos²q =

 

 

 

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