LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  B.Sc. DEGREE EXAMINATION – PHYSICS

AA 03

THIRD SEMESTER – NOV 2006

         MT 3100 – ALLIED MATHEMATICS FOR PHYSICS

(Also equivalent to MAT 100)

 

 

Date & Time : 28-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section A

Answer ALL the questions (10 x 2 =20)

• Evaluate .
• Expand and .
• Prove that .
• Find when .
• Find
• Find
• Show that, in the curve , the polar subtangent varies as the square of the radius vector.

8)  Find the coefficient of  in the expansion of .

9)  Find the rank of the matrix

10) Mention a relation between binomial and Poisson distribution.

Section B

Answer any FIVE questions (5 x 8 = 40)

11) Find

12) Find

13) If , prove that.

14) If , prove that .

15) Find the slope of the tangent with the initial line for the cardioid

at   q = .

16) Find the inverse of the matrix using Cayley Hamilton

theorem.

17) Show that  .

18)  Suppose on an average one house in 1000 in Telebakkam city needs television

service  during  a  year. If  there  are  2000  houses  in  that  city,  what  is  the

probability  that exactly  5  houses will need  television service during the year.

Section  C

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

19) (a) Using Laplace transform solve  when x (0) = 7.5,

x’(0) = -18.5.

(b) Prove that cosh5x = 16cosh⁵x – 20cosh³x + 5coshx.                         (12+8)

20) (a) Expand cos⁴q sin³q in terms of sines of multiples of angle.

(b) If  prove that

(8+12)

21) (a) Find the eigen values and eigen vectors of the matrix

(b) Find the greatest term in the expansion of when .       (15+5)

 

22) (a) Find the angle of intersection of the cardioid and

.

(b) Eight coins are tossed at a time, for 256 times. Number of heads observed at

each throw is recorded and results are given below.

 

No of heads at a throw	0	1	2	3	4	5	6	7	8
frequency	2	6	30	52	67	56	32	10	1

 

What are the theoretical values of mean and standard deviation? Calculate also

the mean and standard deviation of the observed frequencies.                 (10+10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

CV 08

DEGREE EXAMINATION –PHYSICS

THIRD SEMESTER – APRIL 2007

MT 3100 – ALLIED MATHEMATICS FOR PHYSICS

 

 

Date & Time: 28/04/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

 

 

 

Section A

Answer ALL the questions (10 x 2 =20)

1) Evaluate.

2) Find .

3) Find .

4) If cos⁵q = Acosq + Bcos3q+ Ccos5q , prove that

sin⁵q = A sinq -Bsin3q + Csin5q.

5) If  , prove that .

6) If y = e ^2x, prove that .

7) Show that in the curve r = e ^{q cot a} , the polar subtangent is rtana.

8) Derive the expansion of  .

9) Find the coefficient of  in the series

10) Explain the concept of mutually exclusive events with an example.

 

Section B

Answer any FIVE questions (5 x 8 = 40)

11) Determine a, b, c so that .

12) Find  L[t² e ^-t cost].

13) Using Laplace transform solve , given.

14) Prove that  .

15) If , prove that .

16)  Verify Cayley Hamilton theorem for the matrix .

17)  Sum the binomial series  .

 

18)  Assuming that half the population own a two wheeler in a city so that the chance

of an individual  having a two  wheeler is ½  and assuming that 100 investigators

can  take sample of 10 individuals  to see  whether  they own a two wheeler, how

many  investigators  would you expect  to report  that  three  people or  less  were

having two wheelers.?

 

Section  C

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

19) Solve , given x(0) = 0: y(0)=2 using Laplace

transform.

20) (a) Find the real and imaginary parts of sin (x+iy) and tan(u + iv).

If , prove that .

(b) If  , prove that .                                  (10+10)

21) a) Find the angle at which the radius vector cuts the curve .

1. b) A person is  known to hit  target in 3 out  of 4 shots, whereas another person is

known to hit the target 2 out of 3 shots. Find the probability of the targets being

hit at all shots when they both try.

1. c) A bag contains 5 white  and  3 black balls. Two balls are drawn at random one

after the other without  replacement. Find  the probability that both balls drawn

are black.                                                                                                (10+5+5)

22) a) Find the eigen values and eigen vectors of the matrix

1. b) Find the first term with a negative coefficient in the expansion of

(15+5)

 

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