LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
THIRD SEMESTER – APRIL 2012
MT 3102/3100 - MATHEMATICS FOR PHYSICS
Date : 28-04-2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
Section A
(10  2 = 20)
Answer ALL questions:
1.
2.
3.
4.
Find the nth derivative of e4x.
Show that in the curve r = a, the polar sub tangent is constant.
Expand 𝑎 𝑥 in ascending powers of x, ‘a’ being positive.
Define a symmetric matrix and give an example.
5. Find the Laplace transform of t2 + 2t + 3.
1
).
𝑠(𝑠+𝑎)
6. Find 𝐿−1 (
7.
8.
9.
10.
Prove that 2 sinh 𝑥 cosh 𝑥 = sinh 2𝑥.
Write down the expansion of sin 𝜃 and cos 𝜃 in a series of ascending powers of 𝜃.
Two dice are thrown. What is the probability that the sum of the numbers is greater than 8?
Write a short note on binomial distribution.
Section B
(5  8 = 40)
Answer any FIVE questions:
11. Find the nth differential coefficient of sinx sin2x sin3x.
12. Find the angle of intersection of curves rn = ancosn and rn = an sinn.
ℎ2
ℎ4
ℎ6
13. Show that 𝑙𝑜𝑔(𝑥 + 2ℎ) = 2𝑙𝑜𝑔(𝑥 + ℎ) − 𝑙𝑜𝑔𝑥 − [(𝑥+ℎ)2 + 2(𝑥+ℎ)4 + 3(𝑥+ℎ)6 + ⋯ ∞].
−1
2
2
1
14. Show that the matrix 𝐴 = 3 ( 2 −1
2 ) is orthogonal.
2
2 −1
0 𝑤ℎ𝑒𝑛 0 < 𝑡 ≤ 2
15. Find the Laplace transform of 𝑓(𝑡) = {
3
𝑤ℎ𝑒𝑛 𝑡 > 2
16. Separate into real and imaginary parts of 𝑡𝑎𝑛(𝑥 + 𝑖𝑦).
−1
17. Prove that 𝑠𝑖𝑛7 𝜃 = (𝑠𝑖𝑛7𝜃 − 7𝑠𝑖𝑛5𝜃 + 21𝑠𝑖𝑛3𝜃 − 35𝑠𝑖𝑛𝜃).
64
18. Calculate the mean and standard deviation for the following frequency distribution:
Class
Interval
0-8
8 - 16
16 - 24
24 – 32
32 - 40
40 - 48
Frequency
8
7
16
24
15
7
Section C
(2  20 = 40)
Answer any TWO questions:
19. a) If y = acos(logx) + bsin(logx), prove that x2yn + 2 + (2n + 1)xyn + 1 + (n2 + 1)yn = 0.
2.4
2.4.6
2.4.6.8
b) Find the sum to infinity of the series 3.6 + 3.6.9 + 3.6.9.12 + ⋯ ∞.
(12 + 8)
2 −2 3
20.a) Find the characteristic roots of the matrix 𝐴 = [1 1
1 ].
1 3 −1
2 −1 1
b) Verify Cayley Hamilton Theorem for matrix 𝐴 = [−1 2 −1] and also find 𝐴−1.
1 −1 2
(6 + 14)
1−𝑒 𝑡
21. a) Find 𝐿 (
𝑡
).
b) Solve the equation
𝑑2 𝑦
𝑑𝑡 2
𝑑𝑦
+ 2 𝑑𝑡 − 3𝑦 = 𝑠𝑖𝑛𝑡 given that 𝑦 =
𝑑𝑦
𝑑𝑡
= 0 when t = 0.
(5 + 15)
22. a) If cos(𝑥 + 𝑖𝑦) = cos 𝜃 + 𝑖 sin 𝜃 prove that cos 2𝑥 + cosh 2𝑦 = 2.
b) Express 𝑠𝑖𝑛3 𝜃𝑐𝑜𝑠 4 𝜃 in a series of sines of multiples of θ.
c) A car hire firm has two cars, which it hires out day by day. The number of demands for a car on each
day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on which
(i) neither car is used, and (ii) the proportion of days on which some demand is refused.
(8+5+7)
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