LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – CHEMISTRY
FIRST SEMESTER – NOVEMBER 2012
MT 1102 - MATHEMATICS FOR CHEMISTRY
Date : 03/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Part A
Answer ALL questions:
(10 x 2 =20)
−1
1. Differentiate 𝑒 𝑠𝑖𝑛 𝑥 with respect to 𝑠𝑖𝑛−1 𝑥.
2. Find the slope of the tangent at (2,4) to the curve 𝑦 = 10 − 4𝑥 + 10𝑥 2
𝑎
3. Change the limits of 𝐼 = ∫−𝑎 𝑓(𝑥) 𝑑𝑥 as lower limit to 0 and upper limit to 𝑎.
4. Solve
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
− 5 𝑑𝑥 + 4𝑦 = 0.
5.
6.
7.
8.
9.
State Binomial theorem.
Solve the partial differential equation 𝑝 = 𝑦 2 𝑞 2 .
State DeMoivre’s theorem.
Show that cos ℎ2 𝑥 + 𝑠𝑖𝑛ℎ2 𝑥 = cos ℎ2𝑥.
From a well-shuffled pack of 52 cards, one card is drawn at random. What is the probability that it
will be (i) a jack (ii) a spade?
10. Define Normal distribution.
Part B
Answer any 5 questions:
(5 x8 = 40)
11. Find the angle of intersection of the cardioids 𝑟 = 𝑎(1 + cos 𝜃) and
𝑟 = 𝑏(1 − cos 𝜃).
12. If 𝑦 = 𝑐𝑜𝑠ℎ−1 𝑥 , then show that 𝑦 = log(𝑥 + √𝑥 2 − 1).
13. Evaluate: (i)  x 2e x dx , using Bernoulli’s formula.
(ii) Evaluate:

2
1
(sin x) 2
 (sin x)
0
1
2
1
 (cos x) 2
dx. .
(4+4)
14. Solve the differential equation (3𝐷2 + 𝐷 − 14)𝑦 = 13𝑒 2𝑥 .
15. Find the sum of the series 1 +
1+3
2!
+
1+3+32
3!
𝑢
+
1+3+32 +33
4!
sin 2𝑥
+⋯ ∞
16. If tan(𝑥 + 𝑖𝑦) = 𝑢 + 𝑖𝑣, then show that 𝑣 = sinh 2𝑦
17. Obtain the expansion of tan 𝑛𝜃.
18. Find the maxima and minima of the function 2𝑥 3 − 3𝑥 2 − 36𝑥 + 10.
Part C
Answer any TWO questions:
(2 x 20 = 40)
19. (i) Evaluate: ∫(log 𝑥)2 𝑑𝑥.
𝜋
𝜋
1
(ii) Prove that ∫02 log sin 𝑥 𝑑𝑥 = 2 log(2).
(10+10)
20. (i)Obtain the characteristic roots and the associated characteristic vectors of the
6 −2 2
matrix = [−2 3 −1] .
2 −1 3
(ii) Solve the partial differential equation with usual notations 𝑝 + 𝑞 = 𝑥 + 𝑦.
(15+5)
1
21. Given that 𝑓(𝑥) = 2 (𝜋 − 𝑥) in the interval 0 to 2𝜋. Find the Fourier coefficients 𝑎𝑛 and 𝑏𝑛 .Hence
1
1
1
𝜋
deduce that 1 − 3 + 5 − 7 + ⋯ = 4 .
22. (i)Expand 𝑠𝑖𝑛4 𝜃. 𝑐𝑜𝑠 2 𝜃 in series of cosines of multiples of 𝜃.
(ii)The mean mark of 100 students were found to be 40. Later it was observed that a score of 53
was misread as 83. Find the correct mean using the correct score.
(14+6)