LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – APRIL 2012
MT 2100 - MATHEMATICS FOR COMPUTER SCIENCE
Date : 23-04-2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART A
Answer ALL the questions:
1. Define symmetric matrix with an example.
2. Prove that 𝑡𝑎𝑛ℎ 2𝑥 =
10x2 = 20
2 𝑡𝑎𝑛ℎ𝑥
.
1+𝑡𝑎𝑛ℎ 2 𝑥
1
1
3. Remove the fractional coefficients from the equation 𝑥 3 − 4 𝑥 2 + 3 𝑥 − 1 = 0.
4. Find the partial differential coefficients of 𝑢 = 𝑠𝑖𝑛(𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧).
𝜋⁄
2 𝑐𝑜𝑠 5 𝑥
5. Evaluate ∫0
6. Evaluate
3 1
∫2 ∫0 𝑥𝑦
𝑑𝑥 .
𝑑𝑥 𝑑𝑦.
𝑑2 𝑦
𝑑𝑦
7. Solve the equation 𝑑𝑥 2 + 2 𝑑𝑥 + 𝑦 = 0.
8. Derive the partial differential equation by eliminating the arbitrary constants from
(𝑥 + 𝑎)(𝑦 + 𝑏).
9. Find an iterative formula to √N, where N is a positive integer.
1𝑟𝑑
10. Write Simpson’s 3
rule.
PART B
Answer any FIVE questions:
5x8 = 40
11. Show that the equations 𝑥 + 2𝑦 − 𝑧 = 3, 3𝑥 − 𝑦 + 2𝑧 = 1, 2𝑥 − 2𝑦 + 3𝑧 = 2, 𝑥 − 𝑦 + 𝑧 = −1. are
consistent and solve them.
12. Prove that
sin 7𝜃
sin 𝜃
= 7 − 56 𝑠𝑖𝑛2 𝜃 + 112 𝑠𝑖𝑛4 𝜃 − 64 𝑠𝑖𝑛6 𝜃.
13. Find the condition that the roots of the equation 𝑎𝑥 3 + 3𝑏𝑥 2 + 3𝑐𝑥 + 𝑑 = 0 may be in geometric
progression.
𝑥 24
14. Integrate 𝑥 10+1 with respect to x.
15. (i) Evaluate ∫ 𝑥 2 cos 2𝑥 𝑑𝑥.
𝜋
(𝑠𝑖𝑛𝑥)3⁄2
𝜋
(ii) Prove that ∫02 (𝑠𝑖𝑛𝑥)3⁄2+(𝑐𝑜𝑠𝑥)3⁄2 𝑑𝑥 = 4 .
(4 + 4)
16. Solve the equation (𝐷 2 + 4𝐷 + 5)𝑦 = 𝑒 2𝑥 + 𝑥.
17. Solve (i) 𝑝2 + 𝑞 2 = 𝑛𝑝𝑞 (ii) 𝑝 + 𝑞 = 𝑥 + 𝑦.
(4 + 4)
𝑥
18. Determine the root of 𝑥𝑒 − 3 = 0 correct to three decimals using, Regula Falsi method.
PART C
Answer any TWO questions:
2x20 = 40
19. (i) Find all the characteristic roots and the associated characteristic vectors of the matrix
𝑧=
8 −6 2
A =[−6 7 −4].
2 −4 3
(ii) If cos(𝑥 + 𝑖𝑦) = cos 𝜃 + 𝑖 𝑠𝑖𝑛 𝜃, then prove that cos 2𝑥 + cosh 2𝑦 = 2.
20. (i) Solve the equation 6𝑥 5 − 𝑥 4 − 43𝑥 3 + 43𝑥 2 + 𝑥 − 6 = 0.
𝑥 3 +𝑦 3
𝑥+𝑦
(ii) If 𝑢 = 𝑡𝑎𝑛−1 (
𝜕𝑢
𝜕𝑢
), prove that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 𝑠𝑖𝑛 2𝑢.
(14+6)
(14+6)
𝑥+4
21. (i) Integrate 6𝑥−7−𝑥2 with respect to x.
𝑑2 𝑦
𝑑𝑦
(ii) Solve (5 + 2𝑥)2 𝑑𝑥 2 − 6(5 + 2𝑥) 𝑑𝑥 + 8𝑦 = 6𝑥.
(6+14)
22. (i) Solve (𝑦 2 + 𝑧 2 )𝑝 − 𝑥𝑦𝑞 = −𝑥𝑧.
10 𝑑𝑥
1+𝑥 2
(ii) Evaluate ∫0
using trapezoidal rule and Simpson’s rule.
*********************
$$$$$$$
(8+12)