# MT 1103 - Loyola College ```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – COMPUTER SCI. &amp; APPL.
FIRST SEMESTER – NOVEMBER 2012
MT 1103 - MATHEMATICS FOR COMPUTER SCIENCE
Date : 03/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Part A
1.
2.
3.
4.
(10X2 =20)
Define Unitary Matrix.
Write down the expansion of cos 3𝛳 in terms of cosθ.
If α and β are the roots of 2x2 + 3x +5 = 0, find α+β and αβ.
Find partial differential coefficients of u = sin (ax + by + cz) with respect to x, y and z.
3
5. Evaluate ∫1 (𝑥 2 + 3𝑥 + 2)𝑑𝑥.
𝜋
6. Evaluate ∫02 𝑠𝑖𝑛7 𝑥 𝑑𝑥.
7. Solve the differential equation (D2 +2D + 1)y = 0.
8. Find the complete integral of z  px  qy  c (1  p 2  q 2 )
9. Write the formula for Trapezoidal rule.
10. Write Newton’s backward difference formula for first and second order derivatives.
Part B
(5 x8 = 40)
11. Test the consistency of the following system of equations and if consistent solve
2x-y-z = 2, x+2y+z = 2, 4x-7y-5z = 2.
12. Show that
sin 6𝜃
sin 𝜃
4
= 32 cos 5 𝜃 − 32 cos3 𝜃 + 6 cos 𝜃.
13. Solve 𝑥 5 + 4𝑥 + 𝑥 3 + 𝑥 2 + 4𝑥 + 1 = 0.
14. What is the radius of curvature of the curve 𝑥 4 + 𝑦 4 = 2 at the point (1,1).
𝜋
𝜋
15. Show that ∫04 log(1 + tan 𝜃) 𝑑𝜃 = 8 log 2 .
16. Evaluate: ∫ x 2 cos 2𝑥 𝑑𝑥.
𝑑2 𝑦
𝑑𝑦
17. Solve the equation 𝑑𝑥 2 − 5 𝑑𝑥 + 6𝑦 = 𝑥 + 3𝑒 3𝑥 .
18. Find by Newton-Raphson method, the real root of 𝑥 3 − 2𝑥 − 5 = 0, correct to three decimal
places.
Part C
(2 x 20 = 40)
1
19. Verify Cayley-Hamilton theorem for the matrix 𝐴 = [2
1
20. (i) Evaluate: ∫
2𝑥+3
dx
𝑥 2 +𝑥+1
𝜋
√sin 𝑥
2
(ii) Evaluate: ∫0
0
3
1 −1] and hence find its inverse.
−1 1
√sin 𝑥+√cos 𝑥
𝑑𝑥.
(15+5)
2
21. (a) Solve the equation x 2
d y
dy
 7 x  12 y  x 2 .
2
dx
dx
(b) Solve q2 - p = y – x.
(14+6)
22. (i) Solve 𝑥 3 − 3𝑥 + 1 = 0 upto 3 decimals by using Regula-flasi method.
1 1
(ii) Evaluate ∫0
1+𝑥
𝑑𝑥 using Simpson’s 1/3rd rule with ℎ =
1
6
(12+8)
*****
```