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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – COMPUTER SCI. & 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 Answer ALL questions: 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 Answer any FIVE questions: (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 Answer any TWO questions: (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) *****