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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034 B.C.A. DEGREE EXAMINATION THIRD SEMESTER APRIL 2003 CA 3100/CAP 100 APPLICABLE MATHEMATICS 07.04.2003 Max.: 100 Marks 9.00 12.00 PART A (10 2 = 20 Marks) Answer ALL the questions. 1 1 1 ...... log 2. 01. Show that 2 1.2 2.2 3.2 3 02. Define characteristic equation and eigen values. 03. Write down the expansion of cos 8 using Demoive’s theorem. 04. If , , one the roots of the equation x3 7x + 6 = 0 find the value of 3 + 3 + 3. u xy u , show that x y u. 05. If u x y x y 06. Write the Cartesian formula for radius of curvature. dx 07. Find 2 . x 2x 3 08. Evaluate x 3 cos 2 x dx . 09. Solve pq = x. 10. Find the solution of (D2 6D +13) y = 0. PART B (5 8 = 40 Marks) Answer ALL the questions 11. Find the inverse of the matrix using Cayley Hamilton theorem 1 1 2 3 2 1 3 2 3 (OR) Find the sum to infinity of the series 2.5 2.5.8 2.5.8.11 ... 6.12 6.12.18 6.12.18.24 12. If the sum of two roots of the equation x4 + px3 + qx2 + rx + s = 0 equals the sum of the other two prove that p3+8r = 4pq. (OR) sin 6 Express interms of cos . sin 1 13. Find the radius of curvature at the point a 4 , a 4 to the curve to the curve x y a . (OR) Find the maximum and minimum value of the function f (x,y) = x2 y2 x2 y2 . 14. Evaluate xydx dy taken over the positive quadrant of the circle x2 + y2 = a2 (OR) 5x 1 Integrate 2 with respect to x. x 2 x 35 15. Solve (y + z)p + (z +x)q = x + y. (OR) d 2 y dy Find the solution of 3 2 14 y 13e 2 x dx dx PART C (2 20 = 40 Marks) Answer any TWO questions 16. a) Find the eigen values and eigen vectors of the matrix 1 1 3 1 5 1 3 1 1 b) Sum the series to infinity 1 3 1 3 3 2 1 3 3 2 33 ...... 2! 3! 4! 1 17. a) Show that cos5 sin7 11 [sin 12 2sin 10 4sin 8 +10 sin 6 2 + 5 sin 4 20sin 2]. 1 b) Diminish by 1, the roots of x4 4x3 7x2 + 22x + 24 = 0 and hence solve it. 17. a) By changing the order of integration, evaluate 0 x e y dx dy y b) Solve (D2 + 4D + 5) y = ex + x3+cos 2x. * * * * * 2