LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSFIRST SEMESTER – APRIL 2007MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I
Date & Time: 24/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION –AAnswer all: 2 x 10 = 20
1. If y = a cos5x + b sin5x, show that . 2. Write down the nth derivative of eax. 3. What is the formula for radius of curvature in parametric form. 4. Find the sub tangent and the sub normal for y = 3×3. 5. If x = sin 2ө and y = cos 2ө, find . 6. Determine the quadratic equation having 3-2i as one of its roots. 7. Derive the relation sin ix = i sinh x. 8. Separate into real and imaginary parts for cos (x+iy). 9. Define conjugate diameters.10. Write the polar form of the conic.
SECTION –BAnswer any five: 5x 8 = 40
11. Find the nth derivative of sin2x sin4x sin6x.12. Find the angle of intersection of the cardioids r = a(1+cosө) and r = b(1-cosө).13. Find the lengths of the sub tangent and the sub normal at the point (a, a) for the curve y = x3+ 3x+4. 14. Show that the roots of the equation x3+px2+qx+r =0 are in A.P if 2p3-9pq+27r =0.15. Solve the equation 6×5+11×4-33×3-33×2+11x+6= 0.
16. Prove that = 7 – 56 sin2ө + 11 2 sin4ө – 64 sin6ө. 17. Prove that 32 cos6ө = cos6ө + 6 cos4ө +15 cos2ө + 10. 18. Prove that the product of the focal distances of a point on an ellipse is equal to the square of the semi-diameter which is conjugate to the diameter through the point.
SECTION –CAnswer any two: 2x 20 = 40
19. State and prove Leibnitz theorem and prove that (1-x2)y2 –xy1+m2y = 0 and (1-x2) yn+2 –(2n+1)xyn+1+(m2-n2)yn = 0 for y = sin( msin-1x). (P.T.O)20 a) Find the evolute of the ellipse . b) Find the p-r equation of rm = am sinm ө. (10+10)
21 a) Find the equation whose roots are the roots of the equation x4-x3-10×2+4x+24 = 0 increased by 2 and hence solve the equation. b) Find the sum of the fourth power of the roots of the equation x3-2×2+x+1 = 0. (10+10) 22 a) Prove that .
b) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact and encloses a triangle of constant area. (10+10)
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