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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – MATHEMATICS FIRST SEMESTER – APRIL 2012 MT 1500 - ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY Date : 28-04-2012 Time : 1:00 - 4:00 Dept. No. Max. : 100 Marks PART - A Answer ALL the questions: (10 X 2 = 20 Marks) 1 1. Find the nth derivative of . ax b 2 2. Find the slope of the straight line l cos e cos . r 3. Write the formula for the radius of curvature in Cartesian form. 4. Define Cartesian equation of the circle of the curvature. 2 5. If , , are the roots of the equation x3+px2+qx+r=0. Find the value of . 6. Diminish the roots x4+x3-3x2+2x-4 =0 by 2. tan 2 x 2 tan x . x 0 x3 7. Evaluate Lt 8. Prove that cosh x 1 tanh 2 x . 1 tanh 2 x 9. Define Pole and Polar of a ellipse. 10. In the hyperbola 16x2-9y2 = 144, find the equation of the diameter conjugate to the diameter x =2y. PART - B Answer any FIVE questions: (5 X 8 = 40 Marks) 2x 1 11. Find the nth derivative of . 2 x 12 x 3 12. Find the angle between the radius vector and tangent for the curve r a1 cos at 2 . 13. Solve the equation x3-4x2-3x+18=0 given that two of its roots are equal. 14. Solve the equation x4-5x3+4x2+8x-8=0 given that 1- is a root. 15. Expand cos 6 in terms sin . 16. Separate real and imaginary parts tan 1 x iy . 17. P and Q are extremities of two conjugate diameters of the ellipse PQ 2 SP SQ 2b 2 . x2 y2 1 and S is a focus. Prove that a2 b2 2 18. The asymptotes of a hyperbola are parallel to 2x+3y=0 and 3x-2y =0 . Its centre is at (1,2) and it passes through the point (5,3). Find its equation and its conjugate. PART - C Answer any TWO questions: 19. (a) If y x 1 x 2 m (2 x 20=40 Marks) , show that 1 x 2 y2 xy1 m 2 y 0. (b) Prove that the sub-tangent at any point on y be x a is constant ant the subnormal is y2 . a (10 +10) 20. (a) Find the radius of curvature at any point on the curve r 2 a 2 cos 2 . (b) Show that the evolute of the cycloid x a sin , y a1 cos is another cycloid . (10+10) 21. (a) Solve 6x5+11x4-33x3-33x2+11x+6=0. (b) Find by Horner’s method, the roots of the equation x3 3x 1 0, which lies between 1 and 2 correct to two decimal places. (10+10) 22. (a) Prove that 32 cos 6 cos 6 6 cos 4 15 cos 2 10. (b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its asymptotes is constant. (10+10) $$$$$$$