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              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions:                                                                      (10 X 2 = 20 Marks)

1. Find the n^th derivative of .
2. Find the slope of the straight line .
3. Write the formula for the radius of curvature in Cartesian form.
4. Define Cartesian equation of the circle of the curvature.
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.
7. Evaluate
8. Prove that
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)    

11. Find the n^th derivative of .
12. Find the angle between the radius vector and tangent for the curve at

.

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 in terms .
16. Separate real and imaginary parts .
17. P and Q are extremities of two conjugate diameters of the ellipse and S is a focus. Prove that
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:                                                                                         (2 x 20=40 Marks)

 

19. (a) If , show that

(b) Prove that the sub-tangent at any point on                is constant ant the subnormal is

(10 +10)

20. (a) Find the radius of curvature at any point on the curve

 

(b) Show that the evolute  of the cycloid   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  which lies between 1 and 2

correct to two decimal places.                                                                                                                             (10+10)

22. (a) Prove that

 

(b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its

asymptotes is constant.                                                                                                   (10+10)

 

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