LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 1

FIRST SEMESTER – APRIL 2008

MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I

 

 

 

Date : 07/05/2008                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 e^ax.
2. Prove that the subtangent to the curve is of constant length.
3. Find the coordinates of the center of curvature of the curve at .
4. What is the curvature of a (i) circle (ii) straight line.
5. Determine the quadratic equation having (3-2i) as a root.
6. If are the roots of the equatim . Show that .
7. Prove that .
8. Prove that .
9. Find the pole of the line  with respect to the parabola y²=4ax.
10. If are the eccentricities of a hyperbola and its conjugate,

prove that .

 

PART – B

Answer any FIVE questions.                                                           (5 x 8 = 40 marks)

 

1. At which point is the tangent to the curve  parallel to the line
2. Final the angle at which the radius vector cuts the curve .
3. Prove that the radius of curvature at any point of the cycloid

and is .

1. Show that if the roots of the equation are in arithmetic progression then .
2. If  show that .

 

1. If  prove that

1. i)
2. ii)

1. Find the locus of poles of chords of the parabola which subtend a right angle at the focus.
2. Find the equation of a rectangular hyperbola referred to its asymptotes as axes.

 

PART – C

Answer any TWO questions.                                                           (2 x 20 = 40 marks)

1. a) If prove that and

.

1. b) Find the (p,r) –equation of the curve and hence show that the radius of curvature at any point varies as the cube of the focal distance.

1. a) Find the equation of the evolute of the parabola .

1. b) Solve .

1. a) Find the real root of to two places of decimals using Horner’s method.

1. b) Evaluate .

1. a) Prove that .

1. b) Derive the equation of the tangent at the point whose rectorial angle is on the conic .

 

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