LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2004

MT – 1500/MAT 500 – ALGEBRA, ANAL. GEOMETRY, CALCULUS & TRIGONOMETRY

01.11.2004                                                                                                           Max:100 marks

1.00 – 4.00 p.m.

 

SECTION – A

 

Answer ALL Questions.                                                                                (10 x 2 = 20 marks)

 

1. If y = sin (ax + b), find y_n.
2. Show that in the parabola y² = 4ax, the subnormal is constant.
3. Prove that cos h2x = cos h²x + sin h²
4. Write the formula for the radius of curvature in polar co-ordinates.
5. Find the centre of the curvature xy = c² at (c, c).
6. Prove that .
7. Form a rational cubic equation which shall have for roots 1, 3 – .
8. Solve the equation 2x³ – 7x² + 4x + 3 = 0 given 1+is a root.
9. What is the equation of the chord of the parabola y² = 4ax having (x, y) as mid – point?
10. Define conjugate diameters.

 

SECTION – B

 

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

 

11. Find the n^th derivative of cosx cos2x cos3x.
12. In the curve x^m yⁿ = a^m+n , show that the subtangent at any point varies as the abscissa of the point.
13. Prove that the radius of curvature at any point of the cycloid

x = a (q + sin q) and  y = a  (1 – cos q) is 4 a cos .

14. Find the p-r equation of the curve r^m = a^m sin m q.
15. Find the value of a,b,c such that .
16. Solve the equation

6x⁶ – 35x⁵ + 56x⁴ – 56x² + 35x – 6 = 0.

17. 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.
18. Show that in a conic, the semi latus rectum is the harmonic mean between the segments of a focal chord.

SECTION -C

 

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

 

19. a) If y = , prove that

(1 – x²) y₂ – xy₁ – a²y = 0.

Hence show that (1 – x²) y_n+2 – (2n +1) xy_n+1 – (m² + a²) y_n = 0.                     (10)

 

1. Find the angle of intersection of the cardioid r = a (1 + cos q) and r = b (1 – cos q).

(10)

 

20. a) Prove that  = 64 cos⁶ q – 112 cos⁴q + 56 cos²q –                                       (12)

 

1. b) Show that (8)
2. a) If  a + b + c + d = 0, show that

.                               (12)

 

1. b) Show that the roots of the equation x³ + px² + qx + r = 0 are in Arithmetical

progression if 2 p³ – 9pq + 27r = 0.                                                                             (8)

 

22. a) 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.                     (8)

1. b) P and Q are extremities of two conjugate diameters of the ellipse and S is

a focus.  Prove that PQ² – (SP – SQ)² = 2b².                                                              (12)

 

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