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.

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__SECTION – A__

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

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

__SECTION – B__

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

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- Find the n
^{th}derivative of cosx cos2x cos3x. - In the curve x
^{m}y^{n}= a^{m+n}, show that the subtangent at any point varies as the abscissa of the point. - 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 .

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

6x^{6} – 35x^{5} + 56x^{4} – 56x^{2} + 35x – 6 = 0.

- If the sum of two roots of the equation x
^{4}+ px^{3}+ qx^{2}+ rx + s = 0 equals the sum of the other two, prove that p^{3}+ 8r = 4pq. - Show that in a conic, the semi latus rectum is the harmonic mean between the segments of a focal chord.

__SECTION -C__

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

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- a) If y = , prove that

(1 – x^{2}) y_{2} – xy_{1} – a^{2}y = 0.

Hence show that (1 – x^{2}) y_{n+2} – (2n +1) xy_{n+1} – (m^{2} + a^{2}) y_{n} = 0. (10)

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

(10)

- a) Prove that = 64 cos
^{6}q – 112 cos^{4}q + 56 cos^{2}q – (12)

- b) Show that (8)
- a) If a + b + c + d = 0, show that

. (12)

- b) Show that the roots of the equation x
^{3}+ px^{2}+ qx + r = 0 are in Arithmetical

progression if 2 p^{3} – 9pq + 27r = 0. (8)

- 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)

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

a focus. Prove that PQ^{2} – (SP – SQ)^{2} = 2b^{2}. (12)