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)
- If y = sin (ax + b), find yn.
- Show that in the parabola y2 = 4ax, the subnormal is constant.
- Prove that cos h2x = cos h2x + sin h2
- Write the formula for the radius of curvature in polar co-ordinates.
- Find the centre of the curvature xy = c2 at (c, c).
- Prove that .
- Form a rational cubic equation which shall have for roots 1, 3 – .
- Solve the equation 2x3 – 7x2 + 4x + 3 = 0 given 1+is a root.
- What is the equation of the chord of the parabola y2 = 4ax having (x, y) as mid – point?
- Define conjugate diameters.
SECTION – B
Answer any FIVE Questions. (5 x 8 = 40 marks)
- Find the nth derivative of cosx cos2x cos3x.
- In the curve xm yn = am+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 rm = am sin m q.
- Find the value of a,b,c such that .
- Solve the equation
6x6 – 35x5 + 56x4 – 56x2 + 35x – 6 = 0.
- 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.
- Show that in a conic, the semi latus rectum is the harmonic mean between the segments of a focal chord.
Answer any TWO Questions. (2 x 20 = 40 marks)
- a) If y = , prove that
(1 – x2) y2 – xy1 – a2y = 0.
Hence show that (1 – x2) yn+2 – (2n +1) xyn+1 – (m2 + a2) yn = 0. (10)
- Find the angle of intersection of the cardioid r = a (1 + cos q) and r = b (1 – cos q).
- a) Prove that = 64 cos6 q – 112 cos4q + 56 cos2q – (12)
- b) Show that (8)
- a) If a + b + c + d = 0, show that
- b) Show that the roots of the equation x3 + px2 + qx + r = 0 are in Arithmetical
progression if 2 p3 – 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 PQ2 – (SP – SQ)2 = 2b2. (12)