PART – A
Answer all the questions (10 X 2 = 20)
- Write down the nth derivative of cos25x.
- Show that for y2=4ax, the subnormal at any point is a constant.
- Give the formula for the radius of curvature in Cartesian form.
- Define evolute.
- If α, β , γ are the roots of x3+px2+qx+r=0 find the value of .
- Give the number of positive roots of x3+2x+3=0.
- Show that sin ix =i sinh x.
- Evaluate
- Find the polar of (3, 4) with respect to y2 = 4ax.
- Define an asymptote of a hyperbola.
PART – B
Answer any FIVE questions. (5 X 8 = 40)
- Show that in the curve by2=(x+a)3 the square of the subtangent varies as the subnormal .
- Find the radius of curvature at ‘θ’ on x = a(cos θ+ θ sin θ), y=a(sin θ – θ cos θ).
- Find the p-r equation of r sin θ + a = 0.
- Solve: x4+2x3-5x2+6x+2=0 given that (1+i) is a root.
- Remove the second term from the equation x3-6x2+11x-6=0.
- Separate into real and imaginary parts tanh (x+iy).
- Find the locus of poles with respect to y2=4ax of tangents to x2+y2=c2.
- Derive the polar equation =1 + e cos θ of a conic.
PART –C
Answer any TWO questions. 2 X 20 = 40
- a) If , show that (1-x2)yn+2 – (2n+1)xyn+1 – (n2+a2)yn=0.
- b) Find the slope of =cos(θ-α) + ecos θ. (12 + 8)
- a) Show that the radius of curvature at any point on r = aeθ cot α is r cosec α.
- b) Solve 6x5-x4-43x3+43x2+x-6 = 0. (10 +10)
- a) Calculate to two places of decimals, the positive root of x3+6x-2 = 0 by Horner’s method.
- b) Expand cosh8θ in terms of hyperbolic cosines of multiples of θ. (12 + 8)
- a) Sum of infinity : …
- b) If e and e1 are two extremities of hyperbola and its conjugate show that
(10 +10)