PART – A

Answer all the questions                                                                                              (10 X 2 = 20)

1. Write down the n^th derivative of cos²5x.
2. Show that for y²=4ax, the subnormal at any point is a constant.
3. Give the formula for the radius of curvature in Cartesian form.
4. Define evolute.
5. If α, β , γ are the roots of x³+px²+qx+r=0 find the value of .
6. Give the number of positive roots of x³+2x+3=0.
7. Show that sin ix =i sinh x.
8. Evaluate
9. Find the polar of (3, 4) with respect to y² = 4ax.
10. Define an asymptote of a hyperbola.

PART – B

Answer any FIVE questions.                                                                                       (5 X 8 = 40)

11. Show that in the curve by²=(x+a)³ the square of the subtangent varies as the subnormal .
12. Find the radius of curvature at ‘θ’ on x = a(cos θ+ θ sin θ), y=a(sin θ – θ cos θ).
13. Find the p-r equation of r sin θ + a = 0.
14. Solve: x⁴+2x³-5x²+6x+2=0 given that (1+i) is a root.
15. Remove the second term from the equation x³-6x²+11x-6=0.
16. Separate into real and imaginary parts tanh (x+iy).
17. Find the locus of poles with respect to y²=4ax of tangents to x²+y²=c².
18. Derive the polar equation =1 + e cos θ of a conic.

PART –C

Answer any TWO questions.                                                                                       2 X 20 = 40

19. a) If , show that (1-x²)y_n+2 – (2n+1)xy_n+1 – (n²+a²)y_n=0.
20. b) Find the slope of =cos(θ-α) + ecos θ.             (12 + 8)
21. a) Show that the radius of curvature at any point on r = ae^{θ cot α} is r cosec α.
22. b) Solve 6x⁵-x⁴-43x³+43x²+x-6 = 0.                                    (10 +10)
23. a) Calculate to two places of decimals, the positive root of x³+6x-2 = 0 by Horner’s method.
24. b) Expand cosh⁸θ in terms of hyperbolic cosines of multiples of θ.              (12 + 8)
25. a) Sum of infinity : …
26. b) If e and e₁ are two extremities of hyperbola and its conjugate show that

(10 +10)

 

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