Loyola College M.Sc. Mathematics April 2011 Math-I Question Paper PDF Download

 

 

PART – A

Answer all the questions                                                                                              (10 X 2 = 20)

  1. Write down the nth 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)

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

PART –C

Answer any TWO questions.                                                                                       2 X 20 = 40

  1. a) If , show that (1-x2)yn+2 – (2n+1)xyn+1 – (n2+a2)yn=0.
  2. b) Find the slope of =cos(θ-α) + ecos θ.             (12 + 8)
  3. a) Show that the radius of curvature at any point on r = aeθ cot α is r cosec α.
  4. b) Solve 6x5-x4-43x3+43x2+x-6 = 0.                                    (10 +10)
  5. a) Calculate to two places of decimals, the positive root of x3+6x-2 = 0 by Horner’s method.
  6. b) Expand cosh8θ in terms of hyperbolic cosines of multiples of θ.              (12 + 8)
  7. a) Sum of infinity : …
  8. b) If e and e1 are two extremities of hyperbola and its conjugate show that

(10 +10)

 

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