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INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Obtain y(0, 2) using Picard's method upto second iteration for the initial value problem $\dfrac {dy}{dx} = x^2 - 2y \ \ y(0)=1$
6 M
1 (b) Solve by Euler's modified method to obtain y(1, 2) given $y' = \dfrac {y+x} {y-x} \ \ y(1)=2.$
7 M
1 (c) Using Adam Bash forth method obtain y at x=0.8 given $\dfrac {dy} {dx} = x -y^2, \ \ y(0)=0, \ \ y(0.2)=0.02, \ \ y(0.4)= 0.0795 \ and \ y(0.6) = 0.1762. 7 M 2 (a) Solve by 4th order Runge-Kutta method simultaneous equations given by \[ \dfrac {dx} {dt} = y-t, \ \ \dfrac {dy} {dt} = x+t \ with \ x=1 = y at \ t=0,$ obtain y(0.1) and x(0.1).
6 M
2 (b) Solve $\dfrac {d^2y}{dx^2} - x \left ( \dfrac {dy}{dx} \right )^2 + y^2 = 0, \ \ y(0)=1 , \ y'(0)=0$ Evaluate y(0.2) correct to four decimal places, using Runge method of fourth order.
7 M
2 (c) Solve for x=0.4 using Milnes predictor corrector formula for the differential equation y''+xy'+y=0 with y(0)=1, y(0.1)=0.995, y(0.2)=0.9802 and y(0.3)=0.956. Also Z(0)=0, z(0.1)=-0.0995, z=(0.2)=-0.196, z(0.3)=0.2863.
7 M

3 (a) Verify whether f(z)=sin2z is analytic, hence obtain the derivative.
6 M
3 (b) Determine the analytic function f(z) whose imaginary part is $\dfrac {y}{x^2+y^2}.$
7 M
3 (c) Define a harmonic function. Prove that real and imaginary parts of an analytic function are harmonic.
7 M

4 (a) Under the mapping w=ez, find the image of $i) \ 1\le x \le 2 \\ ii) \ \pi /3 < y < \dfrac {\pi } {2 }$
6 M
4 (b) Find the bilinear transformation which maps the points 1, i -1 from z plane to 2, i, -2 into w plane. Also find the fixed points.
7 M
4 (c) State and prove Cauchy's integral formula.
7 M

5 (a) Prove $J_n (x) = \dfrac {x} {2n} [ J_{n-1} (x) + J_{n+1} (x)]$
6 M
5 (b) Prove (n+1) Pn(x) = (2n+1) × Pn(x)- n Pn-1(x).
7 M
5 (c) Explain the following in terms of Legendre's polynomials.
x4+3x3-x2+5x-2.
7 M

6 (a) A class has 10 boys and 6 girls. Three students are selected at random one after another. Find the probability that i) first and third are boys, second a girls ii) first and second are of same sex and third is of opposite sex.
6 M
6 (b) If P(A)=0.4, P(B/A)=0.9, P(B/A)=0.6. Find P(A/B), P(A/B).
7 M
6 (c) In a bolt factory machines A, B and C manufacture 20%, 35% and 45% of the total of their output 5%, 4% and 2% are defective. A bolt is drawn at random found to be defective. What is the probability that is is from machine B?
7 M

7 (a) A random variable x has the following distribution:
 X: -2 -1 0 1 2 3 4 P(x): 0.1 0.1 k 0.1 2k k k

Find k, mean and S.D. of the distribution.
6 M
7 (b) The probability that a bomb dropped hits the target is 0.2. Find the probability that our out of 6 bombs dropped
i) Exactly 2 will hit the target
ii) At least 3 will hit the target.
7 M
7 (c) Find the mean and variance of the exponential distribution.
7 M

8 (a) A die is tossed 960 times and 5 appear 184 times. Is the die biased?
6 M
8 (b) Nine items have value 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these differ significantly from assumed of mean of 47.5. (γ=8, t0.05=2.31).
7 M
8 (c) A set of 5 similar coins tossed 320 times gives following table.
 No. of head 0 1 2 3 4 5 Frequency 6 27 72 112 71 32

Test the hypothesis that data follows binomial distribution (Give γ=5, x20.05 = 11.07.
7 M

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