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 4

^{th}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=e

^{z}, 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) P

_{n}(x) = (2n+1) × P_{n}(x)- n P_{n-1}(x).
7 M

5 (c)
Explain the following in terms of Legendre's polynomials.

x

x

^{4}+3x^{3}-x^{2}+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:

Find k, mean and S.D. of the 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.

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, t

_{0.05}=2.31).
7 M

8 (c)
A set of 5 similar coins tossed 320 times gives following table.

Test the hypothesis that data follows binomial distribution (Give γ=5, x

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, x

^{2}_{0.05}= 11.07.
7 M

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