1 (a)
Use modified Euler's method to solve dy/dx=x+y, y(0)=1 at x=0.1 for three iterations taking h=0.1.

6 M

1 (b)
Solve dy/dx=x+y, x=0, y=1 at x=0.2 using Runge-Kutta method. Take h=0.2

7 M

1 (c)
Using Milne's predictor-corrector method find y(0.3) correct
to three decimals given.

x | -0.1 | 0 | 0.1 | 0.2 |

y | 0.908783 | 1.0000 | 1.11145 | 1.25253 |

7 M

2 (a)
Approximate y and z at x=0.2 using Picard's method for the solution of \[ \dfrac {dy}{dx}=z \ \dfrac {dz}{dx}=x^3 (y+z) \] with y(0)=1, z(0)=1/2. Perform two steps (y

_{1}. y_{2}, z_{1}.z_{2}).
10 M

2 (b)
Using Runge-Kutta method solve y"=x(y')

^{2}-y^{2}at x=0.2 with x_{0}=0, y_{0}=1, z_{0}=0 take h=0.2.
10 M

3 (a)
If f(z)=u+iv is analytic prove that Cauchy-Reimann equations u

_{x}=v_{y}, u_{y}=-v_{x}are true.
6 M

3 (b)
If w=z

^{3}find dw/dz
7 M

3 (c)
If the potential function is \[ \phi =\log \sqrt{x^2+y^2} \] Find the stream function.

7 M

4 (a)
Find the bilinear transformation which maps the points z=1, i, -1 onto the points w=j, o, -i.

6 M

4 (b)
Discuss the conformal transformation w=e

^{z}. Any horizontal strip of height 2π in z-plane will map what portion of w-plane.
7 M

4 (c)
State and prove Cauchy's integral formula.

7 M

5 (a)
Prove that \[ \int^{x}_{1/2}=\sqrt{\dfrac {2}{\pi x}}\sin x. \]

6 M

5 (b)
State and prove Rodrigues formula for Legendre's polynomials.

7 M

5 (c)
Express f(x)=x

^{4}+3x^{3}-x^{2}+5x-2 in terms of Legendre polynomials.
7 M

6 (a)
The probabilities of four persons A, B, C, D hitting target are respectively 1/2, 1/3, 1/4, 1/5. What is the probability that target is hit by atleast one person if all hit simultaneously?

6 M

6 (b)
i) State addition law of probability for any two events A and B.

ii) Two different digits from 1 to 9 are selected. What is the probability that the sum of the two selected digits is odd if '2' one of the digits selected.

ii) Two different digits from 1 to 9 are selected. What is the probability that the sum of the two selected digits is odd if '2' one of the digits selected.

7 M

6 (c)
Three machine A, B, C produce 50%, 30%, 20% of the items. The percentage of defective items are 3, 4, 5 respectively. If the item selected is defective what is the probability that it is from machine A? Also find the total probability thatn an item is defective.

7 M

7 (a)
The p.d.f of x is

Find k. Also p(x≤5), p(3

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

p(x) | k | 3k | 5k | 7k | 9k | 11k | 13k |

Find k. Also p(x≤5), p(3

6 M

7 (b)
A die is thrown 8 times. Find the probability that '3' falls,

i) Exactly 2 times

ii) At least once

iii) At te most 7 times.

i) Exactly 2 times

ii) At least once

iii) At te most 7 times.

7 M

7 (c)
In a certain town the duration of shower has mean 5 minutes. What is the probability that shower will last for i) 10 minutes or more; ii) less than 10 minutes; iii) between 10 and 12 minutes.

7 M

8 (a)
What is null hypothesis, alternative hypothesis significance level?

6 M

8 (b)
The nine items of a sample have the following values: 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these differ significantly from the assumed mean 47.5. Apply student's t-distribution at 5% level of significance. (t

_{0.05}for 8df=2.31).
7 M

8 (c)
In experiments on a pea breading. The following frequencies of seeds were obtained:

is the experiment is in the agreement of theory which predicts proportion of frequencies 9:3:3:1 (x

Round-Yellow |
Wrinkled-Yellow |
Round-Green |
Wrinkled- Green |
Total |

315 | 101 | 108 | 32 | 556 |

^{2}_{0.05}, 3df=7.815).
7 M

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