Define an analytic function and derive the necessary conditions for the analyticity of a function.

State Cauchy's integral formula and hence evaluate

Integral 3/2

Evaluate the following integral using contour integration

Integral (sin 8/3+cos 8)d8 with limits(0 to 2n)

Calculate the first four central moments of the following distribution about the mean

0 1 2 3 4 5 6 7 8
Frequency of 1 8 28 56 70 56 28 8 1

Fit a straight line y a bx to the following data:

50 70 100 120
12 15 21 25

A five digit number is formed by using 4 without repetition. Find the probability that the number is divisible by 4.

In 256 sets of 12 tosses of a coin, in how many ways can one expect 8 heads and 4 tails?

If the probability of a bad reaction from a certain injection is 0,001, find the chance that out of 2,000 individuals, more than two will get a bad reaction.

If X is a normal variate with mean 30 and S.D. find the probabilities that

26 X 40

X -30 5

Find a root of xe^x 1 correct to three decimal places using Bisection method.

Show that Newton-Raphson's method has quadratic convergence.

Discuss the Lagrange's and Newton's divided difference formulae for unequal intervals.

Apply Gauss-Seidel iteration method to solve the system of equations

4x y 2z x 5y Z 2x+ 4z =3.

Using Simpson's 31Sth rule, evaluate the integral with limits to

Find y (2·2) from

dy/dx x y 3

using Runge-Kutta's method of fourth order. (Take step size h 0·1).