1. Answer any two of the following:

(a) Show that the function u e^-2xy sin is harmonic. Find the conjugate function v as z iv. Evaluate <img src='./qimages/10895-1b.jpg'>

along the path y

(ii) y Evaluate

<br><br> <img src='./qimages/10895-1c.jpg'>

where Answer any two of the following:

(a) The four moments about the working mean 28·5 of a distribution are 0.294, 7.144, 42·409 and 454·98. Calculate the mean. Also evaluate r31, r32 and comment about the skewness and kurtosis of the distribution. Find the regression coefficients of y on x and of x on y and the correlation coefficient between x and y on the basis of the following records <img src='./qimages/10895-2b.jpg'>

variance of x 4 and variance of y 9 The probability that a bomb dropped from a plane will hit the target is 1/5. If 6 bombs are dropped, find the probability that exactly two will strike the target,

(ii) at least two will hit the target.

3. Answer any two of the following:

(a) A die is tossed thrice. Getting 5 or 6 on the die in a toss is success. Find the mean and variance of number of successes. The distribution of the number of road accidents per day in a city is Poisson with mean 4. Find the number of days out of 100 days, when there will be no accident,

(ii) at least 2 accidents,

(iii) at most 3 accidents,
between 2 and 5 accidents. In a normal distribution of marks, 31% are under 45 and are over 64. Find the mean and standard deviation of the distribution. It is given that, if

<br><br> <img src='./qimages/10895-3c.jpg'>

then,f(0·5) 0·19 and f(1.4) =0·42.

4. Answer any two of the following:

(a) Find a root of the following equation using the Bisection method, correct to three decimal places
x^3 Using Regula-Falsi method, compute the real root of the equation xe^x correct to four decimal places. Determine the root of the equation cos x 0 using the method of Newton-Raphson.

5. Answer any two of the following: Solve by Gauss-Jordan method, the equations x y 2x 3y 4z 13, 3x 4y 5z 40. A rocket is launched from the ground. Its acceleration is noted for the first 80 seconds as given in the following table
t
0 30
10 31·63
20 33.34
30 35·47
40 37·75
50 40·33
60 43·25
70 46·69
80 50·67

Estimate the velocity of the rocket at t =80 sec employing Simpson's one-third rule.
Solve the equation dy/dx 1 for values of y at x 0·1 and x 0.2, using Runge-Kutta method of order three.