April-1999

Part A - (20 x 2 = 40 marks)

Answer ALL questions.

- Convert y = ax
^{b}into linear form. - For which type of equation the points (x, y) plotted in a semi-log graph sheet will lie on a straight line?
- Write the normal equations to fit y = a + bx by the method of least squares.
- If p and q are the relative errors in x and y respectively, what is the relative error in the quotient x/y?
- If a : 2b : 3c = 3 : 2 : 1, find (a - b) : (b - c).
- If y = nx and S = R(c - x)/y, find y when S = 4R, n = 0.25 and c = 0.2.
- A chemical substance disintegrates at a rate proportional to the amount then present at any instant. Write the differential equation governing this law.
- Solve rdq + 2qdr = 0.
- Solve dy/dx = y/x + e
^{y/x}. - If y is a function of x and z = log(x), express xdy/dx in terms of y and z.
- If y = 2 when x = -2 and y = 8 when x = 1, find y when x = 0 using linear interpolation.
- Find D
^{3 }(x^{2}) taking h = 1. - What is hybrid computer?
- Express the binary number 1100 in octal.
- Express 0.001728 in floating-point.
- What is a data?
- What is an algorithm?
- Write a FORTRAN statement to convert Fahrenheit (F) into Celsius (C).
- Write the solution of ¶ u/¶ t = a
^{2}¶^{2}u/¶ x^{2}such that u is finite when t is tending to infinity. - State Laplace equation.
- (a) The heat capacity c of a material (in calories per gram per degree Celsius) could increase with temperature T (in degree Celsius) according to a relationship of the form c = aT
^{2}+ bT + k. Determine the constants a, b and k by the method of least squares from the following data: - (a) In a chemical reaction involving two substances, the velocity of transformation dx/dt at any time t is known to be equal to the product (0.9 - x), (0.4 - x). Express x in terms of t, given that when t = 300, x = 0.3.
- (a) Find the value of c when T = 40 from the following data:
- (a) Use the Newton-Raphson method to estimate the root of e
^{-x}= x employing an initial guess of x_{0}= 0 correct to six decimal places. - (a) A bar of length 30 cm has end temperatures 0
^{o }C and 100^{o }C until steady state conditions prevail. Suddenly the temperature at the first end is raised to 50^{o}C from 0^{o}C and thereafter maintained. Find the temperature distribution at any point of the bar at any subsequent time.

Part B - (5 x 12 = 60 marks)

Answer (a) or (b) in each question.

T 0 50 100 150 200

c 0.13200 0.14046 0.15024 0.16134 0.17376

(b) Fit a least square exponential curve y = ae^{bx} to the following data:

x 4 9 14 23

y 27 73 197 1194

(b) A tank of volume 0.5 m^{3} is filled with brine containing 40 kg of dissolved salt. Water runs into the tank at the rate of 1.5 x 10^{-4} m^{3}/sec and the mixture, kept uniform by stirring, runs out at the same rate. How much salt is in the tank after two hours?

T -50 -20 10 70

c 0.125 0.128 0.134 0.144

(b) Using Runge-Kutta method of fourth order find y(1), y(2) and y(3) given that dy/dx = 4e^{0.8x} -
0.5y, y(0) = 2.

(b) Write a FORTRAN program for the problem given in question 24 (a).

(b) Write a FORTRAN program for the problem given in the question 23 (b).