April - 1997

Part - A (20 X 2 = 40 marks)

Answer ALL questions

- Convert y = ax
^{3}+ b into a linear form. - What type of graph paper is used to get a straight line for the equation y = ax
^{b}? - If y = f(x) and dx is the error in x, what is the corresponding error in y?
- State the principles of group averages in curve fitting.
- What is meant by mass balance in chemical engineering?
- If a chemical disintegrates at a rate proportional the square of the amount present at any instant, what is the differential equation governing this rule?
- Transform y'' + y' + y = x; y(0) = 1, y'(0) = 1 into two simultaneous (equivalent) first order problems.
- State Newton's backward difference interpolation formula.
- Obtain the second degree polynomial through the points (0, 2), (2, 1), (1, 0).
- What is the central difference approximation to U
_{xx}(where U = U(x,y) ) taking h, k as the step sizes, at the point (x_{i}, y_{i}). - Find D (x
^{2}-x +1) taking h = 1. - Express (12.25)
_{10}in binary system. - Write the octal number 156 in the decimal system.
- Distinguish between floating point and fixed point constants.
- Write the Runge Kutta algorithm of second order for solving y' = f(x, y), y(x
_{0}) = y_{0}. - Write the Newton Raphson formula for finding the roots of the equation f(x) = 0.
- What is the purpose of DO statement in FORTRAN?
- Distinguish between arithmetic IF and logical IF in FORTRAN?
- A homogeneous rod of length l, has one end at x = 0 at 0
^{o}C and the far end is insulated. Write the boundary conditions in symbolic form. - Write a solution by the method of separation of variables of the heat equation U
_{t}= U_{xx}.

Part - B (5 X 12 = 60 marks)

Answer ALL questions, choosing 'a' or 'b' from each question.

- (a) Fit a least square exponential curve y = ab
^{x}to the data. - (a) Random decomposes at a proportional to the account present. If a fraction p of the original amount disappears in 1 year, how much will remain at the end of 21 years.
- (a) Solve using RK fourth order method, dy/dx = 1 -2xy y(0) = 0 at x = 0.1 and 0.2.
- (a) Using Newton's Raphson method, solve for a positive root of e
^{x}- 4x = 0, to 5 decimals. - (a) Using method of separation of variables, solve the one dimensional heat equation (boundary value problem)
U _{t}= a^{2}U_{xx}; 0 < x < p ; t > 0 with U_{x}(0, t) = U_{x}(p , t) and U(x, 0) = cos(2x) - 7cos(4x) and a = 1.(b) Write a FORTRAN program for solving the problem 23 (b).

x: 0 1 2 3 4 5 6 y: 32 47 65 92 132 190 275

(b) Write a technical note on treatment of experimental data and interpretation of results in solving through mathematical methods, chemical engineering problems.

(b) A tank contains 1000 gallons of brine in which 500 lb of salt are dissolved. Fresh water runs into the tank at a rate of 10 gallons/min and the mixture kept uniform by stirring, runs out at the same rate. How long will it be before only 5 lb of salt are left in the tank.

(b) For the given data below

x: 1 3 6 10 y: 0.4 9 18 35

estimate, using Lagrange's interpolation, y(5).

(b) Write a complete FORTRAN program to solve the problem 24 (a) through computer implementation.