GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- IV (New) EXAMINATION - WINTER 2019
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Subject Code: 2140505 Date: 07/12/2019Subject Name: Chemical Engineering Maths
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 (a) Define following, 03
- Error
- Truncation error
- Relative error
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(b) Evaluate the sum S = v3 + v5 + v7 to 4 significant digits and find 04 its absolute and relative errors.
(c) Using the secant method, find a real root of equation xex— 1 =0 correct to four decimal places. 07
Q.2 (a) Write an algorithm for Regula Falsi method. 03
(b) Evaluate v12 correct to three decimal places using Newton-Raphson method. 04
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(c) Find root of the equation x4- 2x - 5 = 0 using the bisection method correct upto three decimal places. 07
OR
(c) Use Gauss elimination method to solve the following system, 07 2x+y+z=10, 3x+2y+3z=18, x +4y+9z=16
Q.3 (a) Define Eigen values and Eigen vectors. 03
(b) Describe Jacobi’s method. 04
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(c) Solve the system, 6x +y+z=20, x+4y-z=6, x-y+5z=7 using Gauss Seidel method: 07
OR
Q.3 (a) Find the inverse of the matrix, 03
A= | 1 2 3 | | 0 1 2 | | 0 0 1 |
(b) Find the best values of a0 and a1 if the straight line y = a0 + a1x is fitted to the data (xi, yi): 04
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| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 0.6 | 2.4 | 3.5 | 4.8 | 5.7 |
Find also the correlation coefficient.
(c) Find constants a and b such that the function y = aebx fits the following data: 07
| X | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| y | 2.473 | 16.722 | 18.274 | 49.673 | 135.026 |
Q.4 (a) Derive the formula for Simpson’s 3/8 Rule. 03
(b) Establish Newton’s forward interpolation formula. 04
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| X | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 |
|---|---|---|---|---|---|
| y(x) | 9.75 | 12.45 | 15.70 | 19.52 | 23.75 |
Find y (4.25), using Newton’s backward difference interpolation formula.
OR
Q.4 (a) Write an algorithm for Trapezoidal Rule. 03
(b) Evaluate ? 1/(1+x2) dx using Simpsons 3/8 rule taking h = 1/2 04 0
(c) Evaluate ? e-x dx using four intervals for Simpson’s 1/3 rule and Trapezoidal rule. 07 1 0
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Q.5 (a) Describe the method of finite difference approximations to partial derivatives. 03
(b) Use second order Runge — Kutta method to solve dy/dx =3x +y, given y = 1.3 when x = 1 to approximate y when x = 1.2 taking step size 0.1. 04
(c) Determine the value of y at x=0.3, given that dy/dx =x +y and y(0) =1, using modified Euler’s method. 07
OR
Q.5 (a) Discuss in brief about Milne’s Predictor-Corrector method. 03
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(b) Using Taylor’s series method, obtain the solution of dy/dx = 3x+ y2, given that y (0) = 1. Find the value of y for x = 0.1. 04
(c) Use fourth order Runge — Kutta method to find the value of y when x=0.2, given that y’ =x +y2, and y = 1 when x = 0 taking step size 0.1. 07
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