2008

1) Find the smallest positive root of equation $x{e}^x-\cos x=0$ using Regula Falsi method. Do three iterations.

[12M]

2) The following values of the function $f(x)=\sin x+\cos \$\$\mathrm{x}$ are given: $\begin{array}{lccc}\mathrm{X}& {10}^{\circ }& {20}^{\circ }& {30}^{\circ }\\ \mathrm{f}(\mathrm{x})& 1.1585& 1.2817& 1.3360\end{array}$ Construct the quadratic interpolating polynomial that fits the data. Hence calculate and compare with exact value.

[15M]

3) Apply Gauss-Seidel method to calculate $x$, $y$, $z$ from the system: $-x-y+6z=42$ $6x-y-z=11.33$ $-x+6y-z=32$ with initial values $(4.67,7.62,9.05)$. Carry out computations for two iterations.

[15M]

2007

1) Find the smallest positive root of equation $3x+\sin x-e^x=0,$ correct to five decimal places, using Regula-falsi method

[10M]

2) Write a computer program using BASIC’ to solve the following problem ${\int }_{\pi /4}^{\pi /2}{\displaystyle \frac{\sin x}{x}}dx$ By trapezoidal rule.

[10M]

3) Compute $y(10)$ using Lagrange’s interpolation formula from the following data: $\begin{array}{lllll}x& 3& 7& 11& 17\\ y& 10& 15& 17& 20\end{array}$

[10M]

4) Derive three-point Gaussian quadrature formula and hence evaluate. ${\int }_{0.2}^{1.5}{e}^{-x^2}dx$ calculating weights and residues. Give the result to three decimal places.

[10M]

5) Solve the system
$1.2x_1+21.2x_2+1.5x_3+2.5x_4=27.46$ $0.9x_1+2.5x_2+1.3x_3+32.1x_4=49.72$ $2.1x_1+1.5x_2+19.8x_3+1.3x_4=28.76$ $20.9x_1+1.2x_2+2.1x_3+0.9x_4=21.70$ using Gauss-Seidel iterative scheme correct to three decimal places starting with initial value $(1.04,1.30,1.45,1.55{)}^{\mathrm{\top }}$

[10M]

2006

1) Perform four iterations of the bisection method to obtain a positive root of the equation $f(x)=x^{}-5x+1$

[10M]

2) Write a BASIC program to evaluate a definite integral by Simpson’s one-third rule Adapt it to evaluate ${\int }_0^{}(x^3+\sin x)dx$ by taking 10 subintervals and indicating which lines are to be modified for a specific problem

[10M]

3) Apply Gauss-Seidel iteration method for three iterations to solve the equation $\left[\begin{array}{cccc}10& -2& -1& -1\\ -2& 10& -1& -1\\ -1& -1& 10& -2\\ -1& -1& -2& 10\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}3\\ 15\\ 27\\ -9\end{array}\right]$

[10M]

4) Apply Runge-Kutta method of fourth order to find an approximate value of y when $x=0.1$ and 0.2 , given that ${\displaystyle \frac{dy}{dx}}=x+y^2,y=1\text{ when }x=0$

[10M]

5) Write a program in BASIC to find a root of an equation by Newton-Raphson method. Adapt it to solve $x^3-4x^2+x+6=0$ using initial approximation as $x_0=5$. Indicate which lines are to be changed for a different equation.

[10M]

6) Apply Newton’s fonard and backard difference formulae to evaluate $f(12)$ and $f(3.9)$ respectively from the data $\begin{array}{lllll}x& & & & \\ y& 1& 15& & \end{array}$

[10M]

2005

1) Perform four iterations of the bisection method to obtain a positive root of the equation $f(x)=x^3-5x+1=0$

[10M]

2) Evaluate ${\int }_0^1\sqrt{1+2x}dx$ by applying Gaussian quadrature formula, namely ${\int }_{-1}^{1}f(t)dt=\sum _{i=1}^n{A}_{i}f\left(t_i\right)$ where the coefficients $A_i$ and the roots ti are given below for $n=4$ as $t_1=-0.8611A_2=A_1=0.3478$ $t_2=-0.3399A_2=A_8=0,6521$ $t_3=0.3399$ $t_4=0.8611$

[10M]

3) Apply Gauss-Seidel iterative method for five iterations to solve the equations $\left[\begin{array}{cccc}10& -2& -1& -1\\ -2& 10& -1& -1\\ -1& -1& 10& -2\\ -1& -1& -2& 10\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{r}3\\ 15\\ 27\\ -9\end{array}\right]$

[10M]

4) Write a BASIC program to evaluate a delinite integral ${\int }_0^1(x^3+\sin x)dx$ by Simpson’s one-third rule. Indicate the lines which are to be modified for a different problem.

[10M]

5) Write a program in BASIC to solve the equation $x^3-4x^2+x+6=0$ by Newton-Raphson method by taking the initial approximation as $x_0=5$. Indicate which lines are to be changed for a different equation.

[10M]

6) Apply Runge-Kutta method of fourth order to find an approximate value of $y$ when $x=0.2$. given that ${\displaystyle \frac{dy}{dx}}=x+y^2,y=1$ when $x=0$

[10M]