IFoS PYQs 1
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: \(-x-y+6 z=42\) \(6 x-y-z=11.33\) \(-x+6 y-z=32\) 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 \(3 x+\sin x-e^{x}=0,\) correct to five decimal places, using Regula-falsi method
[10M]
2) Write a computer program using BASIC’ to solve the following problem \(\int_{\pi / 4}^{\pi / 2} \dfrac{\sin x}{x} d x\) By trapezoidal rule.
[10M]
3) Compute \(y(10)\) using Lagrange’s interpolation formula from the following data: \(\begin{array}{llll} x & 3 & 7 & 11 & 17 \\ y & 10 & 15 & 17 & 20 \end{array}\)
[10M]
4) Derive three-point Gaussian quadrature formula and hence evaluate. \(\int_{0.2}^{1.5} e^{-x^{2}} d x\) calculating weights and residues. Give the result to three decimal places.
[10M]
5) Solve the system
\(1.2 x_{1}+21.2 x_{2}+1.5 x_{3}+2.5 x_{4}=27.46\)
\(0.9 x_{1}+2.5 x_{2}+1.3 x_{3}+32.1 x_{4}=49.72\)
\(2.1 x_{1}+1.5 x_{2}+19.8 x_{3}+1.3 x_{4}=28.76\)
\(20.9 x_{1}+1.2 x_{2}+2.1 x_{3}+0.9 x_{4}=21.70\)
using Gauss-Seidel iterative scheme correct to three decimal places starting with initial value \((1.04,1.30,1 .45,1.55)^{\top}\)
[10M]
2006
1) Perform four iterations of the bisection method to obtain a positive root of the equation \(f(x)=x^{}-5 x+1\)
[10M]
2) Write a BASIC program to evaluate a definite integral by Simpson’s one-third rule Adapt it to evaluate \(\int_{0}^{}\left(x^{3}+\sin x\right) d x\) 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 \(\dfrac{d y}{d x}=x+y^{2}, y=1 \text { when } x=0\)
[10M]
5) Write a program in BASIC to find a root of an equation by Newton-Raphson method. Adapt it to solve \(x^{3}-4 x^{2}+x+6=0\) 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 \(\begin{array}{llll} x & & & & \\ y & 1 & 15 & & \end{array}\)
[10M]
2005
1) Perform four iterations of the bisection method to obtain a positive root of the equation \(f(x)=x^{3}-5 x+1=0\)
[10M]
2) Evaluate \(\int_{0}^{1} \sqrt{1+2 x} d x\) by applying Gaussian quadrature formula, namely \(\int_{-1}^{1} f(t) d t=\sum_{i=1}^{n} A_{i} f\left(t_{i}\right)\) where the coefficients \(A _{i}\) and the roots ti are given below for \(n =4\) as \(t_{1}=-0.8611 \quad A _{2}= A _{1}=0.3478\) \(t_{2}=-0.3399 \quad A _{2}= A _{8}=0,6521\) \(t_{3}=0.3399\) \(t_{4}=0.8611\)
[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 \(\int_{0}^{1}\left(x^{3}+\sin x\right) d x\) 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 \(x^{3}-4 x^{2}+x+6=0\) 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 \(\dfrac{d y}{d x}=x+y^{2}, y=1\) when \(x=0\)
[10M]