2000

1) Using Newton-Raphson’s method, show that the iteration formula for finding the reciprocal of the \(\mathrm{p}^{\text {th }}\) root of \(\mathrm{N}\) is \(x_{i+1}=\dfrac{x_{1}\left(p+1-N x_{i}\right)}{p}\)

[6M]

2) Evaluate \(\int_{0}^{1} \dfrac{d x}{1+x^{2}},\) by subdividing the interval (0,1) into 6 equal parts and using Simpson’s one-third rule. Hence find the value of \(\pi\) and actual error, correct to five places of decimals.

[15M]

3) Solve the following system of linear equations, using Gauss elimination method: \(\begin{array}{l} x_{1}+6 x_{2}+3 x_{3}=6 \\ 2 x_{1}+3 x_{2}+3 x_{3}=117 \\ 4 x_{1}+x_{2}+2 x_{3}=283 \end{array}\)

[15M]

4) Let \(A=\left[\begin{array}{lll}5 & 3 & 1 \\ 3 & 7 & 4 \\ 1 & 4 & 9\end{array}\right] B=A^{-1}\) and \(C=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\) Write a BASIC programm that computes the inverse of A, determinant of \(\mathrm{A}\) and the product of the matrix and its inverse.

[15M]

5) Write a BASIC program to evaluate the formula \(y=\left(\dfrac{x-1}{x}\right)+\dfrac{1}{2}\left(\dfrac{x-1}{x}\right)^{2}+\dfrac{1}{3}\left(\dfrac{x-1}{x}\right)^{3}+\dfrac{1}{4}\left(\dfrac{x-1}{x}\right)^{4}+\dfrac{1}{5}\left(\dfrac{x-1}{x}\right)^{5}\)

[15M]

1999

1) Obtain the Simpson’s rule for the integral
\(I=\int^b_a f(x)\,dx\)
and show that this rule is exact for polynomial of degree \(n\geq3\).In general show that the error for approximation for Simpson’s rule in given by \(R=-\dfrac{(b-a)^5}{2880}f^{iv}(\eta),\;\;\eta\in(0,2).\)
Apply this rule to the integral \(\int^1_0 \dfrac{\,dx}{1+x}\) and show that \(\vert R \vert \leq 0.008333\).

[20M]

2) Using fourth order classical Runge-Kutta method for the intial value problem \(\dfrac{du}{dt}=-2tu^2,\;u(0)=1,\)
where \(h=0.2\) on the interval [0,1],calculate \(u(0.4)\) correct to six places of decimal.

[20M]

1998

1) Evaluate \(int^3_1\dfrac{\, dx}{x}\) by Simpson’s rule with 4 strips. Determine the error by direct integration.

2) By the fourth-order Runge-Kutta method, tablulate the solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{xy+1}{10y^2+4},\) y(0)=0 in \([0,0.4]\) with step length 0.1 correct to five places of decimals.

3) Use Regula-Falsi method to show that the real root of \(x log_{10}x-1.2=0\) lies between 3 and 2.740646.

4) Given \(\int^z_{-\infty}\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac{w^2}{2}}dw\)=0.5987,0.9=6915,0.7734,0.8944,0.9772 for z=0.25,0.5,0.75,1.25,2 respectively.

5) Fit a second degree parabola to the following data taking x as the independent variable:

x y 1 2 2 6 3 7 4 8 5 10 6 11 7 11 8 10 9 9 — —-

1997

1) Apply the fourth order Runge-Kutta method to find a value of y correct to four places of decimals at \(\mathrm{x}=0.2\), when \(y^{\prime}=\dfrac{d y}{d x}=x+y, y(0)=1\)

[10M]

2) Show that the iteration formula for finding the reciprocal of \(\mathrm{N}\) is \(\mathrm{x}_{\mathrm{e}-1}=\mathrm{x}_{\mathrm{n}}\left(2-\mathrm{N}_{\mathrm{xn}}\right), \mathrm{n}=0,1 \ldots\)

[10M]

3) Obtain the cubic spline approximation for the function given in the tabular form below: \(\begin{array}{lllll} \mathrm{x}: & 0 & 1 & 2 & 3 \\ \mathrm{f}(\mathrm{x}) & 1 & 2 & 33 & 244 \end{array}\) and \(\quad M_{0}=0, M_{3}=0\)

[10M]

1996

1) Describe Newton-Raphson method for finding the solutions of the equation \(f(x)=0\) and show that the method has a quadratic convergence.

[15M]

2) The following are the measurements 1 made on a curve recorded by the oscillograph representing a change of current i due to a change in the conditions of an electric current: \(\begin{array}{lllll} t: & 1.2 & 2.0 & 2.5 . & 3.0 \\ i: & 1.36 & 0.58 & 0.34 & 0.20 \end{array}\) Applying an appropriate formula interpolate for the value of i when \(\mathrm{t}=1.6\)

[15M]

3) Solve the system of differential equations \(\dfrac{d y}{d x}=x z+1, \quad \dfrac{d z}{d x}=-x y\) for \(x=0.3\) given that \(y=0\) and \(z=1\) when \(x=0,\) using Runge-Kutta method of order four.

[15M]

1995

1) Find the positive root of \(\log _{ e } x =\cos x\) nearest to five places of decimal by Newton-Raphson method.

[15M]

2) Find the value of \(\int_{1}^{3.4} f(x) d x\) from the following data using Simpson’s \(\dfrac{3}{8} t h\) rule for the interval (1.6,2.2) and \(\dfrac{1}{3} r d,\) rule for (2.2,3.4) \(\begin{array}{lllllll}x & : & 1,6 & 1.8 & 2.0 & 2.2 & 2.4 \\ f(x) & : & 4.953 & 6.050 & 7.389 & 9.025 & 11.023 \\ x & : & 2.6 & 2.8 & 3.0 & 3.2 & 3.4 \\ f(x) & : & 13.464 & 16.445 & 20.086 & 24.533 & 29.964\end{array}\)

[15M]

3) For the differential equation \(\dfrac{d y}{d x}=y-x^{2}, y(0)=1\) starting values are given as \(y(0.2)=1.2186, y(0.4)=1.4682 \text { and } y(0.6)=1.7379\) Using Milne’s predictor corrector method advance the solution to \(x =0.8\) and compare it with the analytical solution. (Carry four decimals).

[15M]

1994

1) Find the positive root of the equation \(e^{x}=1+x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6} e^{0.3 x}\) correct to five decimal places.

[10M]

2) Fit the following four points by the cubic splines. \(\begin{tabular}{|ccccc|} \hline\)i\(& 0 & 1 & 2 & 3 \\ \hline\)x_{i}\(& 1 & 2 & 3 & 4 \\\)y_{i}\(& 1 & 5 & 11 & 8 \\ \hline\end{tabular}\) Use the end conditions \(\quad y^{\prime \prime}{ }_{0}={y}^{\prime \prime}{ }_{3}=0\) Hence, compute (i) y (1.5) (ii) \(y^{\prime}(2)\)

[10M]

3) Find the derivative of \(f ( x )\) at \(x =0.4\) from the following table \(\begin{tabular}{|c|c|c|c|c|} \hline\)x\(& 0.1 & 0.2 & 0.3 & 0.4 \\ \hline\)y = f ( x )\(& 1.10517 & 1.22140 & 1.34986 & 1.49182 \\ \hline \end{tabular}\)

[10M]