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 $x_{i+1}={\displaystyle \frac{x_1(p+1-N{x}_i)}{p}}$

[6M]

2) Evaluate ${\int }_0^1{\displaystyle \frac{dx}{1+x^2}},$ 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: $\begin{array}{l}{x}_1+6x_2+3x_3=6\\ 2x_1+3x_2+3x_3=117\\ 4x_1+x_2+2x_3=283\end{array}$

[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 $y=\left({\displaystyle \frac{x-1}{x}}\right)+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x-1}{x}}\right)}^2+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{x-1}{x}}\right)}^3+{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{x-1}{x}}\right)}^4+{\displaystyle \frac{1}{5}}{\left({\displaystyle \frac{x-1}{x}}\right)}^5$

[15M]

1999

1) Obtain the Simpson’s rule for the integral
$I={\int }_a^{b}f(x)dx$
and show that this rule is exact for polynomial of degree $n\ge 3$.In general show that the error for approximation for Simpson’s rule in given by $R=-{\displaystyle \frac{(b-a{)}^5}{2880}}{f}^{iv}(\eta ),\eta \in (0,2).$
Apply this rule to the integral ${\int }_0^1{\displaystyle \frac{dx}{1+x}}$ and show that $|R|\le 0.008333$.

[20M]

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

[20M]

1998

1) Evaluate $in{t}_1^3{\displaystyle \frac{dx}{x}}$ 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 ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{xy+1}{10y^2+4}},$ 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 $xlo{g}_{10}x-1.2=0$ lies between 3 and 2.740646.

4) Given ${\int }_{-\mathrm{\infty }}^z{\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{w^2}{2}}}dw$=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 $y^{\mathrm{\prime }}={\displaystyle \frac{dy}{dx}}=x+y,y(0)=1$

[10M]

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

[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 $M_0=0,M_3=0$

[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 ${\displaystyle \frac{dy}{dx}}=xz+1,{\displaystyle \frac{dz}{dx}}=-xy$ 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 ${\int }_1^{3.4}f(x)dx$ from the following data using Simpson’s ${\displaystyle \frac{3}{8}}th$ rule for the interval (1.6,2.2) and ${\displaystyle \frac{1}{3}}rd,$ 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 ${\displaystyle \frac{dy}{dx}}=y-x^2,y(0)=1$ 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 $e^x=1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}{e}^{0.3x}$ correct to five decimal places.

[10M]

2) Fit the following four points by the cubic splines. \begin{tabular}{|ccccc|}
\hline$\text{begin{tabular}{|ccccc|}hline}$i& 0 & 1 & 2 & 3 \\
\hline$\text{\& 0 \& 1 \& 2 \& 3 hline}$x_{i}& 1 & 2 & 3 & 4 \\$\text{\& 1 \& 2 \& 3 \& 4 }$y_{i}& 1 & 5 & 11 & 8 \\
\hline\end{tabular}$\text{\& 1 \& 5 \& 11 \& 8 hlineend{tabular}}$ Use the end conditions $y^{\mathrm{\prime }\mathrm{\prime }}{}_0=y^{\mathrm{\prime }\mathrm{\prime }}{}_3=0$ Hence, compute (i) y (1.5) (ii) $y^{\mathrm{\prime }}(2)$

[10M]

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

[10M]