2004

1) Using Newton-Raphson method obtain a root near $x=0$ and correct to three decimal places of the equation $x+\sin x=1$.

[10M]

2) Solve the initial value problem ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{1}{x+y}}$, $y(0)=1$ using Rungc-Kutta method of fourth order to evaluate $y(0.5)$ in a single step.

[13M]

3) $$ Using Gauss-seidel iteration method find the solution, correct to three decimal places, of the linear system $7x+52y+13z=104;3x+8y+29z=71;83x+11y-4z=95$ with $(x^0,y^0,z^{\circ })=(1.145,1.846,1.821)$. Only two iterations may be supplied.

[13M]

2003

1) Find the cube root of 10 using Newton-Raphson method, correct to 4 decimal places.

[10M]

2) Apply modified Euler’s method to determine $y(0.1)$, given that ${\displaystyle \frac{dy}{dx}}=x^2+y$ when $y(0)=1$.

[10M]

3) The velocities of a car running on a straight road at intervals of 2 minutes are given below:

$\begin{array}{lllllll}\text{Velocity (in km/hr):}0& 22& 30& 27& 18& 7& 0\end{array}$ Apply simphson’s $1/3$ rule to find the distance covered by the car.

[13M]

4) Apply Runge-kutta method of order 4 to find on approximate value of $y$ when $x=0.2$ given that ${\displaystyle \frac{dy}{dx}}=x+y,y=1$ when $x=0$.

[14M]

2002

1) From the data given below $\begin{array}{lllllll}x& 0& 1& 2& 4& 5& 6\end{array}$ $f(x)\begin{array}{llllll}1& 14& 15& 5& 6& 19\end{array}$ using Lagrange’s interpolation formula calculate $f(3)$.

[10M]

2) Solve the following system of equations by Gauss’s elimination method.

$10x-7y+3z+5w=6;-6x+8y-z-4w=5$ $3x+y+4z+11w=2;5x-9y-2z+4w=7$

[14M]

3) Given the differential equation: ${\displaystyle \frac{dy}{dx}}=xy;y=2$ when $x=1,$ use Runge-Kutta Fourth order rule to find $y$ at $x=1.2$ taking the step length $h=0.2$.

[13M]

2001

1) Find Lagrange’s interpolation polynomial $P_{2}(x)$ Which $f(0)=P_{2}(0)=1$ $f(-1)=P_{2}(-1)=2$ $f(1)=P_{2}(1)=3$ Find $f(0.5)$

[10M]

2) By applying the Newton-Raphson method to $f(x)=x^{2}-a$ Where a $>0,$ Prove that $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$.

[13M]

3) Applying Simphson’s one-third rule compute the value of the definite int egral $\int_{4}^{5 / 2} \log x d x$ with $h=0.2$ and estimate the error.

[13M]

2000

1) By Applying Newton-Raphson Method to $f(x)=1-{\displaystyle \frac{a}{x^n}}$ Prove that $x_{k+1}={\displaystyle \frac{1}{n}}[(n+1)x_k-{\displaystyle \frac{x_k^{n+1}}{a}}]$

2) Define interpolation. Find the polynomial $P_2(x)$ which satisfies $f(-1)=P_2(-1)=2$

Find $f(1.5)$.

3) Discuss simpson’s one-third rule of integration. Use it to find the value of ${\int }_1^{2}xdx$