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 \(\dfrac{d y}{d x}=\dfrac{1}{x+y}\), \(y(0)=1\) 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 \(7 x+52 y+13 z=104 ; 3 x+8 y+29 z=71 ; 83 x+11 y-4 z=95\) with \(\left(x^{0}, y^{0}, z^{\circ}\right)=(1.145,1.846,1.821)\). 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 \(\dfrac{d y}{d x}=x^{2}+y\) 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 \(\dfrac{d y}{d x}=x+y, y=1\) 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.

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

[14M]

3) Given the differential equation: \(\dfrac{d y}{d x}=x y ; y=2\) 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 \(f(x)=1-\dfrac{a}{x^{n}}\) Prove that \(x_{k+1}=\dfrac{1}{n}\left[(n+1) x_{k}-\dfrac{x_{k}^{n+1}}{a}\right]\)

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 \(\int_{1}^{2} x d x\)