2004

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

[10M]

2) Solve the initial value problem dydx=1x+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 (x0,y0,z∘)=(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 dydx=x2+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:

Velocity (in km/hr):02230271870 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 dydx=x+y,y=1 when x=0.

[14M]

2002

1) From the data given below x012456 f(x)114155619 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: dydx=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−axn Prove that xk+1=1n[(n+1)xk−xn+1ka]

2) Define interpolation. Find the polynomial P2(x) which satisfies f(−1)=P2(−1)=2

Find f(1.5).

3) Discuss simpson’s one-third rule of integration. Use it to find the value of ∫21xdx