1993

1) Find correct to 3 decimal places the two positive roots of 2ex−3x2=2.5644.

[10M]

2) Evaluate approximately ∫3−3x4dx Simpson’s rule by taking seven equidistant ordinates. Compare it with the value obtained by using the trapezoidal rule and with exact value.

[10M]

3) Solve dy/dx=xy for x=1.4 by Runge-kutta method, initially x=1,y=2 (Take h=0.2).

`[10M]

1992

1) Compute to 4 decimal placed by using Newton-Raphson method, the real root of x2+4sinx=0

[10M]

2) Solve by Runge-Kutta method dydx=x+y with the initial conditions x0=0,y0=1 correct up to 4 decimal places, by evaluating up to second increment of y. (Take h=0.1 )

[10M]

3) Fit the natural cubic spline for the data x:01234 y:00100

[10M]

1991

1) Using Regula Falsi method, find the real root of the equation xlog10x−1.2=0 correct to 5 decimal places.

2) Apply Lagrange’s formula to find a root of the equation f(x)=0 given that f(30)=−30, f(34)=−13 f(38)=3 and f(42)=18.

1990

1) Using Runge-Kutta method with third order accuracy, solve dydx=y−x with initial condition y=2, x=0.

2) Solve x2−5x+3=0 in the interval [1,2] by the secant method.

1989

1) The polynomial x3−x−1 has a root between 1 and 2. Using the secant method, find this root correct to three significant figures.

2) The integral is defined by

Given that K(1)=1.5709, K(4)=1.5727 and K(6)=1.5751 find K(3.5) using a second degree interpolating polynomial.

3) Use Runge Kutta method to solve 10dydx=x2+y2, y(0)=1 for the interval 0<x≤0.4 with h=0.1.