1993

1) Find correct to 3 decimal places the two positive roots of \(2 e ^{ x }-3 x ^{2}=2.5644\).

[10M]

2) Evaluate approximately \(\int_{-3}^{3} x^{4} d x\) 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 \(x^{2}+4 \sin x=0\)

[10M]

2) Solve by Runge-Kutta method \(\dfrac{d y}{d x}=x+y\) with the initial conditions \(x _{0}=0, y _{0}=1\) 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 \(\begin{array}{llllll} x : & 0 & 1 & 2 & 3 & 4\end{array}\) \(\begin{array}{llllll}y: & 0 & 0 & 1 & 0 & 0\end{array}\)

[10M]

1991

1) Using Regula Falsi method, find the real root of the equation \(x \log _{10} x-1.2=0\) correct to 5 decimal places.

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

1990

1) Using Runge-Kutta method with third order accuracy, solve \(\dfrac{d y}{d x}=y-x\) with initial condition \(y=2\), \(x=0\).

2) Solve \(x^{2}-5 x+3=0\) in the interval [1,2] by the secant method.

1989

1) The polynomial \(x^{3}-x-1\) 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 \(\mathrm{K}(1)=1.5709, \mathrm{~K}(4)=1.5727\) and \(\mathrm{K}(6)=1.5751\) find \(\mathrm{K}(3.5)\) using a second degree interpolating polynomial.

3) Use Runge Kutta method to solve \(10 \dfrac{d y}{d x}=x^{2}+y^{2}\), \(y(0)=1\) for the interval \(0<x \leq 0.4\) with \(\mathrm{h}=0.1\).