1993

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

[10M]

2) Evaluate approximately ${\int }_{-3}^3{x}^{4}dx$ 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 ${\displaystyle \frac{dy}{dx}}=x+y$ 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 ${\displaystyle \frac{dy}{dx}}=y-x$ with initial condition $y=2$, $x=0$.

2) Solve $x^2-5x+3=0$ 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 $10{\displaystyle \frac{dy}{dx}}=x^2+y^2$, $y(0)=1$ for the interval $0<x\le 0.4$ with $\mathrm{h}=0.1$.