IAS PYQs 1
2000
1) Using Newton-Raphson’s method, show that the iteration formula for finding the reciprocal of the pth root of N is xi+1=x1(p+1−Nxi)p
[6M]
2) Evaluate ∫10dx1+x2, by subdividing the interval (0,1) into 6 equal parts and using Simpson’s one-third rule. Hence find the value of π and actual error, correct to five places of decimals.
[15M]
3) Solve the following system of linear equations, using Gauss elimination method: x1+6x2+3x3=62x1+3x2+3x3=1174x1+x2+2x3=283
[15M]
4) Let A=[531374149]B=A−1 and C=[100010001] Write a BASIC programm that computes the inverse of A, determinant of A and the product of the matrix and its inverse.
[15M]
5) Write a BASIC program to evaluate the formula y=(x−1x)+12(x−1x)2+13(x−1x)3+14(x−1x)4+15(x−1x)5
[15M]
1999
1) Obtain the Simpson’s rule for the integral
I=∫baf(x)dx
and show that this rule is exact for polynomial of degree n≥3.In general show that the error for approximation for Simpson’s rule in given by
R=−(b−a)52880fiv(η),η∈(0,2).
Apply this rule to the integral ∫10dx1+x and show that |R|≤0.008333.
[20M]
2) Using fourth order classical Runge-Kutta method for the intial value problem
dudt=−2tu2,u(0)=1,
where h=0.2 on the interval [0,1],calculate u(0.4) correct to six places of decimal.
[20M]
1998
1) Evaluate int31dxx by Simpson’s rule with 4 strips. Determine the error by direct integration.
2) By the fourth-order Runge-Kutta method, tablulate the solution of the differential equation dydx=xy+110y2+4, y(0)=0 in [0,0.4] with step length 0.1 correct to five places of decimals.
3) Use Regula-Falsi method to show that the real root of xlog10x−1.2=0 lies between 3 and 2.740646.
4) Given ∫z−∞1√2πe−w22dw=0.5987,0.9=6915,0.7734,0.8944,0.9772 for z=0.25,0.5,0.75,1.25,2 respectively.
5) Fit a second degree parabola to the following data taking x as the independent variable:
x y 1 2 2 6 3 7 4 8 5 10 6 11 7 11 8 10 9 9 — —-
1997
1) Apply the fourth order Runge-Kutta method to find a value of y correct to four places of decimals at x=0.2, when y′=dydx=x+y,y(0)=1
[10M]
2) Show that the iteration formula for finding the reciprocal of N is xe−1=xn(2−Nxn),n=0,1…
[10M]
3) Obtain the cubic spline approximation for the function given in the tabular form below: x:0123f(x)1233244 and M0=0,M3=0
[10M]
1996
1) Describe Newton-Raphson method for finding the solutions of the equation f(x)=0 and show that the method has a quadratic convergence.
[15M]
2) The following are the measurements 1 made on a curve recorded by the oscillograph representing a change of current i due to a change in the conditions of an electric current: t:1.22.02.5.3.0i:1.360.580.340.20 Applying an appropriate formula interpolate for the value of i when t=1.6
[15M]
3) Solve the system of differential equations dydx=xz+1,dzdx=−xy for x=0.3 given that y=0 and z=1 when x=0, using Runge-Kutta method of order four.
[15M]
1995
1) Find the positive root of logex=cosx nearest to five places of decimal by Newton-Raphson method.
[15M]
2) Find the value of ∫3.41f(x)dx from the following data using Simpson’s 38th rule for the interval (1.6,2.2) and 13rd, rule for (2.2,3.4) x:1,61.82.02.22.4f(x):4.9536.0507.3899.02511.023x:2.62.83.03.23.4f(x):13.46416.44520.08624.53329.964
[15M]
3) For the differential equation dydx=y−x2,y(0)=1 starting values are given as y(0.2)=1.2186,y(0.4)=1.4682 and y(0.6)=1.7379 Using Milne’s predictor corrector method advance the solution to x=0.8 and compare it with the analytical solution. (Carry four decimals).
[15M]
1994
1) Find the positive root of the equation ex=1+x+x22+x36e0.3x correct to five decimal places.
[10M]
2) Fit the following four points by the cubic splines. \begin{tabular}{|ccccc|} \hlinei& 0 & 1 & 2 & 3 \\ \hlinex_{i}& 1 & 2 & 3 & 4 \\y_{i}& 1 & 5 & 11 & 8 \\ \hline\end{tabular} Use the end conditions y′′0=y′′3=0 Hence, compute (i) y (1.5) (ii) y′(2)
[10M]
3) Find the derivative of f(x) at x=0.4 from the following table \begin{tabular}{|c|c|c|c|c|} \hlinex& 0.1 & 0.2 & 0.3 & 0.4 \\ \hliney = f ( x )& 1.10517 & 1.22140 & 1.34986 & 1.49182 \\ \hline \end{tabular}
[10M]