2008

1) Find the smallest positive root of equation xex−cosx=0 using Regula Falsi method. Do three iterations.

[12M]

2) The following values of the function f(x)=sinx+cos$$x are given: X10∘20∘30∘f(x)1.15851.28171.3360 Construct the quadratic interpolating polynomial that fits the data. Hence calculate and compare with exact value.

[15M]

3) Apply Gauss-Seidel method to calculate x, y, z from the system: −x−y+6z=42 6x−y−z=11.33 −x+6y−z=32 with initial values (4.67,7.62,9.05). Carry out computations for two iterations.

[15M]

2007

1) Find the smallest positive root of equation 3x+sinx−ex=0, correct to five decimal places, using Regula-falsi method

[10M]

2) Write a computer program using BASIC’ to solve the following problem ∫π/2π/4sinxxdx By trapezoidal rule.

[10M]

3) Compute y(10) using Lagrange’s interpolation formula from the following data: x371117y10151720

[10M]

4) Derive three-point Gaussian quadrature formula and hence evaluate. ∫1.50.2e−x2dx calculating weights and residues. Give the result to three decimal places.

[10M]

5) Solve the system
1.2x1+21.2x2+1.5x3+2.5x4=27.46 0.9x1+2.5x2+1.3x3+32.1x4=49.72 2.1x1+1.5x2+19.8x3+1.3x4=28.76 20.9x1+1.2x2+2.1x3+0.9x4=21.70 using Gauss-Seidel iterative scheme correct to three decimal places starting with initial value (1.04,1.30,1.45,1.55)⊤

[10M]

2006

1) Perform four iterations of the bisection method to obtain a positive root of the equation f(x)=x−5x+1

[10M]

2) Write a BASIC program to evaluate a definite integral by Simpson’s one-third rule Adapt it to evaluate ∫0(x3+sinx)dx by taking 10 subintervals and indicating which lines are to be modified for a specific problem

[10M]

3) Apply Gauss-Seidel iteration method for three iterations to solve the equation [10−2−1−1−210−1−1−1−110−2−1−1−210][x1x2x3x4]=[31527−9]

[10M]

4) Apply Runge-Kutta method of fourth order to find an approximate value of y when x=0.1 and 0.2 , given that dydx=x+y2,y=1 when x=0

[10M]

5) Write a program in BASIC to find a root of an equation by Newton-Raphson method. Adapt it to solve x3−4x2+x+6=0 using initial approximation as x0=5. Indicate which lines are to be changed for a different equation.

[10M]

6) Apply Newton’s fonard and backard difference formulae to evaluate f(12) and f(3.9) respectively from the data xy115

[10M]

2005

1) Perform four iterations of the bisection method to obtain a positive root of the equation f(x)=x3−5x+1=0

[10M]

2) Evaluate ∫10√1+2xdx by applying Gaussian quadrature formula, namely ∫1−1f(t)dt=∑ni=1Aif(ti) where the coefficients Ai and the roots ti are given below for n=4 as t1=−0.8611A2=A1=0.3478 t2=−0.3399A2=A8=0,6521 t3=0.3399 t4=0.8611

[10M]

3) Apply Gauss-Seidel iterative method for five iterations to solve the equations [10−2−1−1−210−1−1−1−110−2−1−1−210][x1x2x3x4]=[31527−9]

[10M]

4) Write a BASIC program to evaluate a delinite integral ∫10(x3+sinx)dx by Simpson’s one-third rule. Indicate the lines which are to be modified for a different problem.

[10M]

5) Write a program in BASIC to solve the equation x3−4x2+x+6=0 by Newton-Raphson method by taking the initial approximation as x0=5. Indicate which lines are to be changed for a different equation.

[10M]

6) Apply Runge-Kutta method of fourth order to find an approximate value of y when x=0.2. given that dydx=x+y2,y=1 when x=0

[10M]