INTERNAL TEST – I Date: 22/01/2011
Subject:Numerical Methods & Linear Programming Class: III - ECE
Time: hrs Max.Marks:100
PART – A(Answer ALL Questions) (20*2=40)
1. (i). ------is self correcting method.
(a)Iteration method (b) Direct Method (c) both (a) & (b) (d) None of these.
(ii) Order of convergence of Regula Falsi method is ------.
(a)2 (b) 1 (c) 1.618 (d) None of these
2. (i).By Gaussian elimination method solution of x + y = 2, 2x + 3y = 5 is------
(a)1,2 (b)2,1 (c)1,1 (d)2,2
(ii). The rate of convergence in N.R. Method is …………
3. (i). Solution of 3x + y = 2, x +3y =-2, by Gauss-seidal iteration method is------
(ii). Write the Generalized Newton’s formula.
4.Say True (OR) False
(i). Iteration method will always converge.
(ii). Gauss Jacobi’s method converges faster than Gauss Seidel method.
5. Say True (OR) False
(i). Gauss Elimination Method is a Direct Method.
(ii). Gauss Seidal Method is a Direct Method.
6. State the Condition for the convergence of fixed point iteration method.
7. Solving a linear system, Compare Gaussian elimination method and Gauss Jordan method.
8. Diminish the roots of the equation x3-3x+1=0 by 1
9. State the principle used in Gauss – Jordon method
10. State the iterative formula for regula falsi method and also state rate of convergence .
11. (i). Gauss elimination method, the coefficient matrix is transformed to ------form.
(ii). The sufficient condition for convergence of Gauss-seidal method is------
12. State the two difference between direct and indirect methods for solving system of equations.
13. State the sufficient condition for Iterative (Indirect) Method.
14. Write the Iterative formula for Newton’s Method.
15. Write the matrices L & U , in Crouts Method.
16. For all the system of equations Iteration Method will work.(True/False)
17. Write the operation table in Relaxation Method.
18. Write the sufficient condition for convergence of Relaxation Method.
19. Explain the term ‘Pivoting’.
20.Write the Iterative formula to find the reciprocal of a given number N and hence find the value of 1/19.
PART– B (Answer ANY FIVE Questions) (5 * 12 = 60)
21.(a). Find an approximate root of xx – 1.2 = 0 By Newton’s Raphson method.
(b). Using method of false position find a root of the equation - 3x – 5 = 0.
22.(a). Find the positive root of x3+3x-1=0, correct to two decimal places, by Horner’s Method.
(b)Find the real positive root of 3x-cosx-1=0 by Newton’s Method correct to 6 decimal places.
23. (a). Solve the equation + - 1 = 0 for the positive root by iteration method.
(b). Solve by Gauss-Seidel method, the following system :
28x + 4y – z = 32; x + 3y + 10z = 24; 2x + 17y + 4z = 35.
24. (a). Solve by Gauss-Elimination method:
3x + 4y + 5z = 18; 2x – y + 8z = 13; 5x – 2y +7z = 20
(b). Solve the following system of equation using Gauss – Jacobi Method correct to
four decimal places. x + y + 54z = 110, 27x +6y - z = 85, 6x +15y + 2z = 72.
25.(a).Find the real positive root of x3-x-1=0 by Bisection Method, correct to 4 decimal places.
(b). Obtain Newton’s iterative formula for finding where N is a positive real number. Hence evaluate
26. By Crout’s Method , solve the system. 2x+3y+z=-1; 5x+y+z=9; 3x+2y+4z=11
27.(a). Solve the following equations using relaxation method
10x-2y-2z=6;-x+10y-2z=7; -x-y+10z=8
(b). Find the real root of the equation x3+x2=100 , using iteration method
28. (a) Apply Gauss Jordan Method to find the solution of the following system:
10x + y + z = 12, 2x + 10y + z = 13, x + y + 5z = 7
(b). Solve by Gauss-Seidel method, the following system :
x + y +54 z = 110; 27x + 6y -z = 85; 6x + 15y + 2z = 72.
INTERNAL TEST – II Date: /02/11
Subject : Numerical Methods & Linear Programming Class: III - ECE
Time : 3 hrs Max.Marks:100
PART – A (Answer ALL the Questions) 20 x 2 = 40
1. Write Newton’s backward interpolation formula .
2. If y (10) =35.3 y (15) =32.4 y (20) =29.2 , find the value of y (12) using Newton’s Forward Interpolation formula.
3. Write Gauss Forward Interpolation formula.
4. Write Laplace Everett formula.
5. a) Write the another Name of Bessel’s formula.
b) Interpolation is required near the beginning of the tabular values , then use Newton’s Gregory
forward interpolation formula.(True / False)
6. Write any two properties of divided differences.
7. Write Lagrange’s interpolation formula for unequal intervals.
8. Write Newton’s backward interpolation formula for first derivative at x = x0.
9. Write Newton’s forward interpolation formula for second derivative.
10. Write Newton’s backward interpolation formula for third derivative at x = x1.
11. (i) The order of the error in simpson’s rule is ------
(a) 3 (b) 2 (c) 1 (d) None of these.
(ii) ------is the process of finding the intermediate values of function from a set of its values at specific points given in tabulated form.
(a) Interpolation (b) extrapolation (c) both (d) None of these.
12. Evaluate by Trapezoidal rule, dividing the range into 4 equal parts.
13. When does Simpson’s rule give exact result?
14. What are the errors in Trapezoidal and Simpson’s rule?
15. Write Simpson’s 3/8 rule.
16. Use Simpson’s rule to estimate the value of from the following :
x / 1 / 2 / 3 / 4 / 5f(x) / 13 / 50 / 70 / 80 / 100
17. Write the Taylor Series formula at x = x0.
18. Write Improved Euler’s formula.
19. Write the fourth order R.K. Algorithm.
20. Write Adams Predictor and Corrector formulae.
PART – B (Answer ANY FIVE questions) 5 x 12 = 60
21. a) The following data are taken from the steam table.
Tem. / 140 / 150 / 160 / 170 / 180Pressure kgf/cm2 / 3.685 / 4.854 / 6.303 / 8.076 / 10.225
Find the Pressure at Temperature t = 142o and t = 175o using Newton’s Interpolation Formula.
b) Use Gauss ‘s backward interpolation formula find the population for the year 1936 given that
Year / 1901 / 1911 / 1921 / 1931 / 1941 / 1951Popu. in thousands / 12 / 15 / 20 / 27 / 39 / 52
22. From the following table , estimate and correct to fove decimal placesusing Bessel’s formula and Laplace Everett’s Formula.
x / 0.61 / 0.62 / 0.63 / 0.64 / 0.65 / 0.66 / 0.67y / 1.840431 / 1.858928 / 1.877610 / 1.896481 / 1.915541 / 1.934792 / 1.954237
23. a) Given the values:
x / 1.0 / 1.1 / 1.2 / 1.3 / 1.4 / 1.5 / 1.6y / 7.989 / 8.403 / 8.781 / 9.129 / 9.451 / 9.750 / 10.031
Find and at x = 1.1
b) Find the value of x when y = 20 using Lagrange’s interpolation:
x / 1 / 2 / 3 / 4y / 1 / 8 / 27 / 64
24. a) Evaluate by dividing the range into 4 equal parts using Trapezoidal and Simpson’s rule.
b) The following data gives the velocity of a particle for 20 seconds at an interval of 5 seconds. Find initial acceleration(i.e., at t = 0) using the entire data
Time (sec) / 0 / 5 / 10 / 15 / 20Velocity (m/sec) / 0 / 3 / 14 / 69 / 228
25. a) Using Taylor Series Method , find y at x=0.1 and x=0.2 given , y(0) = 1.( correct to 4
decimal places).
b) Solve y(0) =1, by Picards Method upto the third approximation . Hence find the
value of y(0.1), y(0.2).
26) a) Solve, y(0) = 2 using Modified Euler’s method and tabulate the solutions at
x = 0.2, 0.4 and 0.6,taking h = 0.
b) Apply fourth order Runge-Kutta method to determine y(0.1), y(0.2) with h = 0.1 from , y(0) = 1.
27) a) Given , y(0) = 1, y(0.1) = 0.9537, y(0.2) = 0.9145, y(0.3) = 0.8821, find y(0.4) using Milne’s method.
b) Using the following table , apply Gauss Forward formula to get f(3.75).
X / 2.5 / 3.0 / 3.5 / 4.0 / 4.5 / 5.0f(x) / 24.145 / 22.043 / 20.225 / 18.644 / 17.262 / 16.047
28) Find y(0.1),y(0.2) and y(0.3) from , y(0)=1 using Runge – Kutta method and hence obtain y(0.4) using Adam’s Method.
ALL THE BEST