Numerical & Statistical Computations

1.     Find the negative root of the equation x3 – 21x + 3500 = 0 correct to two decimal places by Newton Raphson Method.

2.    Solve the following set of linear equations by Gauss Seidal method
                  1.2x + 2.1y + 4.2z = 9.9
                  5.3x + 6.1y + 4.7z = 21.6
                  9.2x + 8.3y +      z = 15.2

3.    Define Interpolation with the help of suitable example. Derive the relation between divided differences and ordinary differences.

4.    Integrate the function x³ + 2x +1 with respect to x from 0 to 1 by using Trapezoidal. Divide the interval into eight equal intervals.    

5.     Fit a straight line trend by the method of least square to the following data.
 
 
 

     

Estimate the likely product for the year 2000.

6.   Solve the following differential equation by using Runga Kutta fourth order method to find out y(1).
 
                              
7. Find f’ (3) and f” (3) from the following table using Newton’s forward formula.
 

8.    Discuss the three available methods (Bi-Section, Regula Falsi and Newton Raphson Method) and explain the merits and demerits of each method.

Assignment B

1.    Compare and contrast Trapezoidal, Simpson's 1/3 and Simpson's 3/8 rule of integration.

2. Find the value of f(x) at 3.1 and 3.9 for the following data by using the appropriate formula. x 3 3.2 3.4 3.6 3.8 4.0 y -14 -10.032 -5.296 0.256 6.672 14

3.    Define Interpolation. Prove that E-?=1, where E is the shift operator. (b)?4y0=y4-4y3+6y2-4y1+y0

Assignment C
1. Which one is a method for getting solution to non-linear algebraic equation?
 
Options    
Runga Kutta Method    
Newton Raphson Method    
Jacobi Method    
Divided Difference Formula

2. y=mx+c is the equation of a--
 
Options    
Polygon    
Circle    
Line    
None

3. Which one of the following is not a method for finding the root of an algebraic equation?
 
Options    
Newton Raphson Method    
Bi-Section Method    
Gregory’s Method    
Regula Falsi Method

4. The formula for Newton Raphson method is    
 
Options

               

5. For x3 – 5x +3 =0, the root lies in between    
 
Options    
[0, 1]    
[4, 5]    
[3, 4]    
[0, -1]

6. The value of Δ f(x) is    
 
Options    
f(x1) + f(x0)    
f(x1) – f(x0)    
f(x1)    
None of these

7. Which one is not a method for numerical integration    
 
Options    
Trapezoidal Rule    
Gauss Method    
Simpson’s 1/3 Rule    
Simpson’s 3/8 Rule

8. The Formula for Bi-section method is    
 
Options    
(x1+x2)/2    
(x1-x2)/2    
(x1x2)/2    
None of these

9.
 

Options    
y3-3y2+3y1-y0    
y3+3y2+3y1+y0    
y0-3y1+3y2-y3    
None of these

10. In forward difference formula 'h' is    
 
Options    
The difference between two consecutive y.    
The difference between two consecutive x    
The difference between first and last x values    
The difference between first and last y values
Ans- The difference between two consecutive x    

11. In line fitting method, the general equation of a line is    
 
Options    
y = a + bx    
y2 = a + bx    
y = a + bx2    
None of these

12. For Trapezoidal rule the Generalized Quadrature formula uses    
 
Options    
n=1    
n=2    
n=3    
None of these

13. Gauss elimination method is used to solve the set of linear algebraic equations    
 
Options    
True    
False

14. For f(a) and f(b)are of same sign then equation f(x)=0 has at least one root with in [a,b].

Options    
True    
False

15. C (n, r) or nCr. = n! / (n+r)! r!

Options    
True    
False

16. In Gauss Elimination method, coefficient matrix A is reduced to upper triangle matrix by using the elementary row operations    
 
Options    
True    
False

17. Modified Euler is a modified version of Euler Method.

Options    
True    
False

18. Gauss Elimination method reduces the system of equations to an equivalent upper triangular matrix.

Options    
True    
False    

19. Regula Falsi Method converges fastest among Bi-section, Regula Falsi and Newton Raphson Method.
 
Options    
True    
False

20. The number of distinguishable words that can be formed from the letters of MISSISSIPPI is 34650.    
 
Options    
True    
False

21. The set of linear algebraic equations can be arranged in matrix for AX=B, where A is the coefficient matrix, X is the variable matrix.
 
Options    
True    
False

22. Numerical methods give always-exact solutions to the problems    
 
Options    
True    
False

23. Simpson's method is used to interpolate the value of the function at some given point.
 
Options    
True    
False

24. The set of equation 3x+2y = 0 and 2x+7y = 9 can be solved by using Bi-Section method.

 
Options    
True    
False

25. In solving simultaneous equation by Gauss- Jordan method , the coefficient matrix is reduced to ------------- matrix  

 
Options    
Null    
Unit    
Skew    
Diagonal

26. The order of convergence in Newton Raphson method is

 
Options    
2    
3    
0    
None of these

27. Which of the following is a step by step method    
 
Options    
Taylor`s    
Adams-Bashforth    
Picard`s    
Euler`s

28. In the case of Bisection method , the convergence is    
 
Options    
LINEAR    
Quadratic    
Very slow    
None

29. Solutions of simultaneous non- linear equations can be obtained using    
 
Options    
Method of iteration    
Newton-Raphson method    
Bisection method    
None

30. Bessel`s formula is most appropriate when p lies between    
 
Options    
-0.25 and 0.25    
0.25 and 0.75    
0.75 and 1    
None of the above

31. The order of the matrix [473] is.    
 
Options    
3*1    
1*3    
3*3    
1*1

32. If B is square matrix and BT = - B, then B is called    
 
Options    
Symmetric    
Skew symmetric    
Singular    
Non Singular

33. Find the coefficient of x³ in the Taylor series about x = 0 for f(x)  =sin2x ?
 
Options    
-2/3    
-4/3    
4/3    
2/3

34. The bisection method of finding roots of nonlinear equations falls under the category of a (an) ---------------- method.

 
Options    
Open    
Bracketing    
Random    
Graphical

35. A unique polynomial of degree -----------------passes through n+1 data points.

 
Options    
n+1    
n    
n or less    
n+1 or less

36. Interpolation is the technique to find the value of dependent variable for the given value of independent variable    
 
Options    
True    
False

37. By increasing the iterations of any Numerical methods, we increase the correctness of the solution.    
 
Options    
True    
False    

38. Lagrange’s Interpolation method can be used only for equal interval problems.    
 
Options    
True    
False    

39. Trapezoidal Integration Method is derived by putting

 
Options    
n =0    
n=1    
n=2    
n=4

40. 
If f(x) is a real continuous function in [a,b], and f(a)f(b)<0, then for f(x),there is (are).............in the domain [a,b].    
 
Options    
One root    
An undeterminable number of roots    
No root    
At least on root