Department of Mathematics, UEM



[pic]

UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPUR

QUESTION BANK

SUBJECT NAME: NUMERICAL METHOD,

SUBJECT CODE: M(CS)301

B.TECH, 2ND YEAR, 3RD SEMESTER

GROUP-A

(Objective/Multiple type question)

1. The iterative formula of Euler’s method for solving y’=f(x, y) with y(x0) =y0, is

1. yn = yn+1 + hf (xn+1, yn+1)

2. yn = yn-1 + hf(xn-1, yn-1)

3. yn = yn-1 + hf(xn+1, yn+1)

4. yn = yn+1 + hf(xn-1, yn-1)

2. Using Euler’s method, y’ = (y – 2x)/y, y (0) = 1; gives y (0.1) =? (Take h=0.1)

1. 1.1818

2. 2.1818

3. 1.2020

4. 2.2020

3. The formula for the 4th order Runge-Kutta method is……?

1. k = (k1+2k2+2k3+k4)/4

2. k = (k1+2k2+2k3+k4)/2

3. k = (k1+2k2+2k3+k4)/6

4. k = (k1+2k2+4k3+k4)/6

4. Apply Runge-Kutta fourth order method to find value of k1 and k2 given that

dy/dx = x+y and y(0)=1. (h = 0.2)

1. k1=0.2400, k2=0.2000

2. k1=0.2000, k2=0.2400

3. k1=0.2040, k2=0.2020

4. k1=0.2020, k2=0.2040

5. In Runge-Kutta method formula for finding k4 is….?

1. hf(x0 - h, y0 - k3)

2. hf(x0 + 1/2h, y0 + 1/2k3)

3. hf(x0 ,y0 )

4. hf(x0 + h, y0 + k3)

6. In Runge-Kutta method k is..?

1. Sum of k1, k2, k3 and k4.

2. The weighted mean of k1, k2, k3 and k4.

3. Product of k1, k2, k3 and k4

4. None of this

Q.7 Use the regulafalsi to solve equation x2-10=0(onlyaiteration required and accuracy upto four decimal places required).

1. 3.1436

2. 2.1428

3. 3.1428

4. 2.1436

Q.8...............lies in the category of iterative method.

1. Lagrange Method

2. Newton’s divided difference Method

3. Newton Raphson Method

4. all of the given choices

Q.9 If the root of the given equation lies between a and b, then the first approximation to the root of the equation by bisection method is ……

1. a+b/2

2. a-b/2

3. b-a/2

4. None of the given choices

Q.10 The Newton-Raphson method fails when

1. f’(x) is negative

2. f’(x) is too large

3. f’(x) is zero

4. Never fails

Q.11 In which of the following methods proper choice of initial value is very important?

1. Newton RaphsonMethod

2. Bisection Method

3. Iterative Method

4. RegulaFalsi Method

Q.12 Newton-Raphson method has a _____________ convergence.

1. Linear

2. Quadratic

3. cubic

4. Bi quadratic

Q.13 The root of the equation ex=4x lies between________.

1. (0, 1)

2. (1, 2)

3. (2, 3)

4. (3, 4)

Q.14 The following x-y data is given

|x |15 |18 |22 |

|y |24 |37 |25 |

The Newton’s divided difference second order polynomial for the above data is given by 

           f(x) =24+(x-15)[x0,x1]+(x-15)(x-18)[x0,x1,x2]+(x-15)(x-18)(x-22)[x0,x1,x2,x3]

the value of[x0, x1] is

1. -1.048

2. 0.1433

3. 4.333

4. 24.00

Q.15 Lagrange’s interpolation formula is used to compute the values for _______ intervals.

1. Equal

2. Unequal

3. Open

4. Closed

Q.16 By using Newton’s divided difference table form the following data, what is the value of ▲2y0 ?

|X |1 |2 |4 |7 |12 |

|Y |22 |30 |82 |106 |216 |

1. 16

2. 6

3. 1.6

4. 0.19

Q.18 Matrix inversion method fails when

1.[pic] ≠ 0

2.[pic] = 0

3.[pic]Does not exist

4. Never fails

Q.19 As soon as a new value of a variable is found by iteration, it is used immediately in the following equations, this method is called

1. Gauss-elimination method

2. Gauss-Seidel method

3. Inversion method

4. Factorization method

Q.20 In solving simultaneous equations by LU factorization method, the matrix A is

expressed as the?

1. Sum of a lower triangular and an upper triangular matrices

2. Sum of singular matrices

3. Product of a lower triangular and an upper triangular matrices

4. Product of singular matrices

Q.21 When AX=B is solved by gauss-elimination method, the coefficient matrix is

transformed to

1. Upper triangular matrix

2. Scalar matrix

3. Lower triangular matrix

4. Diagonal matrix

Q.22 Gauss-elimination method fails when

1. Non-pivots are zero

2. A is non-singular matrix

3. Any of the pivots are zero

4. Never fails

Q.23 Which of the following is an iterative method for solving simultaneous

equations?

1. Gauss-Elimination method

2. Factorization method

3. Matrix inversion method

4. Gauss-Seidel method

Q.24 If x= 1/20(17-y+2z), y= 1/20(-18-3x+z) and z=1/20(25-2x+3y) then the value of x1, y1, z1

using gauss-Seidel method will be:

1. x1=0.8500, y1=-1.0275, z1=1.0109

2. x1=0.9500, y1=-1.2750, z1=1.1009

3. x1=1.0500, y1=-1.4575, z1=1.0109

4. x1=0.8500, y1= 1.0275, z1=1.0109

Q.25 In AX=B if A = [pic] and B =[pic] then X=?

1. [pic]

2. [pic]

3. [pic]

4. [pic]

Q.26 In factorization method A=LU provided

1. All the principal minors of A are singular

2. All the principal minors of A are non-singular

3. Matrix L and U are singular

4. Matrix L and U are non-singular

Q.27[pic]then using gauss-elimination method z=?

1. z=3

2. z=4

3. z=5

4. z=2

Q.28 The Relative error in taking [pic]= 3.141593 as 22/7 will be:

a) 0.0005

b) 0.0004

c) 0.0040

d) 0.4000

Q.29 . The shifting operator is denoted by ________.

a) E

b) nabla

c) omega

d) T

Q.30 The process of finding the values outside the interval (X0, Xn) is called

a) Interpolation

b) Extrapolation

c) Iterative

d) Polynomial equation

Q.31 Exact solution of 2/3 does not exist.

a) True

b) False

Q.32 Differences methods find the ________ solution of the system.

Select correct option:

a) Analytical

b) Numerical

Q.33 If [pic]is approximated by a polynomial [pic] of degree n then the error is

given by

a) [pic]

b) [pic]

c) [pic]

d) [pic]

Q.34 If we retain r+1 terms in Newton’s forward difference formula, we obtain

a polynomial of degree ---- agreeing with [pic] at [pic]

a) r+2

b) r+1

c) r

d) r-1

Q.35 In case of Newton Backward Interpolation Formula which equation is

correct to find u?

a) (x – xn) h = u

b) x – xn = uh

c) x – xn = u

d) x + xn = uh

Q.36 The following x-y data is given

|x |15 |18 |22 |

|y |24 |37 |25 |

The Newton’s divided difference second order polynomial for the above data is given by 

           f(x) =24+(x-15)[x0,x1]+(x-15)(x-18)[x0,x1,x2]+(x-15)(x-18)(x-22)[x0,x1,x2,x3]

the value of[x0, x1] is

5. -1.048

6. 0.1433

7. 4.333

8. 24.00

Q.37 If the interval of differencing is unity, then [pic]=

a) 144

b) 144h

c) 0

d) None of the above

Q.38 Which of the following methods does not require starting values?

1) Euler’ s method

2) Milne’s method

3) RK method

4) None of these

Q.39 In the geometrical meaning of Euler’s method the curve is approximated as a-

1) Straight Line

2) Circle

3) Parabola

4) Ellipse

Q.40 Predictor-corrector methods are self-starting methods?

1) True

2) False

Q.41 Runge-kutta method is a self-starting method?

1) True

2) False

Q.42 In Euler’s method, if h is small the method is too small, if h is large, it gives inaccurate value.

1) True

2) False

Q.43 yn +1 = yn+ hf (xn,yn) is the iterative formula for:

1) Euler’s method

2) Milne’s method

3) Ranga-kutta method

4) None of these

Q.44 Single step method is:

1) Euler’s method

2) Corrector predictor methods

3) RK method

4) None of these

Q45 For finding the value of y at xi+1 in the corrector method, the numbers of prior values

are required

1) 1

2) 2

3) 3

4) 4

Q.46 A predictor formula is used to predict the value of y at

1) x

2) xi

3) xi+1

4) yi

Q.47 Milne’s corrector formula is:

1) Y4 = y2 +1/3h (f 2+4 f3 + f4)

2) Y4 = y2 +1/3.4h (2f1- f2+2 f3)

3) Y4 = y2 +1/3h (f 1+4 f2 + f3)

4) Y4 = y2 +1/3.4h (2f1+ f2-2 f3)

Q.48 The following x-y data is given

|x |15 |18 |22 |

|y |24 |37 |25 |

The Newton’s divided difference second order polynomial for the above data is given by 

           f(x) = 24+(x-15)[x0,x1]+(x-15)(x-18)[x0,x1,x2]+(x-15)(x-18)(x-22)[x0,x1,x2,x3]

the value of [x0, x1] is

9. -1.048

10. 0.1433

11. 4.333

12. 24.00

Q.49 If the interval of differencing is unity, then [pic]=

e) 144

f) 144h

g) 0

h) None of the above

Q.50 Value of intrigal[pic] using trapezoidal rule of integration with

h = 1/5 will be

a) 0.59

b) 0.69

c) 0.60

d) 1.0

GROUP-B

(Short answer type questions)

1. What is the difference between direct and iterative method and explain rate of convergence.

2. Perform first two steps of Bi-section method to find the roots of [pic]

3. Prove [pic]

4. Construct the table using newton’s divided difference method.

|X |4 |5 |7 |10 |11 |13 |

|Y |48 |100 |294 |900 |1210 |2028 |

5. Prove that [pic]

6. Using Lagrange’s interpolation formula find [pic], for the following data:

[pic]

7. Find the missing values in the following table:

|X |45 |50 |55 |60 |65 |

|Y |3 |? |2 |? |-2.4 |

8. Define error and types of error.

9. What do you mean by Initial and Boundary value problems?

10. Find an approximate value(by Euler’s) of[pic]

11. Write Runge-kutta fourth order method for [pic]

12. Find the relative error if [pic] is approximated by [pic].

13. Given[pic], Evaluate[pic],where[pic]and[pic],usingMatrix inversion method.

14. Explain Steps of Adams Bashforth Predictor - corrector method.

15. Apply Newton’s backward Interpolation to the data below, to obtain a polynomial of degree 3 in [pic]

[pic] 1 2 3 4

[pic] 1 -1 1 -1

16. Prove that[pic].

17. Perform first two steps of Bi-section method to find the roots of [pic]

18. Using Lagrange’s interpolation formula find [pic], for the following data:

[pic]

19. Find the inverse of the matrix [pic]

20. Perform first step of Regula-Falsi method to find the roots of [pic]

21. Find the percentage error in taking [pic]= 3.141593 as 22/7.

22. State Trapezoidal formula of integration.

23. What do you mean by Initial and Boundary value problems?

24. Write Euler’s method for [pic]

25. Write Runge-Kutta third order method for [pic]

26. What is a upper triangular matrix? Explain with example.

27. Explain Steps of Milne’s Predictor - corrector method.

28. What is mean of iterative method?

29. Find f(12) by using Newton’s Backward interpolation formula.

|x: |10 |15 |20 |25 |30 |35 |

|f(x): |35.5 |32.4 |29.2 |26.1 |23.2 |20.5 |

30. Use Lagrange’s interpolation formula, to find the value of f(40):

|x: |30 |35 |45 |55 |

|f(x): |148 |96 |68 |34 |

31. Find the missing term of the following data

|x |2 |4 |6 |8 |10 |

|y |5.6 |8.6 |13.9 |- |35.6 |

32. Perform matrix inversion method to solve following system of equations:

[pic]

33. For the following algebraic equation perform four iterations of Bisection method [pic]

34. Perform Gaussian elimination method to solve

35. [pic]

35. Use Euler’s method with step size .2 to find the value of [pic] at [pic] for the following differential equation:

[pic]

Compare the values with the exact solution.

36 Find f(12) by using Newton’s forward interpolation formula.

|x: |10 |15 |20 |25 |30 |35 |

|f(x): |35.5 |32.4 |29.2 |26.1 |23.2 |20.5 |

37. Use Newton’s divided difference formula, to find the value of f(9):

|x: |2 |4 |5 |7 |8 |

|f(x): |3 |43 |138 |778 |1515 |

38. Find the missing term of the following data

|X |0 |1 |2 |3 |4 |

|Y |-5 |-10 |- |4 |35 |

39. Perform matrix inversion method to solve following system of equations:

[pic]

40. For the following algebraic equation perform four iterations of Bisection method

[pic]

41. Perform Gaussian elimination method to solve

[pic]

42. Apply the Milne’s P-C method to find a solution of the differential equation

[pic]

At x= 1.4, Satisfies the following set of values x and y

|X |1 |1.1 |1.2 |1.3 |

|Y |1 |0.996 |0.986 |0.972 |

43.Using Newton’s forward interpolation formula find the value of [pic] at [pic]

|X |2.0 |2.5 |3 |3.5 |4 |

|Y |246.2 |409.3 |537.2 |636.3 |715.9 |

44. .Use Lagrange’s interpolation formula, to find the value of f(40):

|x: |30 |35 |45 |55 |

|f(x): |148 |96 |68 |34 |

45.Evaluate [pic]correct to three decimal places using trapezoidal rule of integration.

46. Perform matrix inversion method to solve following system of equations:

[pic]

47.For the following algebraic equation perform four iterations of Bisection method [pic]

48. Perform Gaussian elimination method to solve

[pic]

49.Use Euler’s method with step size .2 to find the value of [pic] at [pic] for the following differential equation:

[pic]

Compare the values with the exact solution.

50. Compute the value of the following integral by Simpson’s 1/3 formula:

[pic]

GROUP-C

(Long type questions)

1. Compute the value of the following integral by Simpson’s 1/3 formula by taking seven ordinates:

[pic]

2. In the table below, the values of y are consecutive terms of a series of which 23.6 is the 6th term. Find the tenth term of the series:

|X |3 |4 |5 |6 |7 |8 |9 |

|Y |4.8 |8.4 |14.5 |23.6 |36.2 |52.8 |73.9 |

3. Find the root of the equation [pic] correct up to five decimal places using Newton-Raphson method.[use[pic]]

4. Apply Newton’s divided difference method to find [pic] by the following data:

|X |300 |304 |305 |307 |

|Y |2.4771 |2.4829 |2.4843 |2.4871 |

5. Solve the system using LU Decomposition method:

[pic]

6. Solve the boundary value problem [pic]with [pic],by the finite difference method at[pic].

7. Solve the system using Gauss Seidel method(show two iteration):

[pic]

8. Apply the Milne’s P-C method to find a solution of the differential equation [pic]

At x= 1.5, Satisfies the following set of values x and y

|x |1 |1.1 |1.2 |1.3 |

|y |1 |1.233 |1.548 |1.979 |

9. Using Newton’s forward interpolation formula find the value of [pic] at [pic]

|X |2.0 |2.5 |3 |3.5 |4 |

|Y |246.2 |409.3 |537.2 |636.3 |715.9 |

10. Find the root of the equation [pic] correct up to four decimal places using Newton-Raphson method.

11. Solve the following system of equation using Gauss- Elimination method:

[pic]

12. Apply newton’s divided difference method to find [pic] by the following data:

|X |4 |5 |7 |10 |11 |13 |

|Y |48 |100 |294 |900 |1210 |2028 |

pute the value of the following integral by Simpson’s 1/3 formula:

[pic]

14. Solve the following system of equation using Gauss- Seidel method.(Show three iteration only)

[pic]

15. Using fourth order Runge-Kutta method find the numerical solution of

[pic]

at[pic] taking step size [pic].

16. Using Milne’s predictor formula evaluate the integral of y’-4y=0 at x=0.4 given that

|x: |0 |0.1 |0.2 |0.3 |

|y(x): |1 |1.492 |2.226 |3.320 |

Q.17 Solve by Gauss – Seidel method (Show five approximate)

[pic]

18. Solve the following system by the LU factorization method

[pic].

19. Compute the value of [pic]from the following integral by Simpson’s 1/3rd rule and compare it with the value obtained by actual integration. Also calculate the value of percentage error

[pic]

20. Find a positive root of the following equation [pic] using the following instructions:

a) Find three iterations using Regula-Falsi method,

b) Considering the 3rd iteration of Regula-falsi as the initial value, perform two iterations of Newton-Raphson method.

21. Using fourth order Runge-Kutta method find the numerical solution of

[pic]

at [pic] in one step. Compare the value with exact values and calculate relative error.

22. Solve the following boundary value problem by using finite difference method:

[pic] [pic]

23. Solve by Gauss – Seidel method (Show five approximate)

[pic]

24. Solve the following system by the LU factorization method

[pic].

25. Compute the value of [pic]from the following integral by Trapezoidal rule and compare it with the value obtained by actual integration. Also calculate the value of percentage error

[pic]

26 Find a positive root of the following equation using the following instructions:

[pic]

a) Find three iterations using Regula-Falsi method,

b) Considering the 3rd iteration of Regula-falsi as the initial value, perform two iterations of Newton-Raphson method.

27. Using fourth order Runge-Kutta method find the numerical solution of

[pic]

at [pic] in two steps.

28. Solve the following boundary value problem by using finite difference method:

[pic] [pic]

Solve by Gauss – Seidel method (Show five approximate)

[pic]

29.Solve the following system by the LU factorization method

[pic].

30. Compute the value of [pic]from the following integral by Simpson’s 1/3rd rule and compare it with the value obtained by actual integration.

[pic]

31Find a positive root of the following equation[pic] using the following instructions:

a) Find three iterations using Bisection method,

b) Considering the 3rd iteration of Bisection method as the initial value, perform two iterations of Newton-Raphson method.

32.Given [pic][pic]. Compute [pic],[pic] and [pic]by Runge-kutta method of order 4.

33.Apply newton’s divided difference method to find [pic] by the following data:

|X |4 |5 |7 |10 |11 |13 |

|Y |48 |100 |294 |900 |1210 |2028 |

34 Use Euler’s method with step size [pic] to find the value of [pic] at [pic] for the following

differential equation: [pic]

35 Solve the boundary value problem [pic] with [pic], by the

Finite difference method at[pic].

36 Using fourth order Runge-Kutta method find the numerical solution of

[pic]

at [pic] taking step size [pic].

37. Apply the Milne’s P-C method to find a solution of the differential equation

[pic]

At x= 1.5, Satisfies the following set of values x and y

|x |1 |1.1 |1.2 |1.3 |

|y |1 |1.233 |1.548 |1.979 |

38. Solve the system using LU Decomposition method:

[pic]

39. Using Newton’s forward interpolation formula find the value of [pic] at [pic]

|X |2.0 |2.5 |3 |3.5 |4 |

|Y |246.2 |409.3 |537.2 |636.3 |715.9 |

40. Find the root of the equation [pic] correct up to four decimal places using Newton-Raphson method.

41. Solve the following system of equation using Gauss- Seidel method:

[pic]

42. Apply newton’s divided difference method to find [pic] by the following data:

|X |4 |5 |7 |10 |11 |13 |

|Y |48 |100 |294 |900 |1210 |2028 |

43. Solve the following system of equations:

[pic]

by Gauss elimination method.

44. Solve the below system by Gauss elimination method:

[pic]

45. Find, from the following table the area bounded by the curve and the x-axis from [pic] to [pic],

[pic]

[pic]

46. Evaluate [pic]correct to three decimal places and also find the approximate value of [pic]

47. A solid of revolution is formed by rotating about the x-axis the area between the x-axis, the lines [pic] and [pic]and a curve through the points with the following coordinates:

[pic]

48. Apply simpson’s1/3 rule to find the integral [pic]

49. Derive Simpson’s1/3-rule of integration.

50. Find, from the following table the area bounded by the curve and the x-axis from [pic] to [pic],

[pic]

[pic]

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