maths4uem.files.wordpress.com · web viewsum of k1, k2, k3 and k4. 2. the weighted mean of k1,...

UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION 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. 3.14362. 2.14283. 3.14284. 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/22. a-b/23. b-a/24. None of the given choices

Q.10 The Newton-Raphson method fails when 1. f’(x) is negative2. f’(x) is too large3. f’(x) is zero4. Never fails

Q.11 In which of the following methods proper choice of initial value is very important?1. Newton RaphsonMethod2. Bisection Method3. Iterative Method4. RegulaFalsi Method

Q.12 Newton-Raphson method has a _____________ convergence.1. Linear2. Quadratic3. cubic

4. Bi quadraticQ.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 givenx 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] is1. -1.0482. 0.14333. 4.3334. 24.00

Q.15 Lagrange’s interpolation formula is used to compute the values for _______ intervals.1. Equal2. Unequal3. Open4. 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. 162. 63. 1.6 4. 0.19

Q.18 Matrix inversion method fails when

1. ≠ 0

2. = 0

3. 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=1.01092. x1=0.9500, y1=-1.2750, z1=1.10093. x1=1.0500, y1=-1.4575, z1=1.01094. x1=0.8500, y1= 1.0275, z1=1.0109

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

1.

2.

3.

4.

Q.26 In factorization method A=LU provided

1. All the principal minors of A are singular2. All the principal minors of A are non-singular3. Matrix L and U are singular4. Matrix L and U are non-singular

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

1. z=32. z=4

3. z=54. z=2

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

a) 0.0005b) 0.0004c) 0.0040d) 0.4000

Q.29 . The shifting operator is denoted by ________. a) E b) nablac) 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) Trueb) False

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

Select correct option:

a) Analyticalb) Numerical

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

given by

a)

b)

c)

d)

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

a polynomial of degree ---- agreeing with xy at 0, 1,..., rx x x

a) r+2b) r+1c) r

d) r-1

Q.35 In case of Newton Backward Interpolation Formula which equation is correct to find u?

a) (x – xn) h = ub) x – xn = uhc) x – xn = u d) x + xn = uh

Q.36 The following x-y data is givenx 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. -1.0486. 0.14337. 4.3338. 24.00

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

a) 144b) 144hc) 0d) None of the above

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

1) Euler’ s method2) Milne’s method3) RK method4) None of these

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

1) Straight Line2) Circle3) Parabola4) 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 method2) Milne’s method3) Ranga-kutta method4) None of these

Q.44 Single step method is:

1) Euler’s method2) Corrector predictor methods3) RK method4) 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) 12) 23) 34) 4

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

1) x2) 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] is9. -1.04810. 0.143311. 4.33312. 24.00

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

e) 144f) 144hg) 0h) None of the above

Q.50 Value of intrigal using trapezoidal rule of integration with

h = 1/5 will be a) 0.59b) 0.69c) 0.60d) 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 3. Prove 4.Construct the table using newton’s divided difference method.

X 4 5 7 10 11 13Y 48 100 294 900 1210 2028

5. Prove that 6. Using Lagrange’s interpolation formula find , for the following data:

7.Find the missing values in the following table:X 45 50 55 60 65Y 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

11. Write Runge-kutta fourth order method for

12. Find the relative error if is approximated by .

13. Given , Evaluate ,where and ,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 1 2 3 4

1 -1 1 -1

16. Prove that .

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

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

19. Find the inverse of the matrix

20. Perform first step of Regula-Falsi method to find the roots of 21. Find the percentage error in taking = 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

25. Write Runge-Kutta third order method for

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 35f(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 - 35.632. Perform matrix inversion method to solve following system of equations:

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

34. Perform Gaussian elimination method to solve

35.

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

Compare the values with the exact solution.36 Find f(12) by using Newton’s forward interpolation formula.

x: 10 15 20 25 30 35f(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 4Y -5 -10 - 4 35

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

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


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

At x= 1.4, Satisfies the following set of values x and y X 1 1.1 1.2 1.3Y 1 0.996 0.986 0.972

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

X 2.0 2.5 3 3.5 4Y 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 correct to three decimal places using trapezoidal rule of

integration.

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

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


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

Compare the values with the exact solution.50. Compute the value of the following integral by Simpson’s 1/3 formula:

GROUP-C

(Long type questions)

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

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 9Y 4.8 8.4 14.5 23.6 36.2 52.8 73.9

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

4.Apply Newton’s divided difference method to find 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:

6. Solve the boundary value problem with ,by

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


At x= 1.5, Satisfies the following set of values x and y x 1 1.1 1.2 1.3y 1 1.233 1.548 1.979

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

X 2.0 2.5 3 3.5 4Y 246.2 409.3 537.2 636.3 715.9

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

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

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

X 4 5 7 10 11 13Y 48 100 294 900 1210 2028

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

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

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

at taking step size .

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)

18. Solve the following system by the LU factorization method

.

19. Compute the value of 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

20. Find a positive root of the following equation 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.


at 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:

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

24. Solve the following system by the LU factorization method

.

25. Compute the value of from the following integral by Trapezoidal rule and compare

it with the value obtained by actual integration. Also calculate the value of percentage error

26 Find a positive root of the following equation 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



at in two steps.

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

Solve by Gauss – Seidel method (Show five approximate)

29.Solve the following system by the LU factorization method

.

30. Compute the value of from the following integral by Simpson’s 1/3rd rule and

compare it with the value obtained by actual integration.

31Find a positive root of the following equation 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


32.Given . Compute , and by Runge-kutta method

of order 4.

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

X 4 5 7 10 11 13Y 48 100 294 900 1210 2028

34 Use Euler’s method with step size to find the value of at for the following differential equation:

35 Solve the boundary value problem with , by

the Finite difference method at .36 Using fourth order Runge-Kutta method find the numerical solution of

at taking step size .


At x= 1.5, Satisfies the following set of values x and y x 1 1.1 1.2 1.3y 1 1.233 1.548 1.979

38 Solve the system using LU Decomposition method:

39 Using Newton’s forward interpolation formula find the value of at

X 2.0 2.5 3 3.5 4Y 246.2 409.3 537.2 636.3 715.9

40 Find the root of the equation correct up to four decimal places usingNewton-Raphson method.

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

42 Apply newton’s divided difference method to find by the following data:

X 4 5 7 10 11 13Y 48 100 294 900 1210 2028

43 Solve the following system of equations:

by Gauss elimination method.

44 Solve the below system by Gauss elimination method:

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

46 Evaluate correct to three decimal places and also find the approximate

value of 47 A solid of revolution is formed by rotating about the x-axis the area between the x-

axis, the lines and and a curve through the points with the following coordinates:

48 Apply simpson’s1/3 rule to find the integral

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

to ,

maths4uem.files.wordpress.com · web viewsum of k1, k2, k3 and k4. 2. the weighted mean of k1,...

Documents