# NUMERICAL METHOD

MC0074

Statistical and Numerical Methoods using c++

One Marks Question

1). If A=     1   2   3 2     Rank of A is 2 then B= 1 2 1 Rank of B is-

2   3 5   1                                            2 3 3

1   3 4   5                                            3 5 4

2 1 5

a. 1                              b. 2                              c. 3                              d. 4

2) A system that yields a solution is called-

a. Consistent system              b. Inconsistent system             c. Solution system

d. Convergence

3) If is an eigen value of matrix then eigen value of A-1 is-

a. λ                                          b. λn                                         c. 1/ λ

d. 1/ λn

4) The iterative procedure for finding the dominating eigen value of a matrix is known as-

a. Eigen method                     b. Royleigh’s power method

c. Aitkens    2 process             d. None of the above

5) In false position method the equation to the chord joining the 2 points [ x0, f(x0 ] is given by the equation.

a.  y-f(x1) / x-x1 = f (x0)-f(x1) / x0 - x1                              b. y-f(x0) / x-x0  = f(x0)-f(x1) / x0 – x1

c. y-f(x0o) / x-x1 = f(x1)- f(x0) / x1- x0                        d. y-f(x0) / x-x0 = f(x1)-f(x0) / x1-x0

6). Process of computing the value of the function outside the given range is called-

a. Interpolation           b. Outerpolation          c. Extrapolation         d. Smoothing

7). The     is defined as-

a. E+1                          b. (E+a) yo                  c. (E-1) y0                   d. E-1

8). Choose the correct trapezoidal rule for the function F f(x) dx =

a. ½ n [(y0+y5)+2(y1+y+2z+ _ _ _ + y4)]         b. 1/0.2 n [(y0+y2)+2(y1+y2+_ _ _y5)]

c. ½ n [(y0+y6)+ 2(y1+y2+_ _ _y5)]                  d. None of these

9). 4th order Runga Kutta formula is given as y1 –

a. y0+1/4 [k1+2k3+2k4+k2]                               b. y1+1/6 [k1+2k3+2k4+k2

c. y1+1/6 [k1+2k2+2k3+k4]                               d. y0+1/6 [k1+2k2+2k3+k4]

10). Enter formula is given by yn+1=

a. yn+nf (xn.yn)                                                b. yn+nf (x0.y0)

c. yn-1+(n-1) f (xn.yn)                                        d. yn-1+(n+1) f (xn-1.yn-1)

11).   1 0 0   is an

2 4 0

a. Upper triangular matrix                               b. Lower triangular matrix

c. Scalar matrix                                               d. None of these

12) Evaluate f5 f(x) dx gives

 X 1 2 3 4 5 F(x) 13 50 70 80 100

a. 286.6834                 b. 236.5325                 c. 257.6667                 d. 295.5835

13). From the following table of values of x and y obtain dy/dx for x= 1-2

 X 1 1.2 1.4 y 2.72 3.32 4.06

a. 3.2659                     b. 3.7               c. 3.85             d. 3.3462

14) Find the cubic polynomial which takes values y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10

a. x3+2x2-1                   b. x3+2x2+1     c. x3+2x2-2      d. x3-2x2+1

15). By the method of least square, find the straight line that best fits the following data.

 X 1 2 4 5 Y 2 5 7 10

a. 1.8x+0.5                  b. 0.6x+1.8                  c. 1.8x+0.6                  d. 1.3x+0.3

16). Consider the equation f(x)=x3-x-1=0 then using bisection method find x2

a. 1.3645                     b. 1.375                       c. 1.285                       d. 1.3125

17. Use taylor series method to solve the given equation y1 = x2 x2 + y2 for x=0.25 given that y(o) = 1

a. 1.666                       b. 1.4353                     c. 1.333.                      d. 1.2463

18. From the following table, estimate the number of students who obtained marks between 40 and 45.

 Marks 30-40 40-50 50-60 60-70 70-80 No. of students 31 42 51 35 31

a. 47.63                       b. 48.97                       c. 47.87                       d. 46.23

19. Find the root of the equation f(x) = x3 –2x-5=0

a. 2.0945                     b. 3.43206                   c. 4.3254                     d. 2.93636

20. Solve using Cramer’s rules

x+y+z=7

2x+3y+2z=17              then (x, y, z)=

4x+9y+z=37

a. (1, 3, 2)                    b. (1, 2, 3)                    c. (2, 3, 2)                    d. (1, 4, 3)

21. Solve the following system of equations by matrix inversion method

4x1 + x2 + 2x3 = 4

Ox1 + x2 + Ox3 = 4

8x1 + 4x+ 5x3 = 2

then find x1, x2, and x3

a. 7, 4, -14                   b. 4, 3, -12                   c. 8, 6, -8                     d. 9, 3, -6

22. Find the root of the equation correct to 4 decimal place f (x) = x5 + 5x + 1

a. 0.199                       b. –0.1999                   c. 1.1999                     d. –1.1999

23. _______ are the errors caused by using approximate formulas in computations.

a. Internet errors                                  b. Truncation errors

c. Approximation errors                      d. Relative errors

24. If the numbers ‘x’ is rounded to N decimal places then    x=

a. ½ 10-N                      b. 2 10N                        c. 2 10-N                             d. ½ 10N

25. If the number 3.1416 correct to 4 decimal places, then relative accuracy is given by

A.              0.5 x 10-6                                            B.       1.59 x 10-3

C.              1.59 x 10-5                                         D.        None of these

26.

This is…..Matrix.

A.                            Diagonal

B.                             Scalar

C.                             Both A & B

D.                            None of these

27. Choose the unit matrix from below

Sol B

28   If all minors of matrix ‘A; of order greater than or equal to 2 are zero, then rank of A

is

A      0

B      2

C      1

D      >2

29. Matrix A is said to be skew symmetric matrix f.

A      A1 = -A

B      A1 = A

C      (AB)-1 = B-1 A-1

D      AA-1 = A-1A

30. Choose the echelon matrix from below.

C Both A & B

D Name of these

Sol C

31. Which of the following is not represent transpose of A?

A   AT

B   A1

C   A*

D   A-

32. Choose the formula for Runga-Kutta fourth are.

A   Y0 + 1/6 [K1 + 2K2 + 3K3 + K4]

B   Y0 + 1/6 [K1 + 2K2 + 2K3 + K4]

C   Y0 + 1/4 [K1 + 2K2 + 3K3 + 4K4]

D   Y0 + 1/4 [K1 + 2K2 + 2K3 + K4]

33. The power of the highest order of the derivative is called:

A   Degree

B   Cardinality

C   Valency

D   None of these

34.           y2=

A   y1 – y0

B   y2 – y0

C   y2 – y1

D   y1 – y2

35 Round off the number 37.46235 to 4 significant figures and compute Ep.

A   37.5, 6.27 x 10-5

B   37.46, 6.27 x 10-3

C   37.462, 6.27 x 10-3

D  None of these

36. Consider the Matrix A = 1 2 3 2    , the rank of the matrix is

2 3 5 1

1 3 4 5

a. 1                                                b. 4                              c. 3                              d. 2

37. Find the root of the equation f(x)= x3-2x-5=0, correct to 2 decimal place.

A   2.09045

B  2.0934

C   2.0904

D   None of these

38. Solve y1= -y with y(0)=1 for x= 0.04 and step length= 0.01. find y(0.01), using enter method

A   0.53

B   0.86

C   0.99

D   0.01

39. Given dy/dx = (y-x), when y(0) = 2 find y (0,1) correct to 4 decimal places using Runga Kutta second order formula.

A   2.2400

B   2.2050

C   2.4205

D   2.6253

40 .      Evaluate f 71 f(x) dx using the following table.

 X Y 1 2.105 2 2.808 3 3.614 4 4.604 5 5.875 6 7.451 7 9.467

A   30.58

B   30.63

C   30.12

D   29.2

2 Marks Question

1    Find the cubic polynomial which takes the following values y(o) = 1, y (1) = 0,     y(2)=1 and y(3) = 10.

A   x3 + x – 1

B   2x3 + x – 1

C   2x2 – x + 1

D   x3 – 2x2 + 1

2.   By the method of least square, find the straight line that best fits the foll data.

 X Y 1 2 2 5 4 7 5 10 6 12 8 15 9 16

Given Σx2 = 227 & Σxy = 453

A   y = 1.98x + 0.096

B   y = 1.93x + 0.95

C   y = 1.68x + 0.98

D   y = 1.95x + 0.098

3. Use the Newton Raphson’s method to obtain the value of X n -1 where n = 1 for the equation x sin x + cos x = 0

A   2.7984

B   2.8233

C   2.7986

D   2.7953

4. Evaluate ƒ51 f (x) dx, given

 X F(x) 1 13 2 50 3 70 4 80 5 100

A   293.684

B   286.632

C   236.243

D   257.6667

5.   Solve using cramer’s rule

x + y + z = 7

2x + 3y + 2z =17               find (x, y, z)

4x + 9y + z = 37

A   2, 3, 3

B   3, 2, 2

C  3, 2, 3

D   2, 3, 2

6. A = 3 –1   1    What is the value of λ, λ2, λ3 ( using eigen method )

-1   5 -1

1 -1   3

A         3, 2, 5

B         2, 3, 6

C         1, 4, 3

D         2, 5, 3

7. Find X4 or φ (X3) for the equation cos x = 3x-1, using iteration method.

A         0.6093

B         0.6072

C         0.6067

D         None of these

8. Use tay lor series method to solve the equation y1 = x – y2 for x = 0.1 correct to 4 decimal places given that y(0) = 1.

A         0.9138

B         0.9625

C         0.9813

D         None of these

9. By using simpson’s 3/8th rule evaluate ƒ45.2 log xdx with the of this table.

 X Log X 4.0 1.3863 4.2 1.4351 4.4 1.4816 4.6 1.5261 4.8 1.5686 5.0 1.6094 5.2 1.6487

A         1.8278

B         1.9364

C         1.7856

D         None of these

10.       A function y = f(x) is specified by the following table

 X Y 1.0 0.00 1.2 0.128 1.4 0.544 1.6 1.296 1.8 2.432 2.0 4.0

Find the appropriate values of f (1.2)

A         1.63

B         1.32

C         0.63

D         0.68

11. Fit a 2nd degree parabola to the following data.

 X Y 1.0 1.1 1.5 1.3 2.0 1.6 2.5 2.0 3.0 2.7 3.5 3.4 4.0 4.1

Given Σx = 0, Σy = 16.2, Σxy = 14.3, Σx2 = 28, Σx2y = 69.9, Σx3 = 196

A         y = 1.03 + 0.286x-1.26x2

B         y = 0.5 + 0.196x + 1.24x2

C         y = 1.04 – 0.198x – 0.244x2

D         y = 1.02 – 0.196x + 0.25x2

12. The number 2004800 contains………….significant digits.

A         3

B         5

C         7

D         None of these

13. Rank of 2 1 1     is ________.

4 2 2

8 4 4

A         3

B         2

C         1

D         None of these

14. The eigen values of A = 2 1     are ____.

4 5

A         1, -6

B         1, 6

C         2, 3

D         2, 5

15. If B= 3 1    and C_ 1 2    then BC= _______.

2 3                 3 1

D      None of these

Sol D

16. A real root of equation x3 – 4x – 9 = 0 is…..(with 3 iterations)

A         3.7812

B         2.6875

C         2.7812

D         None of these

17. If A= 1 3 2    then λ + λ2 + λ3 = _______

0 2 4

1 3 -3

where λ, λ2 and λ3 are eign values

A         3

B         0

C         8

D         None of these

18.     3 y3

A         yA-3y3 + 3y2 = y1

B         y6 – 3 y5 + 3y4 = y3

C         y3 – 3y2 + 3y2 = yo

D         None of these

19. 1fy = 2x3 – 3x2 – 6x-10 then    3 =

a. 2                              b. 12                            c. –10                          d. None of these

20. Given dy/dx = y-x with y(o) = 2, then y (0.1) = __________ using R.K. method of order 2.

a. 2.2205                     b. 20.2050                   c. 2.4020                     d. None of these

21. The characteristic equation of the matrix A = 1 2 3    is ______

4 1 2

3 1 5

a. λ3 - 3 λ2 + 1=0          b. λ3 - 3 λ2 +4=0          c. λ3 - 4 λ2 + 2 λ+5=0 d. None of these

4Marks Question

1. If 2/3 is approximated by 0.667, find the absolute error.

a. 1/3 X 10-3               b. 1/3 X 10-2                c. 2/3 X 10-3                d. 2/3 X 10-2

2. How is a matrix of order m x n read?

a. m into n                   b. m by n                     c. m mult n                 d. m to n

3. Which matrix is this? 1 0 0

2 6 0

8 1 3

a. Upper triangular matrix                               b.     matrix

c. Scalar matrix                                               d. Lower triangular matrix

4. If 2 matrices of 3x3 and 3x2 order are multiplied, what will be the order of the resultant matrix?

a. 3x3                          b. 3x2                          c. 2x3                         d. 2x2

5. The matrix obtained from any given matrix A, by interchanging rows and columns is called?

a. Transpose matrix b. Exchanged matrix   c. Eigen matrix            d. Reverse matrix

6. Using Newton’s divided difference formula evaluate f(8) given that

 X 4 5 7 10 11 13 F(x) 48 100 294 900 1210 2028

a. 458                          b. 229                          c. 260                          d. 448

7. A different equation together with the initial condition is called__________.

a. Sort value problem                                      b. Immediate value problem

c. Initial value problem                                 d. Multistep method

8. Use Runge-Kutta method of order four to solve dy/dx = 1/x+y, y(0.4) = 1, at x=0.5

a. 0.9662                     b. 7.875                       c. 1.0674                     d. 9.207

9. The normal equations of the straight lines are 7a+35b=70, 35a+227b=453, then a = ______

a. 1.096                       b. 0.096                       c. 1.98                         d. None of these

10.The probability of finding a 53 Sunday in a leap year is

Sol 2/7

Q11 The probability of finding 53 Sunday and Monday in a leap year

Sol 1/7

12. By using simpson’s 3/8th rule evaluate ƒ45.2 log xdx with the of this table.

 X Log X 4.0 1.3863 4.2 1.4351 4.4 1.4816 4.6 1.5261 4.8 1.5686 5.0 1.6094 5.2 1.6487

A         1.8278

B         1.9364

C         1.7856

D         None of these

13.       A function y = f(x) is specified by the following table

 X Y 1.0 0.00 1.2 0.128 1.4 0.544 1.6 1.296 1.8 2.432 2.0 4.0

Find the appropriate values of f (1.2)

A         1.63

B         1.32

C         0.63

D         0.68

14. Fit a 2nd degree parabola to the following data.

 X Y 1.0 1.1 1.5 1.3 2.0 1.6 2.5 2.0 3.0 2.7 3.5 3.4 4.0 4.1

Given Σx = 0, Σy = 16.2, Σxy = 14.3, Σx2 = 28, Σx2y = 69.9, Σx3 = 196

A         y = 1.03 + 0.286x-1.26x2

B         y = 0.5 + 0.196x + 1.24x2

C         y = 1.04 – 0.198x – 0.244x2

D         y = 1.02 – 0.196x + 0.25x2

Q15 The probability of sure and unsure event are ……….&…………

Sol 1&0