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Computer Oriented Numerical Methods KARROX COLLEGE OF TECHNOLOGY & MANAGEMENT

www.karrox.com Helpdesk: 66710528 [email protected]

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PART A

1. The number 9.0098 contains ___ significantdigits

A) 4B) 2C) 5

D) 1

2. The sum of 1.12x102 and 1.20x101 is ___

A) 1.212x102B) 1.24x102C) 2.32x102D) 2.32x101

3. is ___

A) a real numberB) a complex number

C) a whole numberD) an irrational number

4. Relative error is the ___

A) Total errorB) Absolute errorC) Percentage errorD) None of these

5. The number 2.31485 is rounded off to 4significant figure ___

A) 2.3149B) 2.3415C) 2.314D) None of these

6. The relative and percentage errors are ___ ofthe units

A) IndependentB) EqualC) DependentD) None of these

7. Inherent errors can be minimized by taking

___

A) Higher precisionB) Lower precisionC) Less significant digitD) None of these

8. The inverse of A = 1 0 is ___0 1

A) 0 11 0

B) 1 0

0 1

C) 1 11 1

D) None of the above

9. A = 0 0 0 is a matrix of order ___0 0 0

A) 6B) ZeroC) 2x3

D) 3x2

10. If A is a singular matrix of order 3 then detA=___

A) 3B) ZeroC) NonzeroD) None of these

11. If A is a skew-symmetric matrix then At =____

A) AB) -AtC) AD) None of these

12. The transpose of the matrix 1 1 is ___0 2

A) 1 01 2

B) 1 12 0

C) 0 21 1

D) None of these

13. Rank of B = 2 4 6 8


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1 2 3 4A) 2x4B) 4x2C) 1D) 2

14. If I3 is the unit matrix of order 3 then rankof I3 = ___

A) 1B) I3C) 3D) None of these

15. The inverse of AB is ___

A) 1/ABB) A-1B-1C) B-1A-1D) None of these

16. If 1/3 is the eigen value of A then 3 is the

eigen value of ___

A) AtB) A-1C) A3D) None of these

17. If 2 3 = 2 30 x+1 0 0

A) 0B) 1C)-1D) None of these

18. The sum of the eigen values of 2 0 is___ 1 2

A) 2B) 5C) 4D) 0

19. If there are 3 tabulated values of x and ythen we can construct a unique polynomial ofdegree

A) 3B) 2C) 1D) 4

20. 2y2 = ___

A) y2 + 2y1 + y0B) y2 - 2y1 + y0C) y4 - 2y3 + y2D) y4 + 2y3 + y2

21. R.K. method is a ___ method

A) ImplicitB) Single step

C) Multi stepD) None of these

22. The determinant of 1 3 is ______0 4

A) 0B) 1C) 3D) 4

23. ehD = ___

A) B) DhC) D) E

24. If y is a polynomial of degree 5 then 5y =____

A) 120B) 5C) 0D) None of these

25. yr= ___

A) yr - yr-1/2B) yr + yr-1/2C) yr+1 - yr-1/2D) None of these

26. Iteration method xn+1=(xn) converges if__ for all x in the interval [a, b]

A) |(x)|1D) None of these

27. E= ___

A) 1+


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B) -1C) 1-D) None of these

28. If x+y=1 then its solutions are ___

A) UniqueB) No solutionC) Infinite

D) None of these

29. If y1=20 and y0=19 then y0=___

A) 39B) -1C) 1D) None of these

30. Gauss elimination is ___method

A) matrix inverseB) an iteration

C) DirectD) none of these

31. Modified Eulers method is ___ method

A) Single stepB) Multi stepC) ExplicitD) None of these

32. The error in Trapezoidal rule is ___

A) b-a h2y()12

B) -b-a hn2y()12

C) -1 uh3y()2

D) None of these

33. The backward difference of f(x)is ___

A) f(x+h)-f(x)B) f(x)-f(x-h)C) f(x)-f(x+h)D) None of these

35. If A= 1 2 B= 3 1 then AB=___

0 1 0 2A) 2 1

3 1

B) 0 21 0

C) 0 00 0

D) None of these

36. ex=___

A) (e-1) exB) (e+1) exC) exD) None of these

37. Truncation error in R.K method of order 4 is___

A) 0(h5)B) 0(h4)C) 0(h6)D) None of these

38. Newton Raphson method is ___ order ofconvergenceA) FirstB) SecondC) ThirdD) None of these

39.

=+

1

02x1

dx

A) B) /4C) /4D) None of these

40. The analytical solution of dy = y isdx

A) y = cexB) y = ex+1C) y = ecx

D) None of these

41. The number 0.653978 rounded off to 4significant figures is

A) 0.6539B) 0.6540


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C) 0.653D) None of these

42. The errors caused by using approximateformula in computations are calledA) Truncation ErrorsB) Inherent errorsC) Round off errorsD) None of these

43. The relative error ER is given by

A) X X1X

B) XX X1

C) X X 1

D) X

44. The percentage error EP caused in roundingoff the number 865250 to 4 significant figuresis

A) 6.71 X 10-3B) 8.62 X 10-3C) 5.778 X 10-3D) None of these

45. The relative error if 2/3 is approximated to0;667 is

A) 3.333 X 104B) -3.333 X 104C) 3.333 X 10-4D) -3.333 X 10-4

46. The sum of all diagonal elements of amatrix is called

A) Trace of the matrixB) Determinant of the matrixC) Transpose of the matrixD) None of these

47. A diagonal matrix whose all the leadingdiagonal elements are equal is called a

A) Square matrixB) Inverse matrixC) Scalar matrix

D) None of these

48. Rank of the matrix A = 1 7 -1 is3 0 4


49. If is an eigen value of a matrix A, then1/ is an eigen value of

A) A-1B) ATC) AD) None of these

50. The eigen values of the matrix A = 2 1are 4 5

A) -1, 6

B) 1,-6C) 1, -6D) 1, 6

51. Cayles Hamilton equation for the matrixA = 1 4

3 2

A) A2 + 3A 10 I = 0B) A2 - 3A 10 I = 0C) A2 + 3A + 10 I = 0D) None of this above

52. A root of f(x) = 0 lies between x = a and x= b if

A) f(a) f(b) 0B) f(a) f(b) > 0C) f(a) f(b) 1B) 11(n)1|1D) None of these


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55. The Aitkens 2 - formula for the root isgiven by =

A) XI+1 (Xi)22 Xi 1

B) X1 XI Xi2 Xi 1

C) (Xi)22 Xi 1

D) None of these

56. The formula for the method of false positionis given by X2 =

A) X0 f(x0)f(x1) f(x0)

B) X0 f(x0) (X1 X0)f(x1) f(x0)

C) X0 + f(x0)f(x1) f(x0)

D) None of these

57. Newton = Raphson iteration formula isgiven byXN+1

A) Xn f1(Xn)f(Xn)

B) Xn + f1(Xn)f(Xn)

C) Xn f(Xn)f1(Xn)

D) None of these

58. Generalized Newtons formula for finding arootof multiplicity p is Xn + 1

A) Xn P f(Xn)f1(Xn)

B) Xn + P f(Xn)f1(Xn)

C) Xn f1(Xn)f1(Xn)

D) None of these

59. Given a set of values of y=f(x) for a set ofvaluesof x, the process of finding the values of y foranintermediate value of x is called

A) DifferentiationB) Integration

C) InterpolationD) Intrapolation

60. If Y0, Y1 , Y2 ....... Yn denote a set ofvalues of Y, then 3 Y0 = ---------

A) y3 + 3y2 + 3y1 + y0B) y3 - 3y2 + 3y1 - y0C) y3 - 3y2 - 3y1 - y0

D) None of these

61. If Y0, Y1 , Y2 ....... Yn denote a set ofvalues of Y, then Yn -1

A) y 2n-12

B) y 2n+12

C) y n-12

D) None of these

62. If and E denote the forward differenceand shift operators respectively, then

A) E = 1 - B) E = 1 + C) = 1 + ED) None of these

63. The first difference of a polynomial ofdegree 10 is again a polynomial of degree

A) 11B) Not a polynomialC) 9D) None of these

64. In the Newtons backward differenceformula X and Xn are related by the equationA) x = xn - phB) x = xn + phC) x = xn + hD) None of these

65. The process of finding an equation of bestfit is called

A) Curve fittingB) InterpolationC) Least squaresD) None of these


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66. The operator E and D are related by theequation

A) E = eDB) E = e-DC) E = e-HdD) E = E = ehD

67. dy |

dx x=x0A) y0 + 12y0 + 13y0+

2 3B) 1[y0 +1 2y0 + 1 3y0+. ]

h 2 3C) 1[y0 1 2y0 + 1 3y0-. ]

h 2 3

D) None of these

xn68. Trapezoidal rule is given by ydx = ..

x0

A) h[(y0+yn)+2(y1+y2+.+yn-1)]2

B) h[y1+y2+.+yn]

C) h[(y0+yn)+4(y1+y2+.+yn-1)]2

D) None of these

69. If y11 (x) denotes the maximum value ofthe second derivative, then the error involvedin the trapezoidal rule is E

A) (b-a)2 h2 Y11 (x)12

B) (b-a)2 h2 Y11 (x)6

C) (b-a)2 Y11 (x)12

D) None of these

70. Simpsons 1rd rule is applicable only whenthe number of subintervals n is

3A) An even numberB) An odd numberC) PrimeD) None of these

xn71. Simpsons 3th rule is given by ydx =

x0

A) 3h [(y0 + yn) + 3(y1 + y2 +..+ yn-1) + 2(y3 + y6 +.yn-3)]

8B) 3 [(y0 + yn) + 3(y1 + y2 +..+ yn-1)]

C) 3h [(y0 + yn) + 4(y1 + y2 +..+ yn-1) + 3(y3 + y6 +.yn-3)]

8D) None of these

72. If two different subintervals of width h andh are used in trapezoidal rule and if E and E arethe corresponding errors involved, then

E1 h1A) =

E2 h2

E1 h12B) =

E2 h22

E1 h22C) =

E2 h12

D) None of these

73. To obtain the value of y at some point,correct to 3 decimal places, using Taylorsseries method, the highest order derivative tobe considered is

A) 1B) 2C) 3D) 4

74. Eulers formula to solve dy = f (x, y), y(x0)= 0 is given by dx

A) y1 = y0 + f(x0, y0)B) y1 = y0 - hf(x0, y0)C) y1 = y0 + hf(x0, y0)D) None of these

75. If y1 = x2+y, y(0) = 1, then f(x0, y0) =.


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A) 0B) 1C) 2D) 4

76. In second order Runge - Kutta method thevalue of k2 is given by

A) f(x0+h, y0+k1)B) hf(x0+h, y0+k1)C) f(x0+h, y0+k1)D) None of these

77. Fourth order Runge - Kutta method is givenby y1 = ----

A) 1 [k1+2k2+2k3+k4]6

B) y0 + [k1+2k2+2k3+k4]C) y0 + 1 [k1+2k2+2k3+k4]

6

D) None of these

78. Given dy = y-x, y(0) = 2, then value of k1is dx


1 2

79. The transpose of the matrix A = 3 4is 5 6

A) 2 4 61 3 5

2 1B) 4 3

6 5

C) 1 3 5

2 4 6

D) None of these

80. If In is the unit matrix of order n, then therank of In is

A) 0B) nC) niD) None of these

PART B

4 1 2

81. Inverse of the matrix A = 0 1 0 is8 4 8


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5/4 -3/4 -1/2A) 0 1 0

2 2 -1

5/4 3/4 -1/2B) 0 1 0

-2 -2 1

5/4 3/4 -1/2C) 0 1 0

-2 -2 1

D) None of these

3 1 482. Rank of the matrix A = 0 5 8

-3 4 4

A) 3B) 1

C) 2D) None of these

3 -1 183. The characteristic equation of A = -1 5 1is 1 -1 3

A) 3 - 11 2 + 36 - 36 = 0B) 3 + 11 2 + 36 + 36 = 0C) 3 - 11 2 - 36 - 36 = 0D) None of these

2 -1 084. The largest eigen value of A = -1 2 -1

obtained using power method by taking [1, 0,0]T as the initial vector is

A) 2.5B) 3.6C) 9.8D) None of the these

85. A real root of equation cosx = 3x 1correct to 3 decimal places is


86. A real root of the equation x3 x 1 = 0


87. Using Regula falsi method, a root ofx3 - 2x - 5 = 0 0 between 2 and 3 is


88. Multiplicity of the root of x3 x2 - x + 1 =0 near x0 = 0.8 is


89. A second degree polynomial which takesthe following values

X 0 1 2

y 1 2 1

Is

A) 1 + 2x x2B) 1 - 2x x2C) 1 + 2x x2D) None of these

90. The population of a town in the decennialcensus was given below

Year x 1891 1901 1911 1921

Population y(In 1000s)

46 66 81 93

The population of the year 1895 is

A) 59.456B) 53.528


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91. Newtons forward difference formula isgiven by yn(x) = ..

A) y0 py0 p(p+1) 2y0 + ..

2!

B) y0 + py0 + p(p+1) 2y0 + ..2!

C) y0 py0 p(p-1) 2y0 + ..

2!D) None of these

92. By the method of least squares the straightline that best fits the following data

X 1 2 4 5

y 2 5 7 10

Is given by

A) y = 0.6 + 1.8xB) y = 0.5 + 9xC) y = 1.1 + 1.8xD) None of these

193. Approximate value of dx

0 1+ xusing Simpsons 3th rule (h = 1) is ..

8 6

A) 0.850C) 0.6930

B) 0.7826D) None of these

94. Given the table

X 1.0 1.2 1.4 1.6

y 2.72 3.32 4.06 4.96

Value of dy at x = 1.0 isdx


95. Given

x 0.1 0.2 0.3 0.4

f(x) 1.10517 1.22140 1.34986 1.49182


96. Given the set of values

x 1 2 3 4 5

Y 2.105 2.808 3.614 4.604 5.857

5Value of y dx, using Trapezoidal rule is

1A) 15.007

B) 16.568C) 14.239D) None of these

98. Given y1 = x2 + y2, y (0) =1, Then theTaylor series for y(x) is

A) 1 + x + x2 + B) 1 x x2 - ..C) 1 x + x2 - ..D) None of these

99. Given dy = 1 + y2, y (0) = 0. Value of ydx

(0.2) using Runge-Kutta method of order 2 is


100. Given dy = y - x, y(0) = 2. Using Runge dx

Kutta methods of order 2, value of y (0,1)

A) 2.2050B) 3.5060B) 3.5060


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101. If A= 1 0 B= 1 1 then AB = ___2 1 1 1

A) 3 02 1

B) 2 13 2

C) 2 12 2

D) None of these

102. The determinant of [] is ___

A) 2B) 4C) 0

D) None of these

103. The normal equations of the best fit straight line for the following data are ___

X 1 2 4 5 6 8 9

Y 2 5 7 10 12 15 19

A) 7a+35b=70, 35a+227b=435B) 7a+35b=70, 35a+227b=453C) 7a+35b=62, 35a+227b=435D) None of these

104. The interpolation polynomial for the data y(0)=-1, y(1)=0, y(2) =3 is ___

A) x2-1B) x2+1C) x2-4D) None of these

105. The largest eigen value of the matrixA= 4 2 0

0 1 2 in 5th iteration is ___1 2 0

A) 81B)C)

D)

106. The value of x and y in the system of equations 2x+5y+z=-1,x+7y-6z=-18 and 3y+6z=9are ___

A) (1,2)B) (1,-1)C) (-1,2)D) None of these


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107. A real root of the equation xsinx+cosx=0 by Newton Raphison method at x0=. Carryout 2iterations

A) 2.7986B) 2.9876C) 2.3102D) 2.8698

108. The characteristic equation of the matrix8 -8 -24 -3 -2 is ___3 -4 -1

A) 3 -62 +11+6=0B) 3 -62 +11-6=0C) 3 +62 +11+6=0D) None of these

109. A real root of the equation x3-4x-9=0 using the bisection method in 4 iteration is ___

A) 2.7561

B) 2.6875C) 207508D) None of these

110. A real root of the equation cosx=3x-1 using Aitkens 2 method is ___


111. If y(0) =1, y(2)=1 and y(3)=10 than y(4)= ____


112. The value of +

1

0x1

dx

using trapezoidal rule with n=5 is ___

A) log 2B) 0.6231C) 0.6992D) None of these

113. The solution of dy/dx= -y with y(0)=1 for x= 0.02 using Taylors series is ___



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114. The normal equations of the best fit straight line for the following data are ___

X 0 1 2 3Y 1 4 -1 2

A) 3a-b=2 ha+b=5B) a+b=0 2a-b=3C) 2a-3b=2 2a+3b=1D) None of these

115. The value of1

0

xcos

dx using trapezoidal rule by taking h=0.2 is ___


116. The value of x in the following system of equation is ___x+y+z=32x+3y+2z=6y+3z=0

A) x=3B) x=6C) x=1D) None of these

117. Find the missing term in the following table

X 0 1 2 3 4 5

Y 1 2 4 8 16 -

A) 31.9B) 32.8C) 32

D) None of these

118. Given the set of values

X 10 15 20 25 30

Y 22 31 28 32 38


119. Given dy/dx =y-x where y(0) =2. Find y(0.2) with h=0.2 using R.K. method of order two


120. The eigen vectors of the matrix 2 3 are ___


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4 5A) (2,3) (1,1)B) (1,2) (3,1)C) (1,0) (2,1)D) None of these

121. The inverse of the matrix 2 1 3 is _____4 5 6

1 3/2A) 0 -

0 -1 2

0 1 2B) 3 0 1

- 0 1

1/5 1 2C) 3 1 2

-1 0 1

D) None of these3 -1 1

122. The eigen values of the matrix -1 5 -11 -1 3

A) 3, 6, 2B) 2, 6, 1C) 2, 3, 6D) None of these

123. The solution of the system of equations 2x+5y+3z=31, 5x+3y+2z=4, 3x+2y+5z=32(using Crammers rule) is y= ___


1 2

1 2124. The value of e-x dx using Simpsons 1/3rd rule with n=6 is ___

0A) 0.3125B) 0.6135C) 0.5135D) None of these

125. Given dy/dx =1+y2, y(0) = 0 with h=0.2, find y(0, 4) using R.K method of order four


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A) 2.1356B) 0.4228C) 0.2135D) 0.2248

8 -8 -2126. If A= 4 -3 -2 then A3-6A2+11A = _____

3 -4 -1

1 0 2A) 2 1 0

3 1 8

8 0 0B) 0 8 0

0 0 8

4 0 2C) 2 1 0

3 1 2

5 0 1D) 3 2 8

1 1 3

127. The real root of the equation Sin2x = x2 -1 is ___ (use Newton Raphson method) carryout 3iterations

A) 1.0404B) 1.4040C) 1.6010D) 2.4012

128. The values of a, b and c so that y=a+bx+cx2 is the best fit to the data

X 1 2 4 5 7 8

Y 1 0 3 10 5 21

A) 1,-3, 2B) 1, 3, 2C) -1, 3, -2D) None of these

129. The value of dy/dx when x=0 from the following table isX 0 1 2 3 4

Y 3 2 -1 4 6



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130. By the method of least squares the best fit straight line for the following data is y=___

X 0 1 3 4 5 6 8Y 2 4 2 8 6 8 12

A) 0.096x+2.91B) 1.98x+2.98C) 0.21x+1.16D) None of these

8 -8 0131. If A= 3 1 2 then A3-6A2+2A = _____

4 5 6

1 0 -2A) 1 2 3

4 1 0

8 0 0B) 0 8 0

0 0 8

26 16 14

C) 3 5 64 3 8

D) None of these

132. The smallest root of the equation

1 x + x2 x3 + x4 + . = 0(2!)2 (3!)2 (4!)2

(Carry out 4 iterations)

A) 1.0490


133. The following table gives values of sin 0 10 20 30 40

sin 0.00 0.1736 0.3420 0.5000 0.6428

Then cos100= ___

A) 0.8498B) 0.9848C) 0.9941


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D) None of these

/2134. The value of e-sin d using Simpsons 3/8th rule is ___

0A) 4.1017B) 3.1017C) 2.1017D) 1.1017

135. The value of y and z are in the system of equations

2x-y+2z+t=4x-y+z+2t=32y+2z-3t=14x+2y-4z+t=3Use Gauss elimination method

A) (1,1)B) (2,2)C) (3,1)D) (1,2)

136. The largest eigen value and the corresponding eigen value of

4 1 0A = 1 20 0 with (0, 1, 0)T as the initial vector, are

0 0 4

A) 21.07, (0.06, 1, 0)TB) 28.17, (0.06, 1, 0)TC) 20.06, (0.06, 1 0)TD) None of these

137. Solution of the system x+ y+ z = 7, 2x + 3y+ 2z = 17, 4x+9y+z = 37 is

A) 2,-3, 2B) 2, 3, 2C) -2, 3, -2D) 2, -3, -2

2 0 1138. Eigen values of the matrix A = 0 3 0 are

1 0 2A) -1,-3B) -1, 3C) 1, 3D) None of these

139. A real root of x3 - x - 4 = 0 correct to 4 decimal places using bisection method is


140. A root correct to 3 decimal places by method of false position of the equation x3 + x2 + x + 7 =0 isA) -2.1049


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141. Using iteration method a real root of 1 = x is given by(1+x)2

A) 0.4839B) 0.4736C) 0.4655

D) None of these

142. A cubic polynomial which takes the values y (0)1, y(1)=0, y(2)= 1, y(3) = 10 is

A) x3 + 2x2 + 1B) x3 2x2 - 1C) x3 2x2 + 1D) None of these

143. A line of best fit for the following table

X 0 1 2 3

Y 0.09 0.63 2.15 4.58

A) y = 0.386+l.499xB) y = -0.386 + 1.499xC) y = 1.499+0.386xD) None of these

144. A function y = f(x) is represented by following table

X 1.0 1.2 1.4 1.6

Y 0.00 0.128 0.544 1.296

Approximate value of dy at x = 1.1 isdx


145. Given the table

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

The number of students who obtained marks between 40 and 45



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1146. Numerical value of e-x dx, by applying Trapezoidal rule (h = 10), is equal to

0A) 0.6896B) 0.6023C) 0.6326D) None of these

147. Given dy = 1 + y2, y (0) = 0. Value of y (0.2) with h = 0.2 using Runge - Kutta method of order

d^4 is


148. Given

X 0 1 2 3 4 5 6

Y 0.146 0.161 0.176 0.190 0.204 0.217 0.230

6

Value of y dx, using Simpsons 3th Rule is0 8


149. Given dy = y - x, y (0) = 1. Value of y at x = 0.1 with h = 0.1, using 4 order Runge - Kuttadx

method is


150. Given y1 = x y2, y (0) = 1. Using Taylor series method value of y at x = 0.1 correct to 4decimal places is


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