Engineering Mathematics - Numerical Methods

Question - 1

Let \({ X }^{ 2 }-117=0.\) The iterative steps for the solution using Newton-Raphson's method is given by 

  • A \({ X }_{ k+1 }=\frac { 1 }{ 2 } \left( { X }_{ k }+\frac { 117 }{ { X }_{ k } } \right) \)
  • B \({ X }_{ k+1 }={ X }_{ k }-\frac { 117 }{ { X }_{ k } } \)
  • C \({ X }_{ k+1 }={ X }_{ k }-\frac { { X }_{ k } }{ 117 } \)
  • D \({ X }_{ k+1 }={ X }_{ k }-\frac { 1 }{ 2 } ({ X }_{ k }+\frac { 117 }{ { X }_{ k } } )\)

Question - 2

Equation \({ e }^{ x }-1=0\)  is required to be solved using Newton's method with an initial guess \({ X }_{ o }=-1\) . Then after one step of Newton;s method, estimate x1 of the solution will be given by

  • A 0.71828
  • B 0.36784
  • C 0.2.587
  • D 0.000

Question - 3

The square root of a number N is to be obtained by applying the Newton-Raphson iteration to the equation \({ X }^{ 2 }-N=0\) . If I denotes the iteration index the correct iterative scheme will be

  • A \({ X }_{ i+1 }=\frac { 1 }{ 2 } \left( { X }_{ i }+\frac { N }{ { X }_{ i } } \right) \)
  • B \({ X }_{ i+1 }=\frac { 1 }{ 2 } \left( { X }^{ 2 }_{ i }+\frac { N }{ { X }^{ 2 }_{ i } } \right) \)
  • C \({ X }_{ i+1 }=\frac { 1 }{ 2 } \left( { X }_{ i }+\frac { N^{ 2 } }{ { X }_{ i } } \right) \)
  • D \({ X }_{ i+1 }=\frac { 1 }{ 2 } \left( { X }_{ i }+\frac { N }{ { X }_{ i } } \right) \)

Question - 4

Gauss-Seidel iterative method can be used for solving a set of

  • A linear differential equation only
  • B linear algebraic equations only
  • C Both linear and non-linear algebraic equations 
  • D Both linear and non-linear differential equations

Question - 5

The convergence of the bisection method is 

  • A cubic
  • B quadratic
  • C linear
  • D None of these

Question - 6

For the differential equation \(\frac { dy }{ dx } =x-{ y }^{ 2 }\) is given that

X 0 0.2 0.4 0.6
Y 0 0.02 0.0795 0.1762

Using predictor-correction method, the y at next value of x is

  • A 0.5114
  • B 0.4648
  • C 0.3046
  • D 0.2498

Question - 7

Which one of the following is correct?

  • A Bisection method is used for iteration
  • B Regula-falsi method is direct method
  • C Secant method is direct method
  • D Newton-Raphson method is not iterative method

Question - 8

if n=3, ao=1, a1=0, a2 = -1, a3 = -11, then the root of the equation between 2 and 3 by regular-falsi method is

  • A 2.0
  • B 2.09
  • C 2.9
  • D 2.2

Question - 9

If \({ e }^{ o }=1,{ e }^{ 1 }=2.72,{ e }^{ 2 }=7.39,{ e }^{ 3 }=20.09\)  and \({ e }^{ 4 }=54.60\) , then by Simpson's \(\frac { 1 }{ 3 } rd\)  rule value of \(\int _{ 0 }^{ 4 }{ { e }^{ x } } dx\) is

  • A 52.78
  • B 53.87
  • C 5.278
  • D 5.387

Question - 10

Match the items in Columns I and II using the codes given below the columns.

          Column I            Column II
(P) Gauss-Seidel method (1) Interpolation
(Q) forward Newton method (2) Non-linear differential equation
(R) Runge-Kutta method (3) Numerical integration
(S) Trapezoidal rule (4) Linear algebraic equations
  • A P Q R S 1 2 3 4
  • B P Q R S 2 3 4 1
  • C P Q R S 3 4 2 1
  • D P Q R S 4 1 2 3
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