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
• 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