Numerical Methods MCQs Question and Answer in Engineering Maths Page-2 section-1

11.
Newton-Raphson method is used to compute a root of the equation x² - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

A. 3.575

B. 3.677

C. 3.667

D. 3.607

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12.
Identify the Newton-Raphson iteration scheme for finding the square root of 2.

A. $${{\text{x}}_{{\text{n}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{n}}} + \frac{2}{{{{\text{x}}_{\text{n}}}}}} \right)$$

B. $${{\text{x}}_{{\text{n}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{n}}} - \frac{2}{{{{\text{x}}_{\text{n}}}}}} \right)$$

C. $${{\text{x}}_{{\text{n}} + 1}} = \frac{2}{{{{\text{x}}_{\text{n}}}}}$$

D. $${{\text{x}}_{{\text{n}} + 1}} = \sqrt {2 + {{\text{x}}_{\text{n}}}} $$

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13.
If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

A. f(a) $$ \cdot $$ f(b) = 0

B. f(a) $$ \cdot $$ f(b) < 0

C. f(a) $$ \cdot $$ f(b) > 0

D. $$\frac{{{\text{f}}\left( {\text{a}} \right)}}{{{\text{f}}\left( {\text{b}} \right)}} \leqslant 0$$

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14.
Consider p(s) = s³ + a₂s² + a₁s + a₀ with all real coefficients. It is known that it is derivative p'(s) has no real roots. The number of real roots of p(s) is

A. 0

B. 1

C. 2

D. 3

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15.
The differential equation $$\frac{{{\text{dx}}}}{{{\text{dt}}}} = \left[ {\frac{{1 - {\text{x}}}}{\tau }} \right]$$ is discretised using Euler's numerical integration method with a time step ΔT > 0. What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete time equation?

A. 1

B. $$\frac{\tau }{2}$$

C. $$\tau $$

D. $$2\tau $$

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16.
Match the correct pairs

Numerical Integration Scheme Order of Fitting Polynomial

P. Simpson's 3/8 Rule 1. First

Q. Trapezoidal Rule 2. Second

R. Simpson's 1/3 Rule 3. Third

A. P-2, Q-1, R-3

B. P-3, Q-2, R-1

C. P-1, Q-2, R-3

D. P-3, Q-1, R-2

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17.
The minimum number of equal length subintervals needed to approximate $$\int\limits_1^2 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx}}} $$ to an accuracy of at least $$\frac{1}{3} \times {10^{ - 6}}$$ using the trapezoidal rule is

A. 1000e

B. 1000

C. 100e

D. 100

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18.
Consider the series $${{\text{x}}_{{\text{n}} + 1}} = \frac{{{{\text{x}}_{\text{n}}}}}{2} + \frac{9}{{8{{\text{x}}_{\text{n}}}}},\,{{\text{x}}_0} = 0.5$$ obtained from the Newton-Raphson method. The series converges to

A. 1.5

B. √2

C. 1.6

D. 1.4

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19.
The value of the function f(x) is given at n distinct values of x and its value is to be interpolated at the point x^⋆, using all the n points. The estimate is obtained first by the Lagrange polynomial, denoted by $${I_{\text{L}}}$$ and then by the Newton polynomial, denoted by $${I_{\text{N}}}$$. Which one of the following statements is correct?

A. $${I_{\text{L}}}$$ is always greater than $${I_{\text{N}}}$$

B. No definite relation exists between $${I_{\text{L}}}$$ and $${I_{\text{N}}}$$

C. $${I_{\text{L}}}$$ and $${I_{\text{N}}}$$ are always equal

D. $${I_{\text{L}}}$$ is always less than $${I_{\text{N}}}$$

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20.
The extremum (minimum or maximum) point of a function f(x) is to be determined by solving $$\frac{{{\text{df}}\left( {\text{x}} \right)}}{{{\text{dx}}}} = 0$$ using the Newton-Raphson method. Let f(x) = x³ - 6x and x₀ = 1 be the initial guess of x. The value of x after two iterations (x₂) is

A. 0.0141

B. 1.4142

C. 1.4167

D. 1.5000

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Numerical Integration Scheme	Order of Fitting Polynomial
P. Simpson's 3/8 Rule	1. First
Q. Trapezoidal Rule	2. Second
R. Simpson's 1/3 Rule	3. Third

Numerical Methods MCQs Question and Answer in Engineering Maths

11. Newton-Raphson method is used to compute a root of the equation x2 - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

Answer & Solution

12. Identify the Newton-Raphson iteration scheme for finding the square root of 2.

Answer & Solution

13. If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

Answer & Solution

14. Consider p(s) = s3 + a2s2 + a1s + a0 with all real coefficients. It is known that it is derivative p'(s) has no real roots. The number of real roots of p(s) is

Answer & Solution

Answer & Solution

16. Match the correct pairs Numerical Integration Scheme Order of Fitting Polynomial P. Simpson's 3/8 Rule 1. First Q. Trapezoidal Rule 2. Second R. Simpson's 1/3 Rule 3. Third

Answer & Solution

17. The minimum number of equal length subintervals needed to approximate $$\int\limits_1^2 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx}}} $$ to an accuracy of at least $$\frac{1}{3} \times {10^{ - 6}}$$ using the trapezoidal rule is

Answer & Solution

18. Consider the series $${{\text{x}}_{{\text{n}} + 1}} = \frac{{{{\text{x}}_{\text{n}}}}}{2} + \frac{9}{{8{{\text{x}}_{\text{n}}}}},\,{{\text{x}}_0} = 0.5$$ obtained from the Newton-Raphson method. The series converges to

Answer & Solution

Answer & Solution

20. The extremum (minimum or maximum) point of a function f(x) is to be determined by solving $$\frac{{{\text{df}}\left( {\text{x}} \right)}}{{{\text{dx}}}} = 0$$ using the Newton-Raphson method. Let f(x) = x3 - 6x and x0 = 1 be the initial guess of x. The value of x after two iterations (x2) is

Answer & Solution

11.
Newton-Raphson method is used to compute a root of the equation x² - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is

12.
Identify the Newton-Raphson iteration scheme for finding the square root of 2.

13.
If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?

14.
Consider p(s) = s³ + a₂s² + a₁s + a₀ with all real coefficients. It is known that it is derivative p'(s) has no real roots. The number of real roots of p(s) is

16.
Match the correct pairs

Numerical Integration Scheme Order of Fitting Polynomial

P. Simpson's 3/8 Rule 1. First

Q. Trapezoidal Rule 2. Second

R. Simpson's 1/3 Rule 3. Third

17.
The minimum number of equal length subintervals needed to approximate $$\int\limits_1^2 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx}}} $$ to an accuracy of at least $$\frac{1}{3} \times {10^{ - 6}}$$ using the trapezoidal rule is

18.
Consider the series $${{\text{x}}_{{\text{n}} + 1}} = \frac{{{{\text{x}}_{\text{n}}}}}{2} + \frac{9}{{8{{\text{x}}_{\text{n}}}}},\,{{\text{x}}_0} = 0.5$$ obtained from the Newton-Raphson method. The series converges to