Numerical Methods — Part 1 MCQs with Explanations

11.365 is the correct answer to “What is the cube root of 1468 to 3 decimal places”. This question is standard in competitive examination syllabi for Numerical Methods.

993 is the correct answer to “Torque exerted on a flywheel over a cycle is listed in the table. Flywheel energy (in J per unit cycle) using Simpson's rule is Angle (degree) 0 60 120 180 240 300 360 Torque (Nm) 0 1066 -323 0 323 -355 0”. This question is standard in competitive examination syllabi for Numerical Methods.

(-1, +j. -j) is the correct answer to “Roots of the algebraic equation x 3 + x 2 + x + 1 = 0 are”. This question is standard in competitive examination syllabi for Numerical Methods.

Only I is the correct answer to “In the Newton-Raphson method, an initial guess of x 0 = 2 is made and the sequence x 0 , x 1 , x 2 ... is obtained for the function 0.75x 3 - 2x 2 - 2x + 4 = 0 Consider the statements I. x 3 = 0. II. The method converges to a solution in a finite number of iterations. Which of the following is TRUE”. This question is standard in competitive examination syllabi for Numerical Methods.

0.96 is the correct answer to “While numerically solving the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{x}}{{\text{y}}^2} = 0,\,{\text{y}}\left( 0 \right) = 1$$ using Euler's predictor-corrector (improved Euler-Cauchy) with a step size of 0.2, the value of y after the first step is”. This question is standard in competitive examination syllabi for Numerical Methods.

square root of R is the correct answer to “The Newton-Raphson iteration $${{\text{x}}_{{\text{n}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{n}}} + \frac{{\text{R}}}{{{{\text{x}}_{\text{n}}}}}} \right)$$ can be used to compute the”. This question is standard in competitive examination syllabi for Numerical Methods.

0.705 is the correct answer to “When the Newton-Raphson method is applied to solve the equation f(x) = x 3 + 2x - 1 = 0, the solution at the end of the first iteration with the initial guess value as x 0 = 1.2 is”. This question is standard in competitive examination syllabi for Numerical Methods.

\[\left[ {\begin{array}{*{20}{c}} 2&0 \\ 4&{ - 3} \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}} 1&{0.5} \\ 0&1 \end{array}} \right]\] is the correct answer to “The matrix \[\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}} 2&1 \\ 4&{ - 1} \end{array}} \right]\] is decomposed into a product of a lower triangular matrix x[L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are”. This question is standard in competitive examination syllabi for Numerical Methods.

0.031 is the correct answer to “Consider a differential equation $$\frac{{{\text{dy}}\left( {\text{x}} \right)}}{{{\text{dx}}}} - {\text{y}}\left( {\text{x}} \right) = {\text{x}}$$ with the initial condition y(0) = 0. Using Euler's first order method with a step size of 0.1, the value of y(0.3) is”. This question is standard in competitive examination syllabi for Numerical Methods.

0.88 is the correct answer to “Consider an ordinary differential equation $$\frac{{{\text{dx}}}}{{{\text{dt}}}} = 4{\text{t}} + 4.$$ If x = x 0 at t = 0, the increment in x calculated using Runge-Kutta fourth order multistep method with a step size of Δt = 0.2 is”. This question is standard in competitive examination syllabi for Numerical Methods.

3.607 is the correct answer to “Newton-Raphson method is used to compute a root of the equation x 2 - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{n}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{n}}} + \frac{2}{{{{\text{x}}_{\text{n}}}}}} \right)$$ is the correct answer to “Identify the Newton-Raphson iteration scheme for finding the square root of 2.”. This question is standard in competitive examination syllabi for Numerical Methods.

f(a) $$ \cdot $$ f(b) > 0 is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

1 is the correct answer to “Consider p(s) = s 3 + a 2 s 2 + a 1 s + a 0 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”. This question is standard in competitive examination syllabi for Numerical Methods.

$$2\tau $$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

P-3, Q-1, R-2 is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

1000e is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

1.5 is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

$${I_{\text{L}}}$$ and $${I_{\text{N}}}$$ are always equal is the correct answer to “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”. This question is standard in competitive examination syllabi for Numerical Methods.

1.4167 is the correct answer to “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) = x 3 - 6x and x 0 = 1 be the initial guess of x. The value of x after two iterations (x 2 ) is”. This question is standard in competitive examination syllabi for Numerical Methods.

O(h4) is the correct answer to “The accuracy of Simpson's rule quadrature for a step size h is”. This question is standard in competitive examination syllabi for Numerical Methods.

0.7854 is the correct answer to “The table below gives values of a function F(x) obtained for values of x at intervals of 0.25. x 0 0.25 0.5 0.75 1.0 F(x) 1 0.9412 0.8 0.64 0.50 The value of the integral of the function between the limits 0 to 1 using Simpson's rule is”. This question is standard in competitive examination syllabi for Numerical Methods.

1.694 is the correct answer to “A numerical solution of the equation f(x) = x + √x - 3 = 0 can be obtained using Newton-Raphson method. If the starting value is x = 2 for the iteration, the value of X that is to be used in the next step is”. This question is standard in competitive examination syllabi for Numerical Methods.

1.901 is the correct answer to “To solve the equation 2sin x = x, by Newton Raphson method, the initial guess value is chosen to be x = 2. Consider x in radius only. The value of x (in radius) obtained after one iteration will be closed to”. This question is standard in competitive examination syllabi for Numerical Methods.

$$ - \frac{4}{3}$$ is the correct answer to “A 2 nd degree polynomial, f(x) has values of 1, 4 and 15 at x = 0, 1 and 2, respectively. The integral $$\int\limits_0^2 {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$ is to be estimated by applying the trapezoidal rule to this data. What is the error (defined as "true value - approximate value") in the estimate”. This question is standard in competitive examination syllabi for Numerical Methods.

x1 = 1.5 is the correct answer to “Starting from x 0 = 1, one step of Newton-Raphson method in solving the equation x 3 + 3x - 7 = 0 gives the next value (x 1 ) as”. This question is standard in competitive examination syllabi for Numerical Methods.

$$\frac{4}{3}$$ is the correct answer to “The equation x 3 - x 2 + 4x - 4 = 0 is to be solved using the Newton-Raphson method. If x = 2 is taken as the initial approximation of the solution, then the next approximation using this method will be”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{k}} + 1}} = \frac{1}{3}\left( {2{{\text{x}}_{\text{k}}} + \frac{{\text{N}}}{{{\text{x}}_{\text{k}}^2}}} \right)$$ is the correct answer to “The iteration step in order to solve for the cube roots of a given number N using the Newton-Raphson's method is”. This question is standard in competitive examination syllabi for Numerical Methods.

0.71828 is the correct answer to “Equation e x - 1 = 0 is required to be solved using Newton's method with an initial guess x 0 = -1. Then, after one step of Newton's method, estimate x 1 of the solution will be given by”. This question is standard in competitive examination syllabi for Numerical Methods.

a-1, b-3, c-4, d-2 is the correct answer to “Match List-I with List-II and select the correct answer: List-I List-II a. Newton-Raphson method 1. Solving nonlinear equations b. Runge-Kutta method equations 2. Solving simultaneous linear equations c. Simpson's Rule equations 3. Solving ordinary differential d. Gauss elimination 4. Numerical integration 5. Interpolation 6. Calculation of Eigenvalues”. This question is standard in competitive examination syllabi for Numerical Methods.

0.00000 is the correct answer to “A calculator has accuracy up to 8 digits after decimal place. The value of $$\int\limits_0^{2\pi } {\sin {\text{x dx}}} $$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is”. This question is standard in competitive examination syllabi for Numerical Methods.

4 is the correct answer to “The following table lists an n th order polynomial f(x) = a n x n + a n - 1 x n - 1 + , .... a 1 x + a 0 and the forward difference evaluated at equally spaced values of x. The order of the polynomial is x f(x) Δf Δ 2 f Δ 3 f -0.4 1.7648 -0.2965 0.089 -0.03 -0.3 1.4683 -0.2075 0.059 -0.0228 -0,2 1.2608 -0.1485 0.0362 -0.0156 -0.1 1.1123 -0.1123 0.0206 -0.0084 0 1 -0.0917 0.0122 -0.0012 0.1 0.9083 -0.0795 0.011 0.006 0.2 0.8288 -0.0685 0.017 0.0132”. This question is standard in competitive examination syllabi for Numerical Methods.

0 is the correct answer to “P(0, 3), Q(0.5, 4) and R(1, 5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limits x = 0 and x = 1 for the curve. The difference between the two results will be”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{n}} + 1}} = \left( {1 + {{\text{x}}_{\text{n}}}} \right)\frac{{{{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}}}{{1 + {{\text{e}}^{ - {{\text{x}}_{\text{n}}}}}}}$$ is the correct answer to “The recursion relation to solve x = e -x using Newton-Raphson method is”. This question is standard in competitive examination syllabi for Numerical Methods.

2 and 3 is the correct answer to “Given that one root of the equation x 3 - 10x 2 + 31x - 30 = 0 is 5, the other two roots are”. This question is standard in competitive examination syllabi for Numerical Methods.

3 is the correct answer to “The bisection method is applied to compute a zero of the function f(x) = x 4 - x 3 - x 2 - 4 in the interval [1, 9]. The method converges to a solution after . . . . . . . . iterations.”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{i}} + 1}} = \frac{1}{2}\left( {{{\text{x}}_{\text{i}}} + \frac{{\text{N}}}{{{{\text{x}}_{\text{i}}}}}} \right)$$ is the correct answer to “The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x 2 - N = 0. If i denotes the iteration index, the correct iterative scheme will be”. This question is standard in competitive examination syllabi for Numerical Methods.

h2 is the correct answer to “During the numerical solution of a first order differential equation using Euler method with step size h, the local truncation error is of order of”. This question is standard in competitive examination syllabi for Numerical Methods.

Both I and II is the correct answer to “With respect to the numerical evaluation of the definite integral $${\text{K}} = \int_{\text{a}}^{\text{b}} {{{\text{x}}^2}{\text{dx,}}} $$ where a and b are given, which of the following statements is/are TRUE? I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral. II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.”. This question is standard in competitive examination syllabi for Numerical Methods.

$${{\text{x}}_{{\text{k}} + 1}} = \frac{{2{\text{x}}_{\text{k}}^3 + 9}}{{3{\text{x}}_{\text{k}}^2 + 4}}$$ is the correct answer to “The following equation needs to be numerically solved using the Newton-Raphson method. x 3 + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)”. This question is standard in competitive examination syllabi for Numerical Methods.