List of top Engineering Mathematics Questions on Numerical Methods

If the initial guess of the solution of \[ 1+x-x^3=0 \] is $x_0=1$, then the $2^{\text{nd}}$ approximate root of \[ 1+x-x^3=0 \] using Newton-Raphson method is

Given the following data: x: (-2, 1, 2), y: (28, 4, 16). Let $P_2(x)$ be the quadratic interpolating polynomial passing through the above three points. Find the value of $P_2(0)$.

Consider the function $f(x) = e^{-x} - x$. Using the Newton-Raphson method, obtain the first improved approximation starting from the initial guess $x_0 = 0.5$. Enter the value of the second approximation, correct to two decimal places.}

Consider a nonlinear algebraic equation, $ e^x - 2 = 0 $. Using the Newton-Raphson method, with the initial guess of $ x_0 = 1 $, the approximated value of the root of the equation after one iteration is ________.

The value of $ k $, for which the linear equations $ 2x + 3y = 6 $ and $ 4x + 6y = 3k $ have at least one solution, is ________. (Answer in integer)

An approximate solution of the equation $ x^3 - 17 = 0 $ is to be obtained using the Newton-Raphson method. If the initial guess is $ x_0 = 2 $, the value at the end of the first iteration is $ x_1 = $ __________ (rounded off to two decimal places).

Consider a nonlinear algebraic equation, $ e^x - 2 = 0 $. Using the Newton-Raphson method, with the initial guess of $ x_0 = 1 $, the approximated value of the root of the equation after one iteration is ________.

Use Newton-Raphson method to find the root of $f(x) = x^3 - x - 2 = 0$ starting with $x_0 = 1$ after one iteration.

Use the bisection method to find the root of $f(x) = x^2 - 2 = 0$ in $[1, 2]$ with an error less than 0.01.

The value of $ \alpha $ for which the Euler method with step size $ h = 0.1 $ provides $ y(1.1) = 1.2 $ for the following initial value problem is \[ \frac{dy}{dx} = -2xy^\alpha, \quad y(1) = 2 \]

The value of $ k $, for which the linear equations $ 2x + 3y = 6 $ and $ 4x + 6y = 3k $ have at least one solution, is ________. (Answer in integer)

An approximate solution of the equation $ x^3 - 17 = 0 $ is to be obtained using the Newton-Raphson method. If the initial guess is $ x_0 = 2 $, the value at the end of the first iteration is $ x_1 = $ ________ (rounded off to two decimal places).

An approximate solution of the equation $ x^3 - 17 = 0 $ is to be obtained using the Newton-Raphson method. If the initial guess is $ x_0 = 2 $, the value at the end of the first iteration is $ x_1 = $ \_\_\_\_\_\ (rounded off to two decimal places).

Using Trapezoidal rule with one interval, find the approximate value of $\displaystyle \int_{1}^{2}\frac{dx}{1+x^{2}}$ (rounded off to 2 decimal places).

If $ \dfrac{dy}{dx} = xy + y^2 $, $ y(0) = 1 $, and $ h = 0.1 $, then the value of $ y(0.1) $ using Euler’s method is:

To solve the equation $ x \log x = 1 $, using Newton-Raphson method, the iterative formula and the first approximate $ x_1 $, when $ x_0 = 1 $ is (Assuming $\log$ is natural logarithm $\ln$)

The equation $ x^3 - 6x - 4 = 0 $ has one real root between 2 and 3. The 2ndapproximation to the root of the equation by bisection method will be

The Newton-Raphson iteration $$ x_{n+1} = \frac{1}{3} \left( 2 x_n + \frac{N}{x_n^2} \right) $$ is used to compute

When the IVP $ \frac{dy}{dx} = y + x,\; y(0) = 1 $ is solved by Rung–Kutta fourth order method to find $ y(0.2) $, the approximate value of $ y(0.2) $ by taking step size $ h = 0.2 $ is:

If $f(x)$ is defined on $[a,b]$ and $[a,b]$ is divided into $2n$ equal parts by the points $x_0=a<x_1<x_2<...<x_{2n}=b$. Then by Simpson's $\frac{1}{3}$ rule $\int_a^b f(x)dx =$