# GATE Questions & Answers of Numerical Methods Civil Engineering

#### Numerical Methods 16 Question(s)

The quadratic equation $2x^2-3x+3=0$ is to be solved numerically starting with an initial guess as $x_0=2$ . The new estimate of $x$ after the first iteration using Newton-Raphson method is ______

Consider the equation $\style{font-family:'Times New Roman'}{\frac{du}{dt}=3t^2+1\;\mathrm{with}\;u=0\;\mathrm{at}\;t=0}$. The is numerically solved by using the forward Euler method with a step size, $\style{font-family:'Times New Roman'}{\triangle t=2}$. The absolute error in the solution at the end the first time step is ___________

The integral $\int_{x_1}^{x_2}\;x^2dx$ with x> x1 > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $I$ is the exact value of the integral obtained analytically and $J$ is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship ?

The quadratic equation x2 - 4x + 4=0 is to be solved numerically, starting with the initial guess xo=3. The Newton- Raphson method is applied once to get a new estimate and then the Secant method is applied once using the initial guess and this new estimate. The estimated value of the root after the application of the secant method is ___________.

In Newton-Raphson iterative method, the initial guess value (xini) is considered as zero while finding the roots of the equation: f(x) = -2+6x-4x2+0.5x3. The correction, $\triangle$x, to be added to xini in the first iteration is_____.

For step-size, $\triangle x$ = 0.4, the value of following integral using Simpson’s 1/ 3 rule is _________.

$\int\limits_0^{0.8}(0.2+25x-200x^2+675x^3-900x^4+400x^5)dx$

There is no value of x that can simultaneously satisfy both the given equations.Therefore, find the ‘least squares error’ solution to the two equations, i.e., find the value of x that minimizes the sum of squares of the errors in the two equations. __________
2x=3
4x=1

Find the magnitude of the error (correct to two decimal places) in the estimation of following integral using Simpson’s $\frac{1}{3}$ Rule. Take the step length as 1. __________

$\int\limits_0^4(x^4+10)dx$

The estimate of $\int\limits_{0.5}^{1.5}\frac{dx}x$ obtained using Simpson’s rule with three-point function evaluation exceeds the exact value by

The error in ${\overline{)\frac{d}{dx}f\left(x\right)}}_{x={x}_{0}}$ for a continuous function estimated with h = 0.03 using the central difference formula ${\overline{)\frac{d}{dx}f\left(x\right)}}_{x={x}_{0}}\approx \frac{f\left({x}_{0}+h\right)-f\left({x}_{0}-h\right)}{2h}$, is 2×10−3. The values of x0 and f(x0) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately

The square root of a number N is to be obtained by applying the Newton Raphson iterations to thee quation x2-N=0.if i denotes the iteration index, the correct iterative scheme will be

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 F(x) 1 0.9412 0.8 0.64 0.5

The value of the integral of the function between the limits 0 to 1 using Simpson’s rule is

In the solution of the following set of linear equations by Gauss elimination using partial pivoting

5x +y +2z =34;     4y -3z =12;   and    10x -2y +z =-4;

The pivots for elimination of x and y are

Three values of x and y are to be fitted in a straight line in the form y=a+bx by the method of least squares. Given:∑x=6, ∑y=21, ∑x2=14 and ∑xy=46, the values of a and b are respectively

The following equation needs to be numerically solved using the Newton-Raphson method.

x3 + 4x - 9 =0

The iterative equation for this purpose is (k indicates the iteration level)