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Numerical Methods · Engineering Mathematics · GATE EE

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Marks 1

1

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $$\sigma_0$$. The $$\sigma_0$$ is estimated by randomly drawing out 10,000 numbers of samples ($$x_n$$). The estimates $${\widehat \sigma _1}$$, $${\widehat \sigma _2}$$ are computed in the following two ways.

$$\matrix{ {\widehat \sigma _1^2 = {1 \over {100000}}\sum\nolimits\limits_{n = 1}^{10000} {x_n^2} } & {\widehat \sigma _2^2 = {1 \over {9999}}\sum\nolimits\limits_{n = 1}^{10000} {x_n^2} } \cr } $$

Which of the following statements is true?

GATE EE 2023
2

Three points in the x-y plane are ($$-$$1, 0.8), (0, 2.2) and (1, 2.8). The value of the slope of the best fit straight line in the least square sense is _________ (Round off to 2 decimal places).

GATE EE 2023
3
If a continuous function $$f(x)$$ does not have a root in the interval $$\left[ {a,b} \right],\,\,$$ then which one of the following statements is TRUE?
GATE EE 2015 Set 1
4
The function $$f\left( x \right) = {e^x} - 1\,\,$$ is to be solved using Newton $$-$$ Raphson method. If the initial value of $${x_0}$$ is taken $$1.0,$$ then the absolute error observed at $${2^{nd}}$$ iteration is ___________.
GATE EE 2014 Set 3
5
When the Newton-Raphson method is applied to solve the equation $$\,\,f\left( x \right) = {x^3} + 2x - 1 = 0,\,\,$$ the solution at the end of the first iteration with the initial value as $${x_0} = 1.2$$ is
GATE EE 2013
6
Let $$\,{x^2} - 117 = 0.\,\,$$ The iterative steps for the solution using Newton -Raphson's method is given by
GATE EE 2009
7
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
GATE EE 2008
8
Given the differential equation $${y^1} = x - y$$ with initial condition $$y(0)=0.$$ The value of $$y(0.1)$$ calculated numerically upto the third place of decimal by the $${2^{nd}}$$ order Runge-Kutta method with step size $$h=0.1$$ is
GATE EE 1993

Marks 2

1
Solution, the variable $${x_1}$$ and $${x_2}$$ for the following equations is to be obtained by employing the Newton $$-$$ Raphson iteration method
equation (i) $$10\,{x_2}\,\sin \,{x_1} - 0.8 = 0$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$10\,x_2^2\, - 10\,{x_2}\cos \,{x_1} - 0.6 = 0$$
Assuming the initial values $${x_1} = 0.0$$ and $${x_2} = 1.0$$ the Jacobian matrix is
GATE EE 2011
2
A differential equation $${{dx} \over {dt}} = {e^{ - 2t}}\,\,u\left( t \right)\,\,$$ has to be solved using trapezoidal rule of integration with a step size $$h=0.01$$ sec. Function $$u(t)$$ indicates a unit step function. If $$x(0)=0$$ then the value of $$x$$ at $$t=0.01$$ sec will be given by
GATE EE 2008
3
The value of $$\,\,\,\int\limits_1^2 {{1 \over x}\,\,\,dx\,\,\,\,} $$ computed using simpson's rule with a step size of $$h=0.25$$ is
GATE EE 1998
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