GATE EC Engineering Mathematics questions

The eigenvalues and the corresponding eigen vectors of a $2 \times 2$ matrix are given by $$\begin{array}{cc} \textbf{Eigenvalue}& \textbf{Eigenvector} \\ \lambda_1=8 & \text{v}_{1} = \left[ \begin{array} {} 1 \\ 1 \end{array} \right] \\ \lambda_2=4 & \text{v}_{2} = \left[ \begin{array} {c} 1 \\ – 1 \end{array} \right] \end{array}$$ The matrix …

For the function of a complex variable $\text{W}=\ln \text{Z}\; ($where, $\mathrm{W}=u+j \mathrm{v}$ and $\mathrm{Z}=x+j y),$ the $u=$ constant lines get mapped in $\text{Z}$-plane as

The value of the contour integral $\displaystyle{}\oint_{\mid z-j \mid =2} \;\frac{1}{z^2+4} d z$ in positive sense is

The integral $\displaystyle{}\int_0^\pi \sin ^3 \theta\; d \theta$ is given by

Three companies $\text{X, Y}$ and $\text{Z}$ supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below …

For the matrix $\left[\begin{array}{ll}4 & 2 \\ 2 & 4\end{array}\right]$, the eigen value corresponding to the eigenvector $\left[\begin{array}{l}101 \\ 101\end{array}\right]$ is

For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are $y=0$ for $x=0$, and $y=0$ for $x=a$ The form of non-zero solutions of $y$ (where $m$ varies over all integers) are

As $x$ is increased from $-\infty$ to $\infty$, the function $$f(x)=\frac{e^x}{1+e^x}$$

A uniformly distributed random variable $X$ with probability density function $$f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5))$$ where $u(.)$ is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable $Y$ would be

The following differential equation has $$3 \frac{d^{2} y}{d t^{2}}+4\left(\frac{d y}{d t}\right)^{3}+y^{2}+2=x$$

A fair dice is rolled twice. The probability that an odd number will follow an even number is

A solution of the following differential equation is given by $$\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$$

Given the matrix $\left[\begin{array}{cc}-4 & 2 \\ 4 & 3\end{array}\right]$, the eigenvector is

Let, $\mathrm{A}=\left[\begin{array}{cc}2 & -0.1 \\ 0 & 3\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{ll}\frac{1}{2} & \mathrm{a} \\ 0 & \mathrm{~b}\end{array}\right]$ Then $(a+b)=$

The value of the integral $\text{I}=\frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} \exp \left(-\frac{x^{2}}{8}\right)$ $d x$ is

Match the following and choose the correct combination: …

Given an orthogonal matrix $A=\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right]$ $\left[\mathrm{AA}^{\mathrm{T}}\right]^{-1}$ is

An output of a communication channel is a random variable $v$ with the probability density function as shown in the figure The mean square value of $v$ is

The distribution function $\mathrm{F}_{1}(x)$ of a random variable $\mathrm{X}$ is shown in the figure. The probability that $X=1$ is

A system described by the following differential equation $\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=x(t)$ is initially at rest. For input $x(t)=2 u(t)$, the output $y(t)$ is