GATE EC question library

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

Consider the function $f(t)$ having Laplace transform $$\text{F}(s)=\frac{\omega_0}{s^2+\omega_0^2} \operatorname{Re}[s]>0$$ The final value of $f(t)$ would be

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

A two-port network is represented by $\text{ABCD}$ parameters given by $$\left[\begin{array}{c} \mathrm{V}_1 \\ \mathrm{I}_1 \end{array}\right]=\left[\begin{array}{ll} \mathrm{A} & \mathrm{B} \\ \mathrm{C} & \mathrm{D} \end{array}\right]\left[\begin{array}{c} \mathrm{V}_2 \\ -\mathrm{I}_2 \end{array}\right]$$ if port-$2$ is terminated by …

In the two port network shown in the figure below, $z_{12}$ and $z_{21}$ are, respectively

The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by

A $2 \; \mathrm{mH}$ inductor with some initial current can be represented as shown below, where $s$ is the Laplace Transform variable. The value of initial current is

In the figure shown, assume that all the capacitors are initially uncharged. If $\text{V}_i(t)=10 u(t)$ Volts, then $\text{V}_0(t)$ is given by

Consider two transfer functions $$\mathrm{G}_1(s)=\frac{1}{s^2+a s+b} \text{ and } \mathrm{G}_2(s)=\frac{s}{s^2+a s+b}$$ The $3\text{-dB}$ bandwidths of their frequency responses are, respectively

A negative resistance $R_{\text{neg }}$ is connected to a passive network $N$ having driving point impedance $Z_{1}$ (s) as shown below. For $Z_{2}(s)$ to be positive real,

In the circuit shown below, the switch wa connected to position $1$ at $t<0$ and at $t=0$, it is changed to position $2$. Assume that $y$ the diode has zero voltage drop and a storage time $t_{s}$. For $0 < t \leq t_{s}, \text{V}_{\text{R}}$ is given by (all in volts)

The majority carriers in an $n$-type semiconductor have an average drift velocity $\mathrm{V}$ in a direction perpendicular to a uniform magnetic field $\text{B}.$ The electric field $\mathrm{E}$ induced due to Hall effect acts in the direction

Find the correct match between Group $1$ and Group $2$ …

A heavily doped $n$-type semiconductor has the following data Hole-electron mobility ratio : $0.4$ Doping concentration : $4.2 \times 10^{8} \; \mathrm{atoms/m}^{3}$ Intrinsic concentration : $1.5 \times 10^{4} \; \mathrm{atoms/m}^{3}$ The ratio of conductance of the $n$-type semiconductor to that of the intrinsic semiconductor of same material and …