GATE EC question library

The system of linear equations $\begin{pmatrix} 2 & 1 & 3\\ 3&0 &1 \\ 1& 2 &5 \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} = \begin{pmatrix} 5\\ -4\\ 14 \end{pmatrix}$ has

The real part of an analytic function $f(z)$ where $z = x + jy$ is given by $e^{-y} \cos(x)$. The imaginary part of $f(z)$ is

The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.

If $\overrightarrow {r}= x\hat{a_{x}}+y\hat{a_{y}}+z\hat{a_{z}}$ and $\mid \overrightarrow{r} \mid= r$ , then $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .

A series LCR circuit is operated at a frequency different from its resonant frequency. The operating frequency is such that the current leads the supply voltage. The magnitude of current is half the value at resonance. If the values of L, C and R are $1$ H, $1$ F and $1 \Omega$, respectively, the operating angular frequency (in rad/s) is _________.

In the h-parameter model of the $2$-port network given in the figure shown, the value of $h_{22}$ (in S) is ______ .

In the figure shown, the capacitor is initially uncharged. Which one of the following expressions describes the current $I(t)$ (in mA) for $t > 0$?

In the magnetically coupled circuit shown in the figure, $56 \%$ of the total flux emanating from one coil links the other coil. The value of the mutual inductance (in H) is ____ .

Assume electronic charge $q= 1.6 \times 10^{-19} \: C$, $kT/q=25 \: mV$ and electron mobility $\mu_{n}= 1000cm^{2}/ V-s$. If the concentration gradient of electrons injected into a $P$-type silicon sample is $1 \times 10^{21}/cm^{4}$, the magnitude of electron diffusion current density (in $A/cm^{2}$) is _________ .

Consider an abrupt PN junction (at $T = 300 K$) shown in the figure. The depletion region width $X_{n}$ on the $N$-side of the junction is $0.2 \mu m$ and the permittivity of silicon $( \varepsilon _{gi} )$ is $1.044 \times 10^{-12} \: F/cm$. At the junction, the approximate value of the peak electric field (in $kV/cm$) is ___________.

When a silicon diode having a doping concentration of $N_{A}= 9\times 10^{16}cm^{-3}$ on $p$-side and $N_{D}= 1 \times 10^{16}cm^{-3}$ on $n$-side is reverse biased, the total depletion width is found to be $3 \mu$. Given that the permittivity of silicon is $1.04 \times 10^{-12}$ F/cm, the depletion width on the p-side and the maximum electric …

The diode in the circuit shown has $V_{on}$$= 0.7$ Volts but is ideal otherwise. If $V_{i}= 5 \sin ( \omega t )$Volts, the minimum and maximum values of $V_{o}$ (in Volts ) are, respectively,

For the n-channel MOS transistor shown in the figure, the threshold voltage $V_{Th}$ is $0.8$ V. Neglect channel length modulation effects. When the drain voltage $V_{D}= 1.6 V,$ the drain current $I_{D}$ was found to be $0.5$ mA. If $V_{D}$is adjusted to be $2$ V by changing the values of R and $V_{DD}$, the new value of $I_{D}$ (in mA) is

For the MOSFETs shown in the figure, the threshold voltage $\mid V_{t} \mid= 2$ V and $K= \frac{1}{2}\mu C_{ox} ( \frac{W}{L} ) = 0.1 \: mA/V^{2}$. The value of $I_{D}$ (in mA) is _______________ .

In the circuit shown, choose the correct timing diagram of the output (y) from the given waveforms $W1$, $W2$, $W3$ and $W4$.

The outputs of the two flip-flops $Q1$, $Q2$ in the figure shown are initialized to $0,0$. The sequence generated at $Q1$ upon application of clock signal is

For the $8085$ microprocessor, the interfacing circuit to input $8$-bit digital data $( DI_{0}-DI_{7})$ from an external device is shown in the figure. The instruction for correct data transfer is

Consider a discrete-time signal $$x[n]= \begin{cases} n & \text{for } 0\leq n\leq 10 \\ 0 & \text{otherwise }\end{cases}$$ If $y[n]$ is the convolution of $x[n]$ with itself, the value of $y[4]$ is _______ .

The input-output relationship of a causal stable LTI system is given as $$y[n] = \alpha \: y[n-1] + \beta \: x[n]$$. If the impulse response $h[n]$ of this system satisfies the condition $\sum_{n=0}^{\infty}h[n]= 2$, the relationship between $\alpha$ and $\beta$ is

The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.