GATE EC question library

The ideal long channel $n\text{MOSFET}$ and $p\text{MOSFET}$ devices shown in the circuits have threshold voltages of $1 \; \text{V}$ and $– 1 \; \text{V},$ respectively. The $\text{MOSFET}$ substrates are connected to their respective sources. Ignore leakage currents and assume that the capacitors are initially discharged. For the applied voltages …

Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path. where the gain $K = 2.$ The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed right-half of the complex …

The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from $0$ to $\infty.$ Select the transfer function $G(s)$ that results in the root-locus plot of the closed-loop system as shown in the figure.

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure. Statement $\text{I}:$ The system is necessarily a pure delay system for inputs which are bandlimited to $- \alpha \leq f \leq \alpha.$ Statement $\text{II}:$ For any wide-sense stationary input process with power spectral density $S_{x}(f),$ the …

In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current $\text{(in Amperes)}$ through the resistor is $2\sin (t+\pi/2).$ The displacement current $\text{(in Amperes)}$ through the capacitor is ______________.

Consider the following partial differential equation $\text{(PDE)}$ $$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x, y),$$ where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that …

An ideal $\text{OPAMP}$ circuit with a sinusoidal input is shown in the figure. The $3 \; \text{dB}$ frequency is the frequency at which the magnitude of the voltage gain decreases by $3 \; \text{dB}$ from the maximum value. Which of the options is/are correct?

Select the Boolean function(s) equivalent to $x+yz,$ where $x, y,$ and $z$ are Boolean variables, and $+$ denotes logical $\text{OR}$ operation.

Select the correct statement(s) regarding $\text{CMOS}$ implementation of $\text{NOT}$ gates.

Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true?

Consider the following wave equation, $$\frac{\partial^{2}f(x, t)}{\partial t^{2}} = 10000 \frac{\partial^{2}f(x, t)}{\partial x^{2}}$$ Which of the given options is/are solution(s) to the given wave equation?

A simple closed path $C$ in the complex plane is shown in the figure. If $$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$ where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).

Let $x_{1} (t) = e^{-t}u(t)$ and $x_{2}(t) = u(t) – u(t – 2),$ where $u(\cdot)$ denotes the unit step function. If $y(t)$ denotes the convolution of $x_{1}(t)$ and $x_{2}(t),$ then $\displaystyle \lim_{t \rightarrow \infty} y(t) =$ ______________ (rounded off to one decimal place).

An ideal $\text{MOS}$ capacitor (p-type semiconductor) is shown in the figure. The $\text{MOS}$ capacitor is under strong inversion with $V_{G} = 2 \; \text{V}.$ The corresponding inversion charge density $(Q_{IN})$ is $2.2 \; \mu \text{C /cm}^{2}.$ Assume oxide capacitance per unit area as $C_{\text{ox}} = 1.7 \; \mu \text{F/cm}^{2}.$ For …

A symbol stream contains alternate $\text{QPSK}$ and $\text{16-QAM}$ symbols. If symbols from this stream are transmitted at the rate of $1$ mega-symbols per second, the raw (uncoded) data rate is ____________ mega-bits per second (rounded off to one decimal place).

The function $f(x) = 8 \log_{e} x – x^{2} + 3$ attains its minimum over the interval $[1, e]$ at $x =$____________. (Here $\log_{e}x$ is the natural logarithm of $x$.)

Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T} v_{2} = 0, v_{1}^{T} v_{1} = 1,$ and $v_{2}^{T} v_{2} = 1.$ Let $A$ be the $3 \times 3$ matrix given by: $$A = \alpha v_{1} v_{1}^{T} + \beta v_{2} v_{2}^{T}$$ The …

For the circuit shown, the locus of the impedance $Z (j \omega)$ is plotted as $\omega$ increases from zero to infinity. The values of $R_{1}$ and $R_{2}$ are:

Consider the circuit shown in the figure with input $V(t)$ in volts. The sinusoidal steady state current $I(t)$ flowing through the circuit is shown graphically (where $t$ is in seconds). The circuit element $Z$ can be ____________.

Consider an ideal long channel $n \text{MOSFET}$ (enhancement-mode) with gate length $10 \; \mu \text{m}$ and width $100 \; \mu \text{m}.$ The product of electron mobility $(\mu_{n})$ and oxide capacitance per unit area $(C_{\text{ox}})$ is $\mu_{n} C_{\text{ox}} = 1 \; \text{mA/V}^{2}.$ The threshold voltage of the transistor is …