The components in the circuit shown below are ideal. If the op-amp is in positive feedback and the input voltage $V_i$ is a sine wave of amplitude 1 V , the output voltage $V_0$ is

In the circuit shown below, all the components are ideal. If $V_i$ is +2 V , the current $I_0$ sourced by the OpAmp is $\_\_\_\_$ mA .

Using the incremental low frequency small - signal model of the MOS device, the Norton equivalent resistance of the following circuit is

The base of an $n p n$ BJT $T 1$ has a linear doping profile $N_B(x)$ as shown below. The base of another $n p n$ BJT $T 2$ has a uniform doping $N_B$ of $10^{17} \mathrm{~cm}^{-3}$. All other parameters are identical for both the devices. Assuming that the hole density profile is the same as that of doping, the common - emitter current gain of $T 2$ is

In the voltage regulator shown below, $V_I$ is the unregulated input at 15 V . Assume $V_{B E}=0.7 \mathrm{~V}$ and the base current is negligible for both the BJTs. If the regulated output $V_0$ is 9 V , the value of $R_2$ is $\_\_\_\_$ $\Omega$.

The components in the circuit given below are ideal. If $R=2 \mathrm{k} \Omega$ and $C=1 \mu \mathrm{~F}$, the -3 dB cut-off frequency of the circuit in Hz is

For the BJT in the amplifier shown below, $V_{B E}=0.7 \mathrm{~V}, \frac{k T}{q}=26 \mathrm{mV}$. Assume that the BJT output resistance ( $r_0$ ) is very high and the base current is negligible. The capacitors are also assumed to be short circuited at signal frequencies. The input $V_i$ is direct coupled. The low frequency voltage gain $\frac{V_0}{V_i}$ of the amplifier is

An enhancement MOSFET of threshold voltage 3 V is being used in the sample and hold circuit given below. Assume that the substrate of the MOS device is connected to -10 V . If the input voltage $v_1$ liesbetween $\pm 10 \mathrm{~V}$, the minimum and the maximum value of $v_G$ required for proper sampling and holding respectively, are

The random variable

$$ Y=\int_{-\infty}^{\infty} W(t) \phi(t) d t, \quad \text { where } \phi(t)=\left\{\begin{array}{cc} 1, & 5 \leq t \leq 7 \\ 0, & \text { otherwise } \end{array}\right. $$

and $W(t)$ is a real white Gaussian noise process with two-sided power spectral density $S_W(f)=3 \mathrm{~W} / \mathrm{Hz}$, for all $f$. The variance of $Y$ is $\_\_\_\_$ .

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous of at least one of its bits is decoded wrongly. The probability that the received block is erroneous, is

A binary random variable $X$ takes the value +2 or -2 . The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is $\_\_\_\_$ .

In a digital communication system, a symbol $S$ randomly chosen from the set $\left\{s_1, s_2, s_3, s_4\right\}$ is transmitted. It is given that $s_1=-3, s_2=-1, s_3=+1$ and $s_4=+2$. The received symbol is $Y=S+W . W$ is a zero mean unit - variance Gaussian random variable and is independent of $S . P_i$ is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol $S=s_i$. The index $i$ for which the conditional symbol error probability $P_i$ is the highest is $\_\_\_\_$ .

$S_{P M}(t)$ and $S_{F M}(t)$ are defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.

$$ \begin{aligned} & S_{P M}(t)=\cos \left[1000 \pi t+k_p m(t)\right] \\ & S_{F M}(t)=\cos \left[1000 \pi t+k_f \int_{-\infty}^t m(\tau) d \tau\right] \end{aligned} $$

Where $k_p$ is the phase deviation constant in radians/volt and $k_f$ is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of $S_{P M}(t)$ and $S_{F M}(t)$ are same, then the value of the ratio $\frac{k_p}{k_f}$ is $\_\_\_\_$ seconds.

For the modulated signal $x(t)=m(t) \cos \left(2 \pi f_c t\right)$, the message signal $m(t)=4 \cos (1000 \pi t)$ and the carrier frequency $f_c$ is 1 MHz . The signal $x(t)$ is passed through a demodulator, as shown in figure below. The output $y(t)$ of the demodulator is

The pole-zero map of a rational function $G(s)$ is shown below. When the closed contour $\Gamma$ is mapped into $G(s)$-plane, then the mapping encircles

The loop transfer function of a negative feedback system is

$$ G(s) H(s)=\frac{K(s+11)}{s(s+2)(s+8)} $$

The value of $K$, for which system is marginally stable, is $\_\_\_\_$ .

The characteristic equation of a system is

$$ s^3+3 s^2+(K+2) s+3 K=0 $$

In the root locus plot for the given system, as $K$ varies from 0 to $\infty$, the break-away or break-in point(s) lie within

A system with transfer function $G(s)=\frac{1}{(s+1)(s+a)}, a>0$ is subjected to input $5 \cos 3 t$. The steady state output of the system is $\frac{1}{\sqrt{10}} \cos (3 t-1.892)$. The value of $a$ is

$$ \text { Consider the following closed loop control system } $$

where $G(s)=\frac{1}{s(s+1)}$ and $C(s)=K \frac{s+1}{s+3}$. If the steady state error for a unit ramp input is 0.1 , then the value of $K$ is $\_\_\_\_$ .

The figure below shows a multiplexer where $S_1$ and $S_0$ are the select lines, $I_0$ to $I_3$ are the input data lines, $E N$ is the enable line, and $F(P, Q, R)$ is the output. $F$ is

A 10 bit D/A converter is calibrated over the full range 0 to 10 V . If the input to the D/A converter is 13 A (in hex), the output (rounded off to three decimal places) is $\_\_\_\_$ V.

For the components in the sequential circuit shown below, $t_{p d}$ is the propagation delay, $t_{\text {secup }}$ is the setup time and $t_{\text {hold }}$ is the hold time. The maximum clock frequency (rounded off to the nearest integer), at which the given circuit can operate reliably, is $\_\_\_\_$ MHz.

$P, Q$ and $R$ are the decimal integers corresponding to the 4-bit binary number 1100 considered in signed magnitude, 1 's complement and 2 's complement representation, respectively. The 6 -bit 2 's complement representation of ( $P+Q+R$ ) is

The state diagram of a sequence detector is shown below. State $S_0$ is the initial state of the sequence detector. If the output is 1 , then

A transmission line of length $3 \lambda / 4$ and having a characteristic impedance of $50 \Omega$ is terminated with a load of $400 \Omega$. The impedance (rounded off to two decimal places) seen at the input end of the transmission line is $\_\_\_\_$ $\Omega$.

The impedances $Z=j X$, for all $X$ in the range ( $-\infty, \infty$ ), map to the Smith chart as

The magnetic field of a uniform plane wave in vacuum is given by

$$ \vec{H}(x, y, z, t)=\left(\hat{a}_x+2 \hat{a}_y+b \hat{a}_z\right) \cos (\omega t+3 x-y-z) . $$

The value of $b$ is $\_\_\_\_$ .

For an infinitesimally small dipole in free space, the electric field $E_\theta$ in the far field is proportional to $\frac{e^{-j k r}}{r} \sin \theta$, where $k=\frac{2 \pi}{\lambda}$. A vertical infinitesimally small electric dipole ( $\delta l \ll \lambda$ ) is placed at a distance $h(h>0)$ above an infinite ideal conducting plane, as shown in the figure. The minimum value of $h$, for which one of the maxima in the far field radiation pattern occurs at $\theta=60^{\circ}$, is

Consider the recombination process via bulk traps in a forward biased $p n$ homojunction diode. The maximum recombination rate is $U_{\max }$. If the electron and the hole capture cross sections are equal, which one of the following is FALSE?

A single crystal intrinsic semiconductor is at a temperature of 300 K with effective density of states for holes twice that of electrons. The thermal voltage is 26 mV . The intrinsic Fermi level is shifted from midbandgap energy level by

A $p n$ junction solar cell of area $1.0 \mathrm{~cm}^2$, illuminated uniformly with $100 \mathrm{mWcm}^{-2}$; has the following parameter : Efficiency $=15 \%$, open circuit voltage $=0.7 \mathrm{~V}$, fill factor $=0.8$, and thickness $=200 \mu \mathrm{~m}$. The charge of an electron is $1.6 \times 10^{-19} \mathrm{C}$. The average optical generation rate ( $\mathrm{in} \mathrm{cm}^{-3} \mathrm{~s}^{-1}$ ) is

A one-sided abrupt $p n$ junction diode has a depletion capacitance $C_D$ of 50 pF at a reverse bias 0.2 V . The plot of $\frac{1}{C_D^2}$ versus the applied voltage $V$ for this diode is a straight line as shown in the figure below. The slope of the plot is $\_\_\_\_$ $\times 10^{20} \mathrm{~F}^{-2} \mathrm{~V}^{-1}$.

The band diagram of $p$-type semiconductor with a bandgap of the 1 eV is shown. Using this semiconductor, a MOS capacitor having $V_{T H}$ of $-0.16 \mathrm{~V}, C_{o x}^{\prime}$ of $100 \mathrm{nF} / \mathrm{cm}^2$ and metal work function of 3.87 eV is fabricated. There is no charge within the oxide. If the voltage across the capacitor is $V_{T H}$, the magnitude of depletion charge per unit area (in $\mathrm{C} / \mathrm{cm}^2$ ) is

If $\mathbf{v}_{\mathbf{1}}, \mathbf{v}_{\mathbf{2}} \ldots \mathbf{v}_{\mathbf{6}}$ are six vectors in $\mathbb{R}^4$, which one of the statements is FALSE?

The two sides of a fair coin are labelled as 0 and 1 . The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X=\min (M, N)$, the expected value $E[X]$ (rounded off to two decimal places) is $\_\_\_\_$ .

The general solution of $\frac{d^2 y}{d x^2}-6 \frac{d y}{d x}+9 y=0$ is

The partial derivative of the function

$$ f(x, y, z)=e^{1-x \cos y}+x z e^{\frac{-1}{\left(1+y^2\right)}} $$

with respect to $x$ at the point $(1,0, e)$ is

For a vector field $\vec{A}$, which one of the following is FALSE?

Consider the following system of linear equations.

$$ x_1+2 x_2=b_1 ; 2 x_1+4 x_2=b_2 ; 3 x_1+7 x_2=b_3 ; 3 x_1+9 x_2=b_4 $$

Which one of the following conditions ensures that a solution exists for the above system?

For the solid $S$ shown below, the value of $\iiint_S x d x d y d z$ (rounded off to two decimal places) is $\_\_\_\_$ .

$X$ is a random variable with uniform probability density function in the interval $[-2,10]$. For $Y=2 X-6$, the conditional probability $P(Y \leq 7 \mid X \geq 5)$ (rounded off to three decimal places) is $\_\_\_\_$ .

Which one of the following options contains two solutions of the differential equation $\frac{d y}{d x}=(y-1) x$ ?

In an 8085 microprocessor, the number of address lines required to access a 16 K bytes memory bank is

$\_\_\_\_$ .

In the given circuit, the two-port network has the impedance matrix $[Z]=\left[\begin{array}{cc}40 & 60 \\ 60 & 120\end{array}\right]$. The value of $Z_L$ for which maximum power is transferred to the load is $\_\_\_\_$ $\Omega$.

$$ \text { In the circuit shown below, the Thevenin voltage } V_{\text {th }} \text { is } $$

The current in the RL - circuit shown below is $i(t)=10 \cos (5 t-\pi / 4) \mathrm{A}$. The value of the inductor (rounded off to two decimal places) is $\_\_\_\_$ H.

In the circuit shown below, all the components are ideal and the input voltage is sinusoidal. The magnitude of the steady - state output $V_0$ (rounded off to two decimal places) is $\_\_\_\_$ V.

$$ \text { For the given circuit, which one of the following is the correct state equation? } $$

$$ \text { The current } I \text { in the given network is } $$

For a 2-port network consisting of an ideal lossless transformer, the parameter $S_{21}$ (rounded off to two decimal places) for a reference impedance of $10 \Omega$ is $\_\_\_\_$ .

The output $y[n]$ of a discrete - time system for an input $x[n]$ is

$$ y[n]=\max\limits_{-\infty \leq k \leq n}|x[k]| $$

The unit impulse response of the system is

Which one of the following pole-zero plots corresponds to the transfer function of an LTI system characterized by the input-output difference equation given below?

$$ y[n]=\sum_{k=0}^3(-1)^k x[n-k] $$

$X(\omega)$ is the Fourier transform of $x(t)$ shown below. The value of $\int\limits_{-\infty}^{\infty}|X(\omega)|^2 d \omega$ (rounded off to two decimal places) is $\_\_\_\_$ .

The transfer function of a stable discrete - time LTI system is $H(z)=\frac{K(z-\alpha)}{(z+0.5)}$ where $K$ and $\alpha$ are real numbers. The value of $\alpha$ (rounded off to one decimal place) with $|\alpha|>1$, for which magnitude response of the system is constant over all frequencies, is $\_\_\_\_$ .

A finite duration discrete-time signal $x[n]$ is obtained by sampling a continuous - time signal $x(t)=\cos (200 \pi t)$ at sampling instants $t=\frac{n}{400}, n=0,1, \ldots ., 7$. The 8-point discrete Fourier transform (DFT) is defined as

$$ X[k]=\sum_{n=0}^7 x[n] e^{-j \pi n k / 4} \text { for } k=0,1, \ldots ., 7 $$

Which one of the following statements is TRUE?

He was not only accused of theft $\_\_\_\_$ of conspiracy.

The Canadian constitution requires that equal importance be given to English and French. Last year, Air Canada lost a lawsuit and had to pay a six-figure fine to a French speaking couple after they filed complaints about formal in-flight announcements in English lasting 15 seconds, as opposed to informal 5 second messages in French.

The French - speaking couple were upset at $\_\_\_\_$ .

Select the word that fits the analogy :

Explicit : Implicit : : Express : $\_\_\_\_$ .

The untimely loss of life is a cause of serious global concern as thousands of people get killed $\_\_\_\_$ accidents every year while many other die $\_\_\_\_$ diseases like cardio vascular diseases, cancer, etc.

A superadditive function $f(\cdot)$ satisfies the following property

$$ f\left(x_1+x_2\right) \geq f\left(x_1\right)+f\left(x_2\right) $$

Which of the following functions is a superadditive function for $x>1$ ?

$a, b, c$ are real numbers. The quadratic equation $a x^2-b x+c=0$ has equal roots, which is $\beta$, then

The global financial crisis in 2008 is considered to be the most serious world-wide financial crisis, which started with the sub-prime lending crisis in USA in 2007. The sub-prime lending crisis led to the banking crisis in 2008 with the collapse of Lehman Brothers in 2008. The sub-prime lending refers to the provision of loans to those borrowers who may have difficulties in repaying loans, and it arises because of excess liquidity following the East Asian crisis. Which one of the following sequences shows the correct precedence as per the given passage?

The following figure shows the data of students enrolled in 5 years (2014 to 2018) for two schools P and Q. During this period, the ratio of the average number of the students enrolled in school P to the average of the difference of the number of students enrolled in schools $P$ and $Q$ is

A circle with center $O$ is shown in the figure. A rectangle $P Q R S$ of maximum possible area is inscribed in the circle. If the radius of the circle is $a$, then the area of shaded portion is $\_\_\_\_$ .

It is quarter past three in your watch. The angle between the hour hand and the minute hand is $\_\_\_\_$ .