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1

GATE IN 2024 | GA Question: 1
If ' $\rightarrow$ ' denotes increasing order of intensity, then the meaning of the words [drizzle $\rightarrow$ rain $\rightarrow$ downpour] is analogous to [ $\rightarrow$ quarrel $\rightarrow$ feud]. Which one of the given options is appropriate to fill the blank? bicker bog dither dodge

If ' $\rightarrow$ ' denotes increasing order of intensity, then the meaning of the words[drizzle $\rightarrow$ rain $\rightarrow$ downpour] is analogous to [ $\rightarro...

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2

GATE IN 2024 | GA Question: 10

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The ...

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3

GATE IN 2024 | Question: 1
Let $\boldsymbol{z}=x+i y$ be a complex variable and $\overline{\boldsymbol{z}}$ be its complex conjugate. The equation $\bar{z}^{2}+z^{2}=2$ represents a parabola hyperbola ellipse circle

Let $\boldsymbol{z}=x+i y$ be a complex variable and $\overline{\boldsymbol{z}}$ be its complex conjugate. The equation $\bar{z}^{2}+z^{2}=2$ represents aparabolahyperbol...

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4

GATE IN 2024 | Question: 28
Consider a discrete-time sequence \[ x[n]=\left\{\begin{array}{ll} (0.2)^{n}, & 0 \leq n \leq 7 \\ 0, & \text { otherwise } \end{array}\right. \] The region of convergence of $X(z)$, the $z$-transform of $x[n]$, consists of all values of $z$ except $z=0.2$ all values of $z$ all values of $z$ except $z=0$ all values of $z$ except $z=\infty$

Consider a discrete-time sequence\[x[n]=\left\{\begin{array}{ll}(0.2)^{n}, & 0 \leq n \leq 7 \\0, & \text { otherwise }\end{array}\right.\]The region of convergence of $X...

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5

GATE IN 2024 | GA Question: 2
Statements: 1. All heroes are winners. 2. All winners are lucky people. Inferences: I. All lucky people are heroes. II. Some lucky people are heroes. III. Some winners are heroes. Which of the above inferences can be logically deduced from statements 1 and 2 ? Only I and II Only II and III Only I and III Only III

Statements:1. All heroes are winners.2. All winners are lucky people.Inferences:I. All lucky people are heroes.II. Some lucky people are heroes.III. Some winners are hero...

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6

GATE IN 2024 | Question: 55
A random variable $X$ has a probability density function \[ f_{X}(x)=\left\{\begin{array}{cc} e^{-x}, & x \geq 0 \\ 0, & \text { otherwise } \end{array}\right. \] The probability of $X>2$ is (rounded off to three decimal places).

A random variable $X$ has a probability density function\[f_{X}(x)=\left\{\begin{array}{cc}e^{-x}, & x \geq 0 \\0, & \text { otherwise }\end{array}\right.\]The probabilit...

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7

GATE IN 2024 | GA Question: 3
A student was supposed to multiply a positive real number $p$ with another positive real number $q$. Instead, the student divided $p$ by $q$. If the percentage error in the student's answer is $80 \%$, the value of $q$ is $5$ $\sqrt{2}$ $2$ $\sqrt{5}$

A student was supposed to multiply a positive real number $p$ with another positive real number $q$. Instead, the student divided $p$ by $q$. If the percentage error in t...

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8

GATE IN 2024 | Question: 34
Consider the feedback control system shown in the figure. The steady-state error $e_{s s}=\lim _{t \rightarrow \infty}(r(t)-y(t))$ due to unit step reference $r(t)$ is $\frac{K-1}{K}$ $\frac{1}{2}$ $0$ $\frac{1-K}{K}$

Consider the feedback control system shown in the figure. The steady-state error $e_{s s}=\lim _{t \rightarrow \infty}(r(t)-y(t))$ due to unit step reference $r(t)$ is$\f...

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9

GATE IN 2024 | GA Question: 6
In the given text, the blanks are numbered (i)-(iv). Select the best match for all the blanks. Yoko Roi stands (i) as an author for standing (ii) as an honorary fellow, after she stood (iii) her writings that stand (iv) the freedom of speech. (i) out (ii) down (iii) in (iv) for (i) down (ii) out (iii) by (iv) in (i) down (ii) out (iii) for (iv) in (i) out (ii) down (iii) by (iv) for

In the given text, the blanks are numbered (i)-(iv). Select the best match for all the blanks.Yoko Roi stands (i) as an author for standing (ii) as an honorary fellow, af...

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10

GATE IN 2024 | Question: 39
The complex functions $f(z)=u(x, y)+i v(x, y)$ and $\overline{f(z)}=u(x, y)-i v(x, y)$ are both analytic in a given domain. Choose the correct option(s) from the following. $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=0$ $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \neq 0$ $\frac{d f(z)}{d z}=0$ $\frac{d f(z)}{d z} \neq 0$

The complex functions $f(z)=u(x, y)+i v(x, y)$ and $\overline{f(z)}=u(x, y)-i v(x, y)$ are both analytic in a given domain. Choose the correct option(s) from the followin...

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11

GATE IN 2024 | Question: 26
A wire of circular cross section with radius $a$ is shown in the figure. The current density is given by $\mathbf{J}=k s^{2}$, where $k$ is a constant, $s$ is the radial distance from the axis and $0 \leq s \leq a$. The total current $I$ in the wire is $\frac{\pi k a^{4}}{2}$ $\frac{2 \pi k a^{3}}{3}$ $\frac{\pi k a^{3}}{2}$ $\frac{\pi k a^{4}}{4}$

A wire of circular cross section with radius $a$ is shown in the figure. The current density is given by $\mathbf{J}=k s^{2}$, where $k$ is a constant, $s$ is the radial ...

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12

GATE IN 2024 | Question: 54
The solution of an ordinary differential equation $y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=30 e^{-t}$ is \[ y(t)=\left(c_{0}+c_{1} t-c_{2} t^{2}+c_{3} t^{3}\right) e^{-t} \] Given $y(0)=3, y^{\prime}(0)=-3$ and $y^{\prime \prime}(0)=-47$, the value of $\left(c_{0}+c_{1}+c_{2}+c_{3}\right)$ is (rounded off ... $c_{0}, c_{1}, c_{2}, c_{3}$ are constants.

The solution of an ordinary differential equation $y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=30 e^{-t}$ is\[y(t)=\left(c_{0}+c_{1} t-c_{2} t^{2}+c_{3} t...

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13

GATE IN 2024 | Question: 27
The measured values from a flow instrument, whose range is between 0 and 2 flow units, are shown in the histogram. The systematic error (bias) and the maximum error (in flow units), respectively are 0.12 and 0.14 0.01 and 0.10 0.10 and 0.14 0.04 and 0.12

The measured values from a flow instrument, whose range is between 0 and 2 flow units, are shown in the histogram. The systematic error (bias) and the maximum error (in f...

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14

GATE IN 2024 | Question: 16
A $3 \times 3$ matrix $P$ with all real elements has eigenvalues $\frac{1}{4}, 1$, and -2 . The value of $\left|P^{-1}\right|$ is (rounded off to nearest integer).

A $3 \times 3$ matrix $P$ with all real elements has eigenvalues $\frac{1}{4}, 1$, and -2 . The value of $\left|P^{-1}\right|$ is (rounded off to nearest integer).

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15

GATE IN 2024 | Question: 8
The sinusoidal transfer function corresponding to the polar plot shown in the figure, for $T>0$, is $1-j \omega T$ $\frac{1-j \omega T}{1+j \omega T}$ $1+j \omega T$ $\frac{1}{1+j \omega T}$

The sinusoidal transfer function corresponding to the polar plot shown in the figure, for $T>0$, is$1-j \omega T$$\frac{1-j \omega T}{1+j \omega T}$$1+j \omega T$$\frac{1...

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16

GATE IN 2024 | Question: 44
Consider the figure shown. For zero deflection in the galvanometer, the required value of resistor $R_{x}$ is $\Omega$ (rounded off to nearest integer).

Consider the figure shown. For zero deflection in the galvanometer, the required value of resistor $R_{x}$ is $\Omega$ (rounded off to nearest integer).

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17

GATE IN 2024 | Question: 31
In the figure shown, both the opamps $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ are ideal, except that the opamp $\mathrm{A}_{1}$ has an offset voltage $\left(\mathrm{V}_{\text {os }}\right)$ of $1 \mathrm{mV}$. For $\mathrm{V}_{\text {in }}=0 \mathrm{~V}$, the values of the output voltages $V_{\text {out1 }}$ ... $-1 \mathrm{mV}$ $1 \mathrm{mV}$ and $0 \mathrm{mV}$ $1 \mathrm{mV}$ and $-1 \mathrm{mV}$ $2 \mathrm{mV}$ and $0 \mathrm{mV}$

In the figure shown, both the opamps $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ are ideal, except that the opamp $\mathrm{A}_{1}$ has an offset voltage $\left(\mathrm{V}_{\tex...

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18

GATE IN 2024 | Question: 9
A matrix $M$ is constructed by stacking three column vectors $v_{1}, v_{2}, v_{3}$ as \[ M=\left[\begin{array}{lll} v_{1} & v_{2} & v_{3} \end{array}\right] \] Choose the set of vectors from the following options such that $\operatorname{rank}(M)=3$ ...

A matrix $M$ is constructed by stacking three column vectors $v_{1}, v_{2}, v_{3}$ as\[M=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\]Choose the set o...

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19

GATE IN 2024 | Question: 48
An ADC has a full scale voltage of $1.4 \mathrm{~V}$, resolution of $200 \mathrm{mV}$, and produces binary output data. The input signal of the ADC has a bandwidth of $500 \mathrm{MHz}$, and it samples the data at the Nyquist rate. The parallel data output is converted to a serial bit stream using a parallel-to-serial converter. The data rate at the output of the parallel-to-serial converter is Gbps (rounded off to nearest integer).

An ADC has a full scale voltage of $1.4 \mathrm{~V}$, resolution of $200 \mathrm{mV}$, and produces binary output data. The input signal of the ADC has a bandwidth of $50...

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20

GATE IN 2024 | Question: 12
Consider a filter defined by the difference equation \[ y[n]-0.5 y[n-2]=a x[n-4] \] where $x[n]$ and $y[n]$ represent the input and output, respectively. If the magnitude response of the filter at $\omega=\frac{\pi}{2}$ is $\left|H\left(\frac{\pi}{2}\right)\right|=0.5$, the value of $a$ is (rounded off to two decimal places).

Consider a filter defined by the difference equation\[y[n]-0.5 y[n-2]=a x[n-4]\]where $x[n]$ and $y[n]$ represent the input and output, respectively. If the magnitude res...

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21

GATE IN 2024 | Question: 30
Laplace transform of a signal $x(t)$ is \[ X(s)=\frac{1}{s^{2}+13 s+42} \] Let $u(t)$ be the unit step function. Choose the signal $x(t)$ from the following options if the region of convergence is $-7

Laplace transform of a signal $x(t)$ is\[X(s)=\frac{1}{s^{2}+13 s+42}\]Let $u(t)$ be the unit step function. Choose the signal $x(t)$ from the following options if the re...

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22

GATE IN 2024 | Question: 20
The voltage applied and the current drawn by a circuit are \[ \begin{array}{c} v(t)=95+200 \cos (120 \pi t)+90 \cos \left(360 \pi t-60^{\circ}\right) \mathrm{V} \\ i(t)=4 \cos \left(120 \pi t-60^{\circ}\right)+1.5 \cos \left(240 \pi t-75^{\circ}\right) \mathrm{A} \end{array} \] The average power absorbed by the circuit is W (rounded off to nearest integer).

The voltage applied and the current drawn by a circuit are\[\begin{array}{c}v(t)=95+200 \cos (120 \pi t)+90 \cos \left(360 \pi t-60^{\circ}\right) \mathrm{V} \\i(t)=4 \co...

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23

GATE IN 2024 | Question: 43
The 4-point DFTs of two sequences $x[n]$ and $y[n]$ are $X[k]=[1,-j, 1, j]$ and $Y[k]=[1,3 j, 1,-3 j]$, respectively. Assuming $z[n]$ represents the 4 -point circular convolution of $x[n]$ and $y[n]$, the value of $z[0]$ is (rounded off to nearest integer). Note: The DFT of a $N$-point sequence $x[n]$ is defined as \[ X[k]=\sum_{n=0}^{N-1} x[n] e^{\frac{-j 2 \pi n k}{N}} \]

The 4-point DFTs of two sequences $x[n]$ and $y[n]$ are $X[k]=[1,-j, 1, j]$ and $Y[k]=[1,3 j, 1,-3 j]$, respectively. Assuming $z[n]$ represents the 4 -point circular con...

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24

GATE IN 2024 | Question: 41
In the figure shown, $\mathrm{R}=4.5 \mathrm{k} \Omega, \Delta \mathrm{R}=1.5 \mathrm{k} \Omega$, and $\mathrm{INA}$ is assumed to be ideal. The equivalent resistance between $\mathrm{A}$ and $\mathrm{B}$ is $\mathrm{k} \Omega$ (rounded off to nearest integer).

In the figure shown, $\mathrm{R}=4.5 \mathrm{k} \Omega, \Delta \mathrm{R}=1.5 \mathrm{k} \Omega$, and $\mathrm{INA}$ is assumed to be ideal. The equivalent resistance bet...

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25

GATE IN 2024 | Question: 4
Match the following measuring devices with their principle of measurement. \begin{tabular}{|l|l|} \hline Measuring Device & Principle of Measurement \\ \hline (P) Optical pyrometer & (I) Variation in mutual inductance \\ \hline (Q) Thermocouple & (II) Change in resistance \\ \hline (R) Strain gauge & (III) Wavelength of radiated energy \\ ... $(\mathrm{P})-(\mathrm{II}),(\mathrm{Q})-(\mathrm{IV}),(\mathrm{R})-(\mathrm{I}),(\mathrm{S})-(\mathrm{III})$

Match the following measuring devices with their principle of measurement.\begin{tabular}{|l|l|}\hline Measuring Device & Principle of Measurement \\\hline (P) Optical py...

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26

GATE IN 2024 | Question: 53
Two magnetically coupled coils, when connected in series-aiding configuration, have a total inductance of $500 \mathrm{mH}$. When connected in series-opposing configuration, the coils have a total inductance of $300 \mathrm{mH}$. If the self-inductance of both the coils are equal, then the coupling coefficient is (rounded off to two decimal places).

Two magnetically coupled coils, when connected in series-aiding configuration, have a total inductance of $500 \mathrm{mH}$. When connected in series-opposing configurati...

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27

GATE IN 2024 | Question: 21
The current $i(t)$ drawn by a circuit is given as \[ i(t)=4+30 \cos (t)-20 \sin (t)+15 \cos (3 t)-10 \sin (3 t) \mathrm{A} \] The root-mean-square value of $i(t)$ is A (rounded off to one decimal place).

The current $i(t)$ drawn by a circuit is given as\[i(t)=4+30 \cos (t)-20 \sin (t)+15 \cos (3 t)-10 \sin (3 t) \mathrm{A}\]The root-mean-square value of $i(t)$ is A (round...

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28

GATE IN 2024 | Question: 18
The resistance of a $20 \mathrm{k} \Omega$ resistor is measured six consecutive times using an LCR meter. The first five readings are $19 \mathrm{k} \Omega, 18 \mathrm{k} \Omega, 23 \mathrm{k} \Omega, 21 \mathrm{k} \Omega$ and $17 \mathrm{k} \Omega$. If the mean of the measurements and the true value are equal, the last reading is $k \Omega$ (rounded off to nearest integer).

The resistance of a $20 \mathrm{k} \Omega$ resistor is measured six consecutive times using an LCR meter. The first five readings are $19 \mathrm{k} \Omega, 18 \mathrm{k}...

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29

GATE IN 2024 | Question: 29
In the bridge circuit shown in the figure, under balanced condition, the values of $R$ and $C$ respectively, are $1.010 \Omega$ and $19.802 \mu \mathrm{F}$ $9.901 \Omega$ and $0.505 \mu \mathrm{F}$ $19.802 \Omega$ and $1.01 \mu \mathrm{F}$ $39.604 \Omega$ and $2.02 \mu \mathrm{F}$

In the bridge circuit shown in the figure, under balanced condition, the values of $R$ and $C$ respectively, are$1.010 \Omega$ and $19.802 \mu \mathrm{F}$$9.901 \Omega$ a...

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30

GATE IN 2024 | Question: 15
The number of complex multiplications required for computing a 16-point DFT using the decimation-in-time radix-2 FFT is (in integer).

The number of complex multiplications required for computing a 16-point DFT using the decimation-in-time radix-2 FFT is (in integer).

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31

GATE IN 2024 | Question: 25
Indian Premier League has divided the sixteen cricket teams into two equal pools: Pool-A and Pool-B. Four teams of Pool-A have blue logo jerseys while the rest four have red logo jerseys. Five teams of Pool-B have blue logo jerseys while the rest three have red logo jerseys. If one team from each pool reaches the final, the probability that one team has a blue logo jersey and another has a red logo jersey is (rounded off to one decimal place).

Indian Premier League has divided the sixteen cricket teams into two equal pools: Pool-A and Pool-B. Four teams of Pool-A have blue logo jerseys while the rest four have ...

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32

GATE IN 2024 | Question: 32
In the figure shown, the positive edge triggered $\mathrm{D}$ flip-flops are initially reset to $Q=0$. The logic gates and the multiplexers have no propagation delay. After reset, a train of clock pulses (CLK) are applied. The logic-states of the inputs DIN, $\mathrm{S}$ and the clock pulses are also shown in the figure. Assuming no timing violations, the ... clock to the $5^{\text {th }}$ clock, $\mathrm{Y}_{3} \mathrm{Y}_{4} \mathrm{Y}_{5}$ is $001$ $010$ $000$ $011$

In the figure shown, the positive edge triggered $\mathrm{D}$ flip-flops are initially reset to $Q=0$. The logic gates and the multiplexers have no propagation delay. Aft...

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33

GATE IN 2024 | Question: 33
In the figure shown, $R=1 \mathrm{k} \Omega$ and $C=0.1 \mu \mathrm{F}$. For a dc gain of -10 , the $3 \mathrm{~dB}$ cut-off frequency (rounded off to one decimal place) is Assume the opamp is ideal. $159.1 \mathrm{~Hz}$ $1591.5 \mathrm{~Hz}$ $1750.7 \mathrm{~Hz}$ $175.0 \mathrm{~Hz}$

In the figure shown, $R=1 \mathrm{k} \Omega$ and $C=0.1 \mu \mathrm{F}$. For a dc gain of -10 , the $3 \mathrm{~dB}$ cut-off frequency (rounded off to one decimal place) ...

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34

GATE IN 2024 | GA Question: 7
Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder. The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is $5 / 2$ $7 / 2$ $9 / 2$ $3$

Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.The...

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35

GATE IN 2024 | GA Question: 9
Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction $X$, the planar image of the assembly appears as Figure B. Figure $\mathrm{A}$ Figure B If viewed from direction $Y$, the planar image of the assembly (Figure A) will appear as \begin{tabular}{|l|l|} \hline & \\ \hline & \\ \hline \end{tabular}

Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction $X$, the planar image of the assembly appears as Figure...

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36

GATE IN 2024 | Question: 38
Consider an ultrasonic measurement system shown in the figure. The ultrasonic transmitter $(T)$ sends a continuous wave signal $x(t)=\cos \left(2 \pi f_{1} t\right)$ volts towards an object whose vibration is modeled as $m(t)=0.5 \sin \left(2 \pi f_{2} t\right)$ volts. Neglecting the phase shift due to any other effect, the received signal at ... 25 and $\pm 250 \mathrm{~Hz}$ 0.5 and $\pm 500 \mathrm{~Hz}$ 1 and $\pm 1000 \mathrm{~Hz}$ 0.75 and $\pm 1000 \mathrm{~Hz}$

Consider an ultrasonic measurement system shown in the figure. The ultrasonic transmitter $(T)$ sends a continuous wave signal $x(t)=\cos \left(2 \pi f_{1} t\right)$ volt...

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37

GATE IN 2024 | Question: 45
Consider a unity negative feedback system with its open-loop pole-zero map as shown in the figure. If the point $s=j \alpha, \alpha>0$, lies on the root locus, the value of $\alpha$ is (rounded off to nearest integer). Note: The poles are marked with $x$ in the figure.

Consider a unity negative feedback system with its open-loop pole-zero map as shown in the figure. If the point $s=j \alpha, \alpha>0$, lies on the root locus, the value ...

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38

GATE IN 2024 | Question: 13
Consider the circuit shown in the figure. The CMOS digital logic circuit has infinite input impedance. Assume the opamp is ideal. A 1.8 V Zener diode with a minimum Zener current of $2 \mathrm{~mA}$ is used. The corresponding maximum value of resistance $\mathrm{Rz}$ is $\mathrm{k} \Omega$. (rounded off to one decimal place).

Consider the circuit shown in the figure.The CMOS digital logic circuit has infinite input impedance. Assume the opamp is ideal. A 1.8 V Zener diode with a minimum Zener ...

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39

GATE IN 2024 | Question: 11
A linear transducer is calibrated for the ranges shown in the figure. The gain of the transducer is $\mathrm{mA} /{ }^{\circ} \mathrm{C}$ (rounded off to two decimal places).

A linear transducer is calibrated for the ranges shown in the figure. The gain of the transducer is $\mathrm{mA} /{ }^{\circ} \mathrm{C}$ (rounded off to two decimal plac...

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40

GATE IN 2024 | Question: 14
Figure shows an amplifier using an NMOS transistor. Assume that the transistor is in saturation with device parameters, $\mu_{n} C_{o x}=250 \mu \mathrm{A} / \mathrm{V}^{2}$, threshold voltage $\mathrm{VT}_{\mathrm{T}}=0.65 \mathrm{~V}$ and $\mathrm{W} / \mathrm{L}=4$. Ignore the channel length modulation effect. The drain current of the transistor at the operating point is $\mu \mathrm{A}$ (rounded off to nearest integer).

Figure shows an amplifier using an NMOS transistor. Assume that the transistor is in saturation with device parameters, $\mu_{n} C_{o x}=250 \mu \mathrm{A} / \mathrm{V}^{...

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