List of top Questions asked in GATE EC

For the Boolean function: \[ F(A, B, C, D) = \Sigma m(0, 2, 5, 7, 8, 10, 12, 13, 14, 15), \] the essential prime implicants are:

Examples of mirror and water reflections are shown in the figures below:

An object appears as the following image after first reflecting in a mirror and then reflecting on water:

The original object is:

In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of $R_1$ and the corresponding frequency of oscillation are $\_\_\_\_$.

Consider a system $S$ represented in state space as: \[ \frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix} x + \begin{bmatrix} 1 \\ 0 \end{bmatrix} r, \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix} x. \] Which of the state space representations given below has/have the same transfer function as that of $S$?

A full-scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1 W, and the total noise and distortion normalized power is $10 \, \mu {W}$. The effective number of bits (rounded off to the nearest integer) of the ADC is $\_\_\_\_$.

The propagation delay of the $2 \times 1$ MUX shown in the circuit is $10 \, {ns}$. Consider the propagation delay of the inverter as $0 \, {ns}$. If $S$ is set to 1, then the output $Y$ is $\_\_\_\_$.

In the network shown below, maximum power is to be transferred to the load $R_L$. The value of $R_L$ (in $\Omega$) is $\_\_\_\_$.

Consider two continuous-time signals $x(t)$ and $y(t)$ as shown below. If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_$.

A causal and stable LTI system with impulse response $h(t)$ produces an output $y(t)$ for an input signal $x(t)$. A signal $x(0.5t)$ is applied to another causal and stable LTI system with impulse response $h(0.5t)$. The resulting output is:

Let $ X(t) = A \cos(2\pi f_o t + \theta) $ be a random process, where amplitude $ A $ and phase $ \theta $ are independent of each other, and are uniformly distributed in the intervals [−2, 2] and [0, 2π], respectively. $ X(t) $ is fed to an 8-bit uniform mid-rise type quantizer.
Given that the autocorrelation of $ X(\tau) $ is \[ R_x(\tau) = \frac{2}{3} \cos(2\pi f_o \tau), \] the signal to quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is ___.

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X≥Y is_____.

In a number system of base r, the equation x²-12x+37=0 has x=8 of its solutions. The value of r is____.

In the context of Bode magnitude plots, $40 \, {dB/decade}$ is the same as:

In the circuit shown below, the transistors $M_1$ and $M_2$ are biased in saturation. Their small signal transconductances are $g_{m1}$ and $g_{m2}$, respectively. Neglect body effect, channel length modulation, and intrinsic device capacitances. Assuming that capacitor $C_1$ is a short circuit for AC analysis, the exact magnitude of small signal voltage gain $\left|\frac{v_{{out}}}{v_{{in}}}\right|$ is $\_\_\_\_$.

The sequence of states ($Q_1 Q_0$) of the given synchronous sequential circuit is $\_\_\_\_$.

A satellite attitude control system, as shown below, has a plant with transfer function: $ G(s) = \frac{1}{s^2}, $ cascaded with a compensator: $ C(s) = \frac{K(s + \alpha)}{s + 4}, $ where $ K $ and $ \alpha $ are positive real constants. In order for the closed-loop system to have poles at $ -1 \pm j\sqrt{3} $, the value of $ \alpha $ must be $\_\_\_\_\_$.

A lossless transmission line with characteristic impedance Z_o = 50 Ω is terminated with an unknown load. The magnitude of the reflection co-efficient is |T| = 0.6. As one moves towards the generator from the load, the maximum value of the input impedance magnitude looking towards the load (in Ω) is____.

An NMOS transistor operating in the linear region has $I_{DS} = 5 \, \mu{A}$ at $V_{DS} = 0.1 \, {V}$. Keeping $V_{GS}$ constant, the $V_{DS}$ is increased to $1.5 \, {V}$. Given that: \[ \mu_n C_{ox} \frac{W}{L} = 50 \, \mu{A}/{V}^2, \] the transconductance at the new operating point (in $\mu{A}/{V}$, rounded off to two decimal places) is $\_\_\_\_$.

Five years ago, the ratio of Aman's age to his father's age was 1:4, and five years from now, the ratio will be 2:5. What was his father's age when Aman was born?

Let R and R³ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of a for which the set of vectors
{[2-3 α], [3 -1 3], [1 −5 7]}
does not form a basis of R³ is____.

Consider the matrix: \[ \begin{bmatrix} 1 & k \\ 2 & 1 \end{bmatrix}, \] where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

In the circuit shown, the $n:1$ step-down transformer and the diodes are ideal. The diodes have no voltage drop in forward-biased condition. If the input voltage (in Volts) is $V_s(t) = 10\sin\omega t$ and the average value of load voltage $V_L(t)$ (in Volts) is $2.5/\pi$, the value of $n$ is $\_\_\_\_$.

For the closed-loop amplifier circuit shown below, the magnitude of open-loop low-frequency small signal voltage gain is 40. All the transistors are biased in saturation. The current source $I_{SS}$ is ideal. Neglect body effect, channel length modulation, and intrinsic device capacitances. The closed-loop low-frequency small signal voltage gain $\frac{v_{out}}{v_{in}}$ (rounded off to three decimal places) is: \begin{center} \includegraphics[width=6cm]{18.png} \end{center}

Consider a lossless transmission line terminated with a short circuit as shown in the figure below. As one moves towards the generator from the load, the normalized impedances $z_{inA}$, $z_{inB}$, $z_{inC}$, and $z_{inD}$ (indicated in the figure) are: \begin{center} \includegraphics[width=8cm]{15.png} \end{center}

Four identical cylindrical chalk-sticks, each of radius $ r = 0.5 \, {cm} $ and length $ l = 10 \, {cm} $, are bound tightly together using a duct tape as shown in the following figure.

The width of the duct tape is equal to the length of the chalk-stick. The area (in cm$^2$) of the duct tape required to wrap the bundle of chalk-sticks once, is: