List of top Questions asked in GATE EC

Consider the following series:
(i) \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \)
(ii) \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \)
(iii) \( \sum_{n=1}^{\infty} \frac{1}{n!} \)
Choose the correct option.

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

Consider a frequency-modulated (FM) signal \[ f(t) = A_c \cos(2\pi f_c t + 3 \sin(2\pi f_1 t) + 4 \sin(6\pi f_1 t)), \] where \( A_c \) and \( f_c \) are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency \( f_1 \) is in Hz, and assume that \( f_c>100 f_1 \). The peak frequency deviation of the FM signal in Hz is _________.

Consider an additive white Gaussian noise (AWGN) channel with bandwidth \( W \) and noise power spectral density \( \frac{N_0}{2} \). Let \( P_{{av}} \) denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity \( C \) on the bandwidth \( W \) (keeping \( P_{{av}} \) and \( N_0 \) fixed)?

The Nyquist plot of a system is given in the figure below. Let \( \omega_P, \omega_Q, \omega_R, \) and \( \omega_S \) be the positive frequencies at the points \( P, Q, R, \) and \( S \), respectively. Which one of the following statements is TRUE?

Consider a continuous-time, real-valued signal \( f(t) \) whose Fourier transform \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \exp(-j \omega t) \, dt { exists.} \] Which one of the following statements is always TRUE?

Consider a part of an electrical network as shown below. Some node voltages, and the current flowing through the \( 3\,\Omega \) resistor are as indicated.
The voltage (in Volts) at node \( X \) is _________.

Let \( i_C, i_L, \) and \( i_R \) be the currents flowing through the capacitor, inductor, and resistor, respectively, in the circuit given below. The AC admittances are given in Siemens (S).
Which one of the following is TRUE?

A simplified small-signal equivalent circuit of a BJT-based amplifier is given below.
The small-signal voltage gain \( \frac{V_o}{V_S} \) (in V/V) is _________.

The ideal BJT in the circuit given below is biased in the active region with a \( \beta \) of 100. If \( I_B \) is 10 µA, then \( V_{CE} \) (in Volts, rounded off to two decimal places) is _________.

A 3-input majority logic gate has inputs \( X \), \( Y \), and \( Z \). The output \( F \) of the gate is logic ‘1’ if two or more of the inputs are logic ‘1’. The output \( F \) is logic ‘0’ if two or more of the inputs are logic ‘0’.
Which one of the following options is a Boolean expression of the output \( F \)?

Consider the function \( f: \mathbb{R} \to \mathbb{R} \), defined as \[ f(x) = 2x^3 - 3x^2 - 12x + 1. \] Which of the following statements is/are correct? (Here, \( \mathbb{R} \) is the set of real numbers.)

Let \( x[n] \) be a discrete-time signal whose \( z \)-transform is \( X(z) \). Which of the following statements is/are TRUE?

Consider a message signal \( m(t) \) which is bandlimited to \( [-W, W] \), where \( W \) is in Hz. Consider the following two modulation schemes for the message signal:
• Double sideband-suppressed carrier (DSB-SC): \[ f_{DSB}(t) = A_c m(t) \cos(2\pi f_c t) \] • Amplitude modulation (AM): \[ f_{AM}(t) = A_c \left( 1 + \mu m(t) \right) \cos(2\pi f_c t) \] Here, \( A_c \) and \( f_c \) are the amplitude and frequency (in Hz) of the carrier, respectively. In the case of AM, \( \mu \) denotes the modulation index. Consider the following statements:
(i) An envelope detector can be used for demodulation in the DSB-SC scheme if \( m(t)>0 \) for all \( t \).
(ii) An envelope detector can be used for demodulation in the AM scheme only if \( m(t)>0 \) for all \( t \).
Which of the following options is/are correct?

Which of the following statements is/are TRUE with respect to an ideal op-amp?

Which of the following statements is/are TRUE with respect to ideal MOSFET-based DC-coupled single-stage amplifiers having finite load resistors?

Which of the following can be used as an n-type dopant for silicon?
Select the correct option(s).

The function \( y(t) \) satisfies \[ t^2 y''(t) - 2t y'(t) + 2y(t) = 0, \] where \( y'(t) \) and \( y''(t) \) denote the first and second derivatives of \( y(t) \), respectively. Given \( y'(0) = 1 \) and \( y'(1) = -1 \), the maximum value of \( y(t) \) over \( [0, 1] \) is _________ (rounded off to two decimal places).

The generator matrix of a \( (6,3) \) binary linear block code is given by

The minimum Hamming distance \( d_{{min}} \) between codewords equals (answer in integer).

All the components in the bandpass filter given below are ideal. The lower -3 dB frequency of the filter is 1 MHz. The upper -3 dB frequency (in MHz, rounded off to the nearest integer) is _________.

A 4-bit weighted-resistor DAC with inputs \( b_3, b_2, b_1, \) and \( b_0 \) (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bits are logic ‘1’ and open otherwise. When the input \( b_3b_2b_1b_0 \) changes from 1110 to 1101, the magnitude of the change in the output voltage \( V_o \) (in mV, rounded off to the nearest integer) is _________.

Let \( G(s) = \frac{1}{10s^2} \) be the transfer function of a second-order system. A controller \( M(s) \) is connected to the system \( G(s) \) in the configuration shown below.

Consider the following statements.

(i) There exists no controller of the form \( M(s) = \frac{K_I}{s} \), where \( K_I \) is a positive real number, such that the closed-loop system is stable.
(ii) There exists at least one controller of the form \( M(s) = K_p + sK_D \), where \( K_p \) and \( K_D \) are positive real numbers, such that the closed-loop system is stable.

Which one of the following options is correct?

Consider the polynomial \[ p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40. \] Let \( (L, I, R) \) be defined as follows: \[ L \text{ is the number of roots of } p(s) \text{ with negative real parts.} \] \[ I \text{ is the number of roots of } p(s) \text{ that are purely imaginary.} \] \[ R \text{ is the number of roots of } p(s) \text{ with positive real parts.} \] Which one of the following options is correct?

Consider a continuous-time finite-energy signal \( f(t) \) whose Fourier transform vanishes outside the frequency interval \( [-\omega_c, \omega_c] \), where \( \omega_c \) is in rad/sec.

The signal \( f(t) \) is uniformly sampled to obtain \( y(t) = f(t) p(t) \). Here, \[ p(t) = \sum_{n=-\infty}^{\infty} \delta(t - \tau - nT_s), \] with \( \delta(t) \) being the Dirac impulse, \( T_s > 0 \), and \( \tau > 0 \). The sampled signal \( y(t) \) is passed through an ideal lowpass filter \( h(t) = \omega_c T_s \frac{\sin(\omega_c t)}{\pi \omega_c t} \) with cutoff frequency \( \omega_c \) and passband gain \( T_s \).

The output of the filter is given by _________.

In the circuit below, \( M_1 \) is an ideal AC voltmeter and \( M_2 \) is an ideal AC ammeter. The source voltage (in Volts) is \( v_s(t) = 100 \cos(200t) \). What should be the value of the variable capacitor \( C \) such that the RMS readings on \( M_1 \) and \( M_2 \) are 25 V and 5 A, respectively?