List of top Engineering Mathematics Questions asked in GATE Electronics and Communication Engineering

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).


 

Consider a system where \( x_1(t), x_2(t), \) and \( x_3(t) \) are three internal state signals and \( u(t) \) is the input signal. The differential equations governing the system are given by: \[ \frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} u(t). \] Which of the following statements is/are TRUE?
Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle \( C \) in the complex plane is/are TRUE?

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?


 

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 _________.
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 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.)

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 _________.

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 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.

Consider the matrix \( A \) below: \[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix} \] For which of the following combinations of \( \alpha, \beta, \) and \( \gamma \), is the rank of \( A \) at least three? (i) \( \alpha = 0 \) and \( \beta = \gamma \neq 0 \). 
(ii) \( \alpha = \beta = \gamma = 0 \). 
(iii) \( \beta = \gamma = 0 \) and \( \alpha \neq 0 \). 
(iv) \( \alpha = \beta = \gamma \neq 0 \).
 

Let \( f(t) \) be a periodic signal with fundamental period \( T_0>0 \). Consider the signal \[ y(t) = f(\alpha t), \quad {where} \, \alpha>1. \] The Fourier series expansions of \( f(t) \) and \( y(t) \) are given by \[ f(t) = \sum_{k=-\infty}^{\infty} c_k e^{j \frac{2\pi}{T_0} k t}, \quad y(t) = \sum_{k=-\infty}^{\infty} d_k e^{j \frac{2\pi}{T_0 \alpha} k t}. \] Which of the following statements is/are TRUE?
Consider a non-negative function \( f(x) \) which is continuous and bounded over the interval [2, 8]. Let \( M \) and \( m \) denote, respectively, the maximum and the minimum values of \( f(x) \) over the interval. Among the combinations of \( \alpha \) and \( \beta \) given below, choose the one(s) for which the inequality \[ \beta \leq \int_2^8 f(x) \, dx \leq \alpha \] is guaranteed to hold.
Find the next term in the sequence: 3, 9, 19, 33, _
Given the matrix: \[A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}\]

 If rank(A) is at least 3, then what are the possible values of \( \alpha, \beta, \gamma \)?

Two fair dice are rolled, and the random variable \( X \) denotes the sum of the outcomes. What is the expected value of \( X \)?
If a complex function \( f(z) \) is analytic everywhere inside a closed contour \( C \) with anti-clockwise direction, then which of the following statements are correct?
Consider three Boolean variables \( x, y, z \). A majority function outputs 1 if the majority of its inputs are 1, otherwise, it outputs 0. Derive the Boolean expression for the majority function and simplify it using Boolean algebra.