List of top Questions asked in GATE EC

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 $.

An ideal p-n junction germanium diode has a reverse saturation current of 10 $\mu A$ at 300 K. The voltage (in Volts, rounded off to two decimal places) to be applied across the junction to get a forward bias current of 100 mA at 300 K is _________. (Consider the Boltzmann constant $ k_B = 1.38 \times 10^{-23} { J/K} $ and the charge of an electron $ e = 1.6 \times 10^{-19} { C} $.

The random variable $ X $ takes values in $ \{-1, 0, 1\} $ with probabilities \[ P(X = -1) = P(X = 1) = \alpha \quad {and} \quad P(X = 0) = 1 - 2\alpha, \quad 0<\alpha<\frac{1}{2}. \] Let $ g(\alpha) $ denote the entropy of $ X $ (in bits), parameterized by $ \alpha $. Which of the following statements is/are TRUE?

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?

The electron mobility $ \mu_n $ in a non-degenerate germanium semiconductor at 300 K is 0.38 m$^2$/Vs. The electron diffusivity $ D_n $ at 300 K (in cm$^2$/s, rounded off to the nearest integer) is _________.

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.

A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and a full scale voltage of 3.3 V. For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB, rounded off to two decimal places) and the data rate (in Mbps) at the output of the ADC are _________, respectively.

A source transmits symbol $ S $ that takes values uniformly at random from the set $ \{-2, 0, 2\} $. The receiver obtains $ Y = S + N $, where $ N $ is a zero-mean Gaussian random variable independent of $ S $. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $ S $. Suppose the probability of symbol estimation error $ P_e $ is expressed as follows: \[ P_e = \alpha P(N>1), \] where $ P(N>1) $ denotes the probability that $ N $ exceeds 1. What is the value of $ \alpha $?

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

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 $?

The table lists the top 5 nations according to the number of gold medals won in a tournament; also included are the number of silver and the bronze medals won by them. Based only on the data provided in the table, which one of the following statements is INCORRECT?

A stick of length one meter is broken at two locations at distances of $ b_1 $ and $ b_2 $ from the origin (0), as shown in the figure. Note that $ 0<b_1<b_2<1 $. Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?
Note: All lengths are in meter. The figure shown is representative.