GATE EC question library

Consider the square region $R$ in the $X-Y$ plane as shown with the dark shading in the Figure. The value of $\iint_{R}\left(x^{2}+y^{2}-1\right) d x d y$ is $\_\_\_\_$ .(rounded off to two decimal places)

Consider an ideal OP-AMP circuit as shown in the Figure. The resistances $R_{1}=R_{2}=R_{3}=R_{4}=50 \mathrm{~k} \Omega$. The magnitude of the closed loop gain is $\_\_\_\_$ . (rounded off to two decimal places)

The average bit error rate at the input of a $(7,4,1)$ Hamming decoder is $0.10$. The probability that the decoder will fail to decode a received word correctly is $\_\_\_\_\_$ (rounded off to two decimal places)

Consider that the concentration of electrons in a semiconductor bar varies linearly from $2 \times 10^{17} \mathrm{~cm}^{-3}$ at $x=1 \mu \mathrm{~m}$ to $1 \times 10^{16} \mathrm{~cm}^{-3}$ at $x=4 \mu \mathrm{~m}$ along the $x$-direction. Assume that the concentration of electrons is not varying along other directions (that is along $y$ - and …

Consider the ideal diodes $\text{D1}$ and $\text{D2}$ as shown in the Figure with cut-in voltage $V_{\gamma}=0$ Volt and $v_{i}(t)$ is in Volt. The maximum voltage (Volt) of the output $v_{o}(t)$ is $\_\_\_\_$ . (rounded off to two decimal places)

Here are two analogous groups, Group-$I$ and Group-$II$, that list words in their decreasing order of intensity. Identify the missing word in Group-$II$. Group-$I$: Abuse $\rightarrow$ Insult $\rightarrow$ Ridicule Group-$II$: ________ $\rightarrow$ Praise $\rightarrow$ Appreciate

Had I learnt acting as a child, I _________ a famous film star. Select the most appropriate option to complete the above sentence.

The $12$ musical notes are given as $C, C^{\#}, D, D^{\#}, E, F, F^{\#}, G, G^{\#}, A, A^{\#}$, and $B$. Frequency of each note is $\sqrt[12]{2}$ times the frequency of the previous note. If the frequency of the note $C$ is $130.8 \: \mathrm{Hz}$, then the ratio of frequencies of notes $F^{\#}$ and $C$ is:

The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after 'Iteration $n$ ' is: Note: The figures shown are representative.

Which one of the following plots represents $f(x)=-\frac{|x|}{x}$, where $x$ is a nonzero real number? Note: The figures shown are representative.

Identify the option that has the most appropriate sequence such that a coherent paragraph is formed: $\mathrm{P} \rightarrow \mathrm{Q} \rightarrow \mathrm{R} \rightarrow \mathrm{S}$ $\mathrm{Q} \rightarrow \mathrm{S} \rightarrow \mathrm{R} \rightarrow \mathrm{P}$ $\mathrm{R} \rightarrow \mathrm{S} \rightarrow \mathrm{Q} \rightarrow \mathrm{P}$ …

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.

Eight students ($\mathrm{P, \:Q, \:R, \:S, \:T, \: U, \:V,}$ and $\mathrm{W}$) are playing musical chairs. The figure indicates their order of position at the start of the game. They play the game by moving forward in a circle in the clockwise direction. After the $1^{\text{st }}$ round, $4^{\text{th }}$ student behind $\mathrm{P}$ leaves the game. …

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? …

An organization allows its employees to work independently on consultancy projects but charges an overhead on the consulting fee. The overhead is $20 \%$ of the consulting fee, if the fee is up to ₹$5,00,000$. For higher fees, the overhead is ₹$1,00,000$ plus $10 \%$ of the amount by which the fee exceeds ₹ $5,00,000$. The government charges a …

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

Consider the following series: $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ $\sum_{n=1}^{\infty} \frac{1}{n!}$ 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 $$f(t)=A_{c} \cos \left(2 \pi f_{c} t+3 \sin \left(2 \pi f_{1} t\right)+4 \sin \left(6 \pi f_{1} t\right)\right)$$ where $A_{c}$ and $f_{c}$ are, respectively, the amplitude and frequency (in $\mathrm{Hz}$ ) of the carrier waveform. The frequency $f_{1}$ is in $\mathrm{Hz}$ , and assume that …

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $\frac{N_{0}}{2}$. Let $P_{a v}$ 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 $P_{a v}$ and $N_{0}$ fixed)?