GATE EC question library

Let $Q(\sqrt{\gamma})$ be the BER of a BPSK system over an AWGN channel with two-sided noise power spectral density $N_{0}/2.$ The parameter $\gamma$ is a function of bit energy and noise power spectral density. A system with two independent and identical AWGN channels with noise power spectral density $N_{0}/2$ is shown in the figure. The BPSK …

A fair coin is tossed repeatedly until a ‘Head’ appears for the first time. Let $L$ be the number of tosses to get this first ‘Head’. The entropy $H(L)$ in bits is _____.

In spherical coordinates, let $\hat{a_{\theta}},\hat{a_{\phi}}$ denote unit vectors along the $\theta,\phi$ directions. $$\textbf{E} = \dfrac{100}{r}\sin\theta \cos (\omega t – \beta r)\hat{a_{\theta}}\: V/m$$ and $$\textbf{H} = \dfrac{0.265}{r}\sin\theta \cos (\omega t – \beta r)\hat{a_{\phi}} \:A/m$$ represent the electric and magnetic field components …

For a parallel plate transmission line, let $v$ be the speed of propagation and $Z$ be the characteristic impedance. Neglecting fringe effects, a reduction of the spacing between the plates by a factor of two results in

The input impedance of a $\dfrac{\lambda}{8}$ section of a lossless transmission line of characteristic impedance $50\: \Omega$ is found to be real when the other end is terminated by a load $Z_{L} (=R + jX)\Omega.$ If $X$ is $30\:\Omega ,$ the value of $R ( \text{in} \:\Omega )$ is _______.

Choose the most appropriate word from the options given below to complete the following sentence. Communication and interpersonal skills are__________ important in their own ways.

Which of the options given below best completes the following sentence? She will feel much better if she ________________.

Choose the most appropriate pair of words from the options given below to complete the following sentence. She could not _________ the thought of _________ the election to her bitter rival.

A regular die has six sides with numbers $1$ to $6$ marked on its sides. If a very large number of throws show the following frequencies of occurrence: $1 \rightarrow 0.167$; $2 \rightarrow 0.167$; $3 \rightarrow 0.152$; $4 \rightarrow 0.166$; $5 \rightarrow 0.168$; $6 \rightarrow 0.180$. We call this die

Fill in the missing number in the series. $2$ $3$ $6$ $15$ __________ $157.5$ $630$

Find the odd one in the following group $Q,W,Z,B \qquad B,H,K,M \qquad W,C,G,J \qquad M,S,V,X$

Lights of four colors (red, blue, green, yellow) are hung on a ladder. On every step of the ladder, there are two lights. If one of the lights is red, the other light on that step will always be blue. If one of the lights on a step is green, the other light on that step will always be yellow. Which of the following statements is not necessarily correct?

The sum of eight consecutive odd numbers is $656$. The average of four consecutive even numbers is $87$. What is the sum of the smallest odd number and second largest even number?

The total exports and revenues from the exports of a country are given in the two charts shown below. The pie chart for exports shows the quantity of each item exported as a percentage of the total quantity of exports. The pie chart for the revenues shows the percentage of the total revenue generated through export of each item. The total quantity of exports …

It takes $30$ minutes to empty a half-full tank by draining it at a constant rate. It is decided to simultaneously pump water into the half-full tank while draining it. What is the rate at which water has to be pumped in so that it gets fully filled in $10$ minutes?

The determinant of matrix $A$ is $5$ and the determinant of matrix B is $40$. The determinant of matrix $AB$ is ________

Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less than $100$. The expectation, $E[X]$, is ________.

For $0 \leq t < \infty ,$ the maximum value of the function $f(t)= e^{-t}-2e^{-2t}$ occurs at

The value of $$\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$$ is

If the characteristic equation of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$$ has two equal roots, then the value of $\alpha$ are