GATE EC question library

The current density in a medium is given by $${\overrightarrow J= \frac{400 \sin\theta}{2\pi(r^2+4)} \hat {a}_r \: Am^{-2}}$$ The total current and the average current density flowing through the portion of a spherical surface $r = 0.8 \: m$, $\frac{\pi}{12} \leq \theta \leq \frac{\pi}{4}$, $0 \leq \phi \leq 2\pi$ are given, respectively, by

An antenna pointing in a certain direction has a noise temperature of $50 \: K$. The ambient temperature is $290 \: K$. The antenna is connected to a pre-amplifier that has a noise figure of $2$ dB and an available gain of $40$ dB over an effective bandwidth of $12$ MHz. The effective input noise temperature $T_e$ for the amplifier and the noise …

Two lossless X-band horn antennas are separated by a distance of $200 \lambda$. The amplitude reflection coefficients at the terminals of the transmitting and receiving antennas are $0.15$ and $0.18$,respectively. The maximum directives of the transmitting and receiving antennas (over the isotropic antenna) are $18$ dB and $22$ dB, respectively. …

The electric field of a uniform plane wave traveling along the negative $z$ direction is given by the following equation: $${\overrightarrow {E}^i_w= ( \hat {a}_x+ j\hat {a}_y)E_0e^{jkz}}$$ This wave is incident upon a receiving antenna placed at the origin and whose radiated electric field towards the incident wave is given by the following equation: …

The far-zone power density radiated by a helical antenna is approximated as: $${\overrightarrow W_{rad}= \overrightarrow W_{average} \approx \hat {a_r}C_0\frac{1}{r^2}\cos^4\theta}$$ The radiated power density is symmetrical with respect to $\phi$ and exists only in the upper hemisphere: $0\leq\theta\leq\frac{\pi}{2}; \: 0\leq\phi\leq 2\pi$; $C_0$ is a …

Based on the given statements, select the appropriate option with respect to grammar and usage. Statements The height of Mr. $X$ is $6$ feet. The height of Mr. $Y$ is $5$ feet.

The students ___________ the teacher on the teachers’ day for twenty years of dedicated teaching.

After India’s cricket world cup victory in $1985$, Shrotria who was playing both tennis and cricket till then, decided to concentrate only on cricket. And the $\underline{\text{rest is history}}$. What does the underlined phrase mean in this context?

Given ($9$ inches)$^{1/2}=(0.25$ yards)$^{1/2}$, which one of the following statements is $TRUE$?

$S$, $M$, $E$ and $F$ are working in the shifts in a team to finish a project. $M$ works with twice the efficiency of others but for half as many days as $E$ worked. $S$ amd $M$ have $6$ hour shifts in a day, whereas $E$ and $F$ have $12$ hours shifts. What is the ratio of contribution of $M$ to contribute of $E$ in the project?

The Venn diagram shows the preference of the students population for leisure activities. From the data given, the number of students who like to read books or play sports is _____.

Social science disciplines were existence in an amorphous form until the colonial period when they were institutionalized. In varying degrees, they were intended to further the colonial interest. In the time of globalization and the economic rise of postcolonial countries like India, conventional ways of knowledge production have become obsolete? Which of …

Two and a quarter hours back, when seen in a mirror, the reflection of a wall clock without umber markings seemed to show $1.30$. What is the actual current time shown by the clock?

$M$ and $N$ start from the same location. $M$ travels $10$ km east ad then $10$ km North-East. $N$ travels $5$ km South and then $4$ km South-East. What is the shortest distance (in km) between $M$ and $N$ at the end of their travel?

A wire of length $340$ mm is to be cut into two parts. One of the parts is to be made into a square and the other into a rectangle where sides are in the ratio of $1:2$. What is the length of the square (in mm) such that the combined area of the square and the rectagle is a $\textbf{MINIMUM?}$

The value of $x$ for which the matrix $A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ -6&-4 &-9+x \end{bmatrix}$ has zero as an eigenvalue is ________

Consider the complex valued function $f(z)=2z^{3}+b\mid z \mid^{3}$ where $z$ is a complex variable. The value of $b$ for which function $f(z)$ is analytic is _________

As $x$ varies from $-1$ to $+3$, which one of the following describes the behaviour of the function $f(x)=x^{3}-3x^{2}+1?$

How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians?

Consider the time-varying vector $\textbf{I}=\hat{x}15\cos(\omega t)+\hat{y}5\sin(\omega t)$ in Cartesian coordinates, where $\omega> 0$ is a constant. When the vector magnitude $\mid \textbf{I} \mid$ is at its minimum value, the angle $\theta$ that $\textbf{I}$ makes with the $x$ axis (in degrees, such that $0\leq \theta \leq 180)$ ________