GATE EC question library

The contour $C$ given below is on the complex plane $z=x+j y,$ where $j=\sqrt{-1}.$ The value of the integral $\displaystyle{}\dfrac{1}{\pi j}\oint _{C}\dfrac{dz}{z^{2}-1}$ is _______.

Let $r=x^{2}+y-z$ and $z^{3}-xy+yz+y^{3}=1.$ Assume that $x$ and $y$ are independent variables. At $\left( x,y,z \right)=\left ( 2,-1,1 \right ),$ the value (correct to two decimal places) of $\dfrac{\partial r}{\partial x}$ is _________ .

Consider the network shown below with $R_{1}=1\:\Omega,R_{2}=2\:\Omega$ and $R_{3}=3\:\Omega.$ The network is connected to a constant voltage source of $11\:V$. The magnitude of the current (in amperes, accurate to two decimal places ) through the source is _________.

A band limited low-pass signal $x(t)$ of bandwidth $5\:kHz$ is sampled at a sampling rate $f_{s}$.The signal $x(t)$ is reconstructed using the reconstruction filter $H(f)$ whose magnitude response is shown below: The minimum sampling rate $f_{s}(\text{in}\: kHz)$ for perfect reconstruction of $x(t)$ is ________.

Let $X\left[ k \right ] = k + 1,0\leq k\leq 7$ be $8$-point $\:\text{DFT}\:$ of a sequence $x[n]$. where $X\left [ k \right ]=\sum_{n=0}^{N-1}x \left [ n \right ]e^{-j2\pi nk/N}$. The value (correct to two decimal places) of $\sum_{n=0}^{3}x \left [ 2n \right ]$ is ________.

She has a sharp tongue and it can occasionally turn__________________

I __________ made arrangements had I _________informed earlier.

In the summer, water consumption is known to decrease overall by $25 \%$. A Water Board official states that in the summer household consumption decreases by $20 \%$, while other consumption increases by $70 \%$. Which of the following statement is correct?

$40 \%$ of deaths on city roads may be attributed to drunken driving. The number of degrees needed to represent this as a slice of a pie chart is

Some tables are shelves. Some shelves are chairs. All chairs are benches. Which of the following conclusion can be deduced from the preceding sentences? Only $\text{i}$ Only $\text{ii}$ Only $\text{ii}$ and $\text{iii}$ Only $\text{iv}$

“If you are looking for a history of India, or for an account of the rise and fall of the British Raj, or for all reason of the cleaving of the subcontinent into two mutually antagonistic parts and the effects this mutilation will have in the respective sections, and ultimately on Asia, you will not find it in these pages; for though I have spent a lifetime …

S, T, U, V, W, X, Y and Z are seated around a circular table .T’s neighbors are Y and V. Z is seated third to the left of T and second to the right of S. U’s neighbors are S and Y; and T and W are not seated opposite each other .Who is third to the left of V?

Trucks ($10$ m long) and cars ($5$ m long) go on a single lane bridge. There must be a gap of at least $20$ m after each truck and a gap of at least $15$ m after each car. Trucks and cars travel at a speed of $36$ km/h. If cars and trucks go alternately, what is the maximum number of vehicles that can use the bridge in one hour?

There are $3$ Indians and $3$ Chinese in a group of $6$ people. How many subgroups of this group can we choose so that every subgroup has at least one Indian?

A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at $25$ m intervals in this plot. The path from $P$ to $Q$ is best described by

Consider the 5 $\times$ 5 matrix $$\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$$ It is given that A has only one real eigenvalue. Then the real eigenvalue of A is

The rank of the matrix $$\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$$ is

Consider the following statements about the linear dependence of the real valued function $y_{1}=1,y_{2}=x$ and $y_{3}=x^{2}$ over the field of real numbers. Which one among the following is correct? Both I and II are true Both I and III are true Both II and IV are true Both III and IV are true

Three fair cubical dice are thrown simultaneously . The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place)________.

Consider the following statements for continuous-time linear time invariant (LTI) systems. There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane. There is no causal and BIBO stable with a pole in the right half of the complex plane. Which one among the following is correct?