List of top Questions asked in GATE CE, GATE CY

Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ________.
In the circuit shown below, \(R\) is \(1\ \text{k}\Omega\) and \(R_f\) is \(10\ \text{k}\Omega\). If \(V_{in}\) is \(100\) mV and the op-amps have supply voltages of \(\pm15\) V, then \(V_{out}\) is ______ V.
An amplitude modulated voltage signal \(x(t)\) drives a load of \(1\,\Omega\).
\[ x(t) = K\cos(300\pi t) + L\cos(240\pi t) + L\cos(360\pi t) \]
If the efficiency is \(60\%\) and the carrier power is \(50\) W, the value of \(L\) is ______ V.
A single phase step-down transformer has a high voltage primary winding and a low voltage secondary winding. The secondary winding is connected to a load impedance \(Z_2\). The impedance \(Z_2\) when referred to the primary winding is denoted by \(Z_2'\). Which of the following conditions is true?
The Kelvin double bridge is used to measure ______ valued resistances and has ______ resistors in its construction.
A digital counter increments its count every \(0.1\) \(\mu\text{s}\). The counter is used in an instrument to measure frequency. The counter is operated for one period of the input signal to the instrument to measure its frequency. The input signal is approximately at \(100\) kHz. The maximum error in the measured frequency is kHz (rounded off to the nearest integer).
In a linear element, the measured current is \(100\) mA \(\pm 2.5\%\) and the measured voltage is \(5\) V \(\pm 5\%\). Which of the following options show(s) the consumed power?
If the definition of SI unit 'X' is required to define SI unit 'Y', the dependency is indicated as \(X \rightarrow Y\). A valid order of dependencies, as per present convention, is __________.
A single phase transformer has a rating of 10 kVA, 50 Hz, 1100 V/ 220 V. In an open circuit test with the high voltage side open, the following are obtained:
Voltage applied to the low voltage winding = 220 V
Measured power = 363 W
Measured current = 2.75 A
Magnetizing reactance \(X_m\) referred to the high voltage winding is ________ \(\Omega\). (rounded off to the nearest integer)
A closed Lissajous pattern observed on an oscilloscope in the \(X-Y\) mode has three horizontal tangencies and two vertical tangencies in steady state. If the signal frequency on channel-\(X\) is 600 Hz, then the signal frequency on channel-\(Y\) is __________ Hz.
Consider a matrix
\[ A=\begin{bmatrix}1 & 0 & 2\\ -1 & 1 & 0\\ 0 & 1 & 2\end{bmatrix} \]
Let \(b=\begin{bmatrix}1\\3\\0\end{bmatrix}\) and \(x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\).
The number of solutions to the linear system of equations \(Ax=b\) is _____.
A matrix \(P\in\mathbb{R}^{3\times3}\) satisfies \(P^3-4P^2+5P-2I=0\), where \(I\) is a \(3\times3\) identity matrix.
The set of all possible eigenvalues of matrix \(P\) is ____.
In a series \(RL\) circuit shown in the figure, the current through the resistor lags the AC source voltage by \(45^{\circ}\). Let \(R = 100\pi\) \(\Omega\) and \(L = 2\) H.

The frequency of the source is ____ Hz. (rounded off to the nearest integer)
A data set is given to be \([1, 2, 0, -1, -3, 1, 2, 0, 1]\).
The median of the data set is ____. (rounded off to the nearest integer)
Which one among the following statements is true for the function \(f(x)=x|x|\)?
Given a differential equation \(\dfrac{d^2x}{dt^2} + x= 0\) with \(x(0) \neq 0\).
Which of the following statements is/are true?
Given \(f(x,y) = x^2 -2xy+ y^2\).
The complete contour of the equation \(f(x,y) = 1\) is described by the option(s) ___.
Consider a function \(f(z)=z^2+z+1\), where \(z\in\mathbb{C}\) is a complex variable. A simple closed contour \(\gamma\) in the \(z\)-plane encloses the point \(z=1+0j\).
The value of the integral
\[\oint_{\gamma}\frac{f(z)}{z-1}\,dz\]
is ______.
A box contains three apples and two oranges. Two fruits are removed randomly in succession. The probability that the first is an apple and the second is an orange is \(\_\_\_\_\_\).
A two-port network has z-parameters \(z_{11}=z_{22}=10\ \Omega\), and \(z_{12}=z_{21}=5\ \Omega\).
The value of \(R_L\) such that maximum power is transferred to \(R_L\) is \(\_\_\_\_\_\_\_\_\) \(\Omega\).
Port 1 of the network is driven by a 5 V source in series with a 5 \(\Omega\) resistor, and the load \(R_L\) is connected across port 2, as shown in the figure.
The slope efficiency of a laser diode is \(0.5\) W/A, and the output optical power at a current of \(100\) mA is \(30\) mW. Assuming piece-wise linear characteristics of the laser diode, the threshold current of the laser is ________ mA.
The potential difference between nodes \(A\) and \(B\) in a circuit is \(v_A - v_B = 5\) V. The work done in moving a charge of \(1\) coulomb from point \(B\) to \(A\) is ____ J. (answer in integer)
An electric field has a potential of \(V(x,y,z)=\sqrt{x^2+y^2+z^2}\) V.
A charge of \(1\) coulomb placed at \((\hat{i}+\hat{j}+\hat{k})\) experiences a force of
\[\vec{F}=(a\hat{i}+b\hat{j}+c\hat{k})\text{ N}\]
The values of \((a,b,c)\) are ______.
The initial state of \(Q_3Q_2Q_1Q_0\) in the digital circuit shown below is 1011.

The new value of \(Q_3Q_2Q_1Q_0\) after two rising edges of the clock (CLK) will be __________.
Which of the following functions represent(s) the output of the CMOS logic-gate shown in the circuit below?