GATE EC question library

Consider a $2\times2$ sqaure matrix $$\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$$ where $x$ is unknown. If the eigen values of the matrix $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma - j\omega)$, then $x$ is equal to

For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______

The probability of getting a “head” in a single toss of a biased coin is $0.3$. The coin is tossed repeatedly till a head is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _______

The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______

Consider the first order initial value problem $$y’= y+2x-x^2 ,\ y(0)=1,\ (0 \leq x < \infty)$$ with exact solution $y(x) = x^2 + e^x$. For $x = 0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runga-Kutta method with step-size $h=0.1$ is _______

Consider the signal $x(t) = \cos(6\pi t) + \sin (8\pi t)$, where $t$ is in seconds. The Nyquist sampling rate (in samples/second) for the signal $y(t) = x(2t + 5)$ is

If the signal $x(t) = \large \frac{\sin(t)}{\pi t}$$*$$\large \frac{\sin(t)}{\pi t}$ with $*$ denoting the convolution operation, then $x(t)$ is equal to

A discrete-time signal $x[n] = \delta[n – 3] + 2 \delta[n – 5]$ has $z$-transform $X(z)$. If $Y(z) = X(-z)$ is the $z$-transform of another signal $y[n]$, then

In the RLC circuit shown in the figure, the input voltage is given by $$v_i(t) = 2\cos (200t) + 4\sin (500t).$$ The output voltage $v_o(t)$ is

The $I-V$ characteristics of three types of diodes at the room temperature, made of semiconductors $X, Y$ and $Z$, are shown in the figure. Assume that the diodes are uniformly doped and identical in all respects except their materials. If $E_{gX}$,$E_{gY}$ and $E_{gZ}$ are the band gaps of $X, Y$ and $Z$, respectively, then

The figure shows the band diagram of a Metal Oxide Semiconductor (MOS). The surface region of this MOS is in

The figure shows the $I-V$ characteristics of a solar cell illuminated uniformly with solar light of power $100\ mW/cm^2$. The solar cell has an area of $3\ cm^2$ and a fill factor of $0.7$. The maximum efficiency (in $\%$) of the device is _______

The diodes $D1$ and $D2$ in the figure are ideal and the capacitors are identical. The product $RC$ is very large compared to the time period of the ac voltage. Assuming that the diodes do not breakdown in the reverse bias, the output voltage $V_o$(in volt) at the steady state is _______

Consider the circuit shown in the figure. Assuming $V_{EE1} = V_{EE2} = 0.7$ volt, the value of the dc voltage $V_{C2}$(in volt) is _______

In the astable multivibrator circuit shown in the figure, the frequency of oscillation (in kHz) at the output pin $3$ is _______

In an $8085$ microprocessor, the contents of the accumulator and the carry flag are $A7$ (in hex) and $0$, respectively. If the instruction $RLC$ is executed, then the contents of the accumulator (in hex) and the carry flag, respectively, will be

The logic functionality realized by the circuit shown below is

The minimum number of $2$-input $NAND$ gates required to implement a $2$-input $XOR$ gate is

The block diagram of a feedback control system is shown in the figure. The overall closed-loop gain of $G$ of the system is

For the unity feedback control system shown in the figure, the open-loop transfer function $G(s)$ is given as $$G(s) = \frac{2}{s(s+1)}$$ The steady state error $e_{ss}$ due to a unit step input is