GATE EC question library

The values of the integrals $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$$ and $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$$ are

An integral $I$ over a counterclockwise circle $\textbf{C}$ is given by $$I=\oint _c \frac{z^2-1}{z^2+1} e^z dz.$$ If $\textbf{C}$ is defined as$\mid z \mid=3$ , then the value of $I$ is

If the vector function $\overrightarrow{F}=\widehat{a_x}(3y-k_1z)+\widehat{a_y}(k_2x-2z)-\widehat{a_z}(k_3y+z)$ is irrotational, then the values of the constants $k_1$,$k_2$ and $k_3$, respectively, are

Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is $40 \%$ chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is __________

The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.

The switch in the circuit, shown in the figure, was open for a long time and is closed at $t=0$. The current $i(t)$ (in ampere) at $t=0.5$ seconds is ____________

Consider the circuit shown in the figure. The Thevenin equivalent resistance (in Ω) across P-Q is _____________

Consider an LTI system with magnitude response $$\mid H(f) \mid=\begin{cases} 1-\frac{\mid f \mid}{20}, & \mid f \mid \leq 20 \\ 0,& \mid f \mid > 20 \end{cases}$$ and phase response $$\arg \{ H(f) \}= – 2f.$$ If the input to the system is …

The transfer function of a causal LTI system is $H(s)=1/s$. If the input to the system is $x(t)=[\sin(t)/\pi t] u(t)$, where $u(t)$ is a unit step function, the system output $y(t)$ as $t\to \infty$ is ____________

Consider the parallel combination of two LTI systems shown in the figure. The impulse responses of the systems are $$\begin{array} {} h_1(t)=2\delta (t+2)-3\delta (t+1) \\ h_2(t)=\delta (t-2). \end{array}$$ If the input $x(t)$ is a unit step signal, then the energy of $y(t)$ is ____________

A MOS capacitor is fabricated on $p$-type Si (Silicon) where the metal work function is $4.1$ eV and electron affinity of Si is $4.0$ eV. $E_C-E_F=0.9$ eV, where $E_C$ and $E_F$ are the conduction band minimum and the Fermi energy levels of Si, respectively. Oxide $\epsilon _r=3.9$, $\epsilon _o=8.85\times 10^{-14}$ F/cm, oxide thickness …

For a particular intensity of incident light on a silicon $pn$ junction solar cell, the photocurrent density $(J_L)$ is $2.5 mA/cm^2$ and the open-circuit voltage ($V_{oc}$) is $0.451$ V. Consider thermal voltage ($V_T$) to be $25$ mV. If the intensity of the incident light is increased by $20$ times, assuming that the temperature remains …

Two n-channel MOSFETs, T1 and T2, are identical in all respects except that the width of T2 is double that of T1. Both the transistors are biased in the saturation region of operation, but the gate overdrive voltage ($V_{GS}-V_{TH}$) of T2 is double that of T1,where $V_{GS}$ and $V_{TH}$ are the gate-to-source voltage and threshold voltage of the …

An abrupt $pn$ junction (located at $x=0$) is uniformly doped on both $p$ and $n$ sides. The width of the depletion region is $W$ and the electric field variation in the $x$-direction is $E(x)$. Which of the following figures represents the electric field profile near the $pn$ junction?

Assuming that transistors $M_1$ and $M_2$ are identical and have a threshold voltage of $1$ V, the state of transistors $M_1$ and $M_2$ are respectively

In the circuit shown, transistors $Q_1$ and $Q_2$ are biased at a collector current of $2.6$ mA. Assuming that transistor current gains are sufficiently large to assume collector current equal to emitter current and thermal voltage of $26$ mV, the magnitude of voltage gain $V_0/V_s$ in the mid-band frequency range is __________(up to second decimal …

In the voltage reference circuit shown in the figure, the op-amp is ideal and the transistors $Q_1, Q_2, \dots ,Q_{32}$ are identical in all respects and have infinitely large values of common-emitter current gain $(\beta)$. The collector current $(Ic)$ of the transistors is related to their base-emitter voltage ($V_{BE}$) by the relation …

The state diagram of a finite state machine (FSM) designed to detect an overlapping sequence of three bits is shown in the figure. The FSM has an input 'In' and an output 'Out'. The initial state of the FSM is $S_0$. If the input sequence is $10101101001101$, starting with the left-most bit, then the number of times 'Out' will be $1$ is ____________

Figure Ⅰ shows a $4$-bit ripple carry adder realized using full adders and Figure Ⅱ shows the circuit of a full-adder (FA). The propagation delay of the XOR, AND and OR gates in Figure Ⅱ are $20$ ns, $15$ ns and $10$ ns, respectively. Assume all the inputs to the $4$-bit adder are initially reset to $0$. Figure I Figure II At $t = 0$, the …

A programmable logic array (PLA) is shown in the figure. The Boolean function $F$ implemented is