GATE EC question library

Let $f(x)=e^{x+x^{2}}$ for real $x$ . From among the following, choose the Taylor series approximation of $f(x)$ around $x=0$, which includes all powers of $x$ less than or equal to $3$.

A three dimensional region $R$ of finite volume is described by $x^2 + y^2 \leq z^3; \: \: 0 \leq z \leq 1,$ where $x,y,z$ are real. The volume of $R$ (up to two decimal places) is _________

Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of $I$ is ____________.

Which one of the following is the general solution of the first order differential equation $$\frac{dy}{dx}=(x+y-1)^{2},$$ where $x,y$ are real?

Starting with $x=1$, the solution of the equation $x^{3}+x=1$, after two iterations of Newton-Raphson’s method (up to two decimal places) is__________.

Let $x(t)$ be a continuous time periodic signal with fundamental period $T=1$ seconds.Let ${a_{k} }$ be the complex Fourier series coefficients of $x(t)$, where $k$ is integer valued. Consider the following statements about $x(3t)$: The complex Fourier series coefficients of $x(3t)$ are ${a_{k}}$ where $k$ is integer valued The complex …

Two discrete-time signals $x[n]$ and $h[n]$ are both non-zero only for $n=0,1,2$, and are zero otherwise .It is given that $$x[0]=1, \: x[1]=2, \: x[2]=1, \: h[0]=1$$ Let $y[n]$ be the linear convolution of $x[n]$ and $h[n]$. Given that $y[1]=3$ and $y[2]=4$, the value of the expression $(10y[3]+y[4])$ is__________.

Let $h[n]$ be the impulse response of a discrete-time linear time invariant(LTI) filter. The impulse response is given by $$h[0]=\frac{1}{3}; \, h[1]=\frac{1}{3}; \, h[2]=\frac{1}{3} \text{; and } h[n]=0 \text{ for } n<0 \text{ and } n>2.$$ Let $H(\omega)$ be the discrete-time Fourier transform (DTFT) of $h[n]$, where $\omega$ is the normalized …

The figure shows an RLC circuit excited by the sinusoidal voltage $100 \cos(3t)$ Volts, where $t$ is in seconds. The ratio $\frac{\text{amplitude of }V_{2}}{\text{amplitude of }V{1}}$ is______.

In the circuit shown, the voltage $V_{IN}(t)$ is described by: $$V_{IN}(t)=\begin{cases} 0, & \text{for } t< 0 \\ 15 \text{Volts} & \text{for } t\geq 0 \end{cases}$$ where $t$ is in seconds. The time (in seconds) at which the current $I$ in the circuit will reach the value $2$ Amperes is ___________.

The dependence of drift velocity of electrons on electric field in a semiconductor is shown below. The semiconductor has a uniform electron concentration of $n=1 \times 10^{16} cm^{3}$ and electronic charge $q=1.6 \times 10 ^{-19}C$. If a bias of $5 V$ is applied across a $1 µm$ region of this semiconductor, the resulting current density in this …

As shown , a uniformly doped Silicon bar of length $L=0.1\mu m$ with a donor concentration $N_{D}=10^{16}cm^{-3}$ is illuminated at $x=0$ such that electron and hole pairs are generated at the rate of $G_{L}=G_{L0}(1-\frac{x}{L}), 0\leq x\leq L$, where $G_{L0}=10^{17}cm^{-3}s^{-1}$ . Hole life time is $10^{-4} s$, electronic charge …

As shown, two Silicon (Si) abrupt $p-n$ junction diodes are fabricated with uniform donor doping concentrations of $N_{D1}=10^{14} cm^{-3}$ and $N_{D2}=10^{16}cm^{-3}$ in the $n$-regions of the diodes, and uniform acceptor doping concentrations of $N_{A1}=10^{14} cm^{-3}$ and $N_{A2}=10^{16} cm{-3}$ in the $p$-regions of the …

In the figure shown, the $npn$ transistor acts as a switch. For the input $V_{in}(t)$ as shown in the figure, the transistor switches between the cut-off and saturation regions of operation, when $T$ is large. Assume collector-to-emitter voltage at saturation $V_{CE(sat)}=0.2V$ and base-to-emitter voltage $V_{BE}=0.7V$.The minimum value of the …

For the circuit shown, assume that the NMOS transistor is in saturation. Its threshold voltage $V_{tn}=1V$ and its transconductance parameter $\mu_{n}C_{ox}\big(\frac{W}{L}\big)=1 \: mA/V^{2}$.Neglect channel length modulation and body bias effects. Under these conditions, the drain current $I_{D}$ in $mA$ is ________.

For the DC analysis of the Common-Emitter amplifier shown, neglect the base current and assume that the emitter and collector currents are equal. Given that $V_{T}=25mV,V_{BE}=0.7V$, and the BJT output resistance $r_{o}$ is practically infinite. Under these conditions, the midband voltage gain magnitude, $A_{v}=\mid v_{o}/v_{i} \mid$, V/V is …

The amplifier circuit shown in the figure is implemented using a compensated operational amplifier (op-amp), and has an open-loop voltage gain, $A_{0}=10^{5}V/V$ and an open-loop cut-off frequency, $f_{c}=8Hz$. The voltage gain of the amplifier at $15$ kHz, in V/V, is____________.

Which one of the following gives the simplified sum of products expression for the Boolean function $F=m_{0}+m_{2}+m_{3}+m_{5}$, where $m_{0},m_{2},m_{3}$ and $m_{5}$ are minterms corresponding to the inputs $A,B$ and $C$ with $A$ as the $MSB$ and $C$ as the $LSB$?

A $4$-bit shift register circuit configured for right-shift operation, i.e. $D_{in}\rightarrow A,A\rightarrow B,B\rightarrow C,C\rightarrow D$, is shown. If the present state of the shift register is $ABCD=1101$, the number of clock cycles required to reach the state $ABCD=1111$ is _________.

The following FIVE instructions were executed on an $8085$ microprocessor. MVI A, $33$H MVI B, $78$H ADD B CMA ANI $32$H The Accumulator value immediately after the execution of the fifth instruction is