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41
GATE2016-5
In the neighborhood of $z=1$, the function $f(z)$ has a power series expansion of the form $f(z)$ = $1$ + $(1-z)$ + $(1-z)^2+ \ldots$ Then $f(z)$ is $\frac{1}{z}$ $\frac{-1}{z-2}$ $\frac{z-1}{z+}$ $\frac{1}{2z-1}$
In the neighborhood of $z=1$, the function $f(z)$ has a power series expansion of the form $f(z)$ = $1$ + $(1-z)$ + $(1-z)^2+ \ldots$Then $f(z)$ is$\frac{1}{z}$$\frac{-1}...
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44
GATE IN 2021 | Question: 1
Consider the row vectors $v=(1,0)$ and $w=(2,0)$. The rank of the matrix $M=2v^{T}v+3w^{T}w$, where the superscript $\text{T}$ denotes the transpose, is $1$ $2$ $3$ $4$
Consider the row vectors $v=(1,0)$ and $w=(2,0)$. The rank of the matrix $M=2v^{T}v+3w^{T}w$, where the superscript $\text{T}$ denotes the transpose, is$1$$2$$3$$4$
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45
GATE IN 2021 | Question: 2
Consider the sequence $x_{n}=0.5x_{n-1}+1,n=1,2, \dots \:\dots$ with $x_0=0$ . Then $\displaystyle \lim_{n\rightarrow \infty} x_n$ is $0$ $1$ $2$ $\infty$
Consider the sequence $x_{n}=0.5x_{n-1}+1,n=1,2, \dots \:\dots$ with $x_0=0$ . Then $\displaystyle \lim_{n\rightarrow \infty} x_n$ is$0$$1$$2$$\infty$
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47
GATE2020: 19
Consider the signal $x(t)=e^{-|t|}$. Let $X(j\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt$ be the Fourier transform of $x(t)$. The value of $X(j0) $is _________
Consider the signal $x(t)=e^{-|t|}$. Let $X(j\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt$ be the Fourier transform of $x(t)$. The value of $X(j0) $is _________
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49
GATE IN 2021 | Question: 38
Given $A=\begin{pmatrix} 2 & 5\\ 0 & 3 \end{pmatrix}$. The value of the determinant $\left | A^{4}-5A^{3}+6A^{2}+2I \right |=$ _______________.
Given $A=\begin{pmatrix} 2 & 5\\ 0 & 3 \end{pmatrix}$. The value of the determinant $\left | A^{4}-5A^{3}+6A^{2}+2I \right |=$ _______________.
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50
GATE2017: 27
The angle between two vectors $X_1=\begin{bmatrix}2 & 6 & 14\end{bmatrix}^T$ and $X_2=\begin{bmatrix}-12 & 8 & 16\end{bmatrix}^T$ in radian is __________.
The angle between two vectors $X_1=\begin{bmatrix}2 & 6 & 14\end{bmatrix}^T$ and $X_2=\begin{bmatrix}-12 & 8 & 16\end{bmatrix}^T$ in radian is __________.
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52
GATE2015-36
The probability that a thermistor randomly picked up from a production unit is defective is $0.1$. The probability that out of $10$ thermistors randomly picked up, $3$ are defective is 0.001 0.057 0.107 0.3
The probability that a thermistor randomly picked up from a production unit is defective is $0.1$. The probability that out of $10$ thermistors randomly picked up, $3$ ar...
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53
GATE2012-30
Consider the differential equation $\frac{d^2y(t)}{dt^2}+2\frac{dy(t)}{dt}+y(t)=\delta(t)$ with $y(t)|_{t=0^-}=-2$ and $\frac{dy}{dt}|_{t=0^-}=0$. The numerical value of $\frac{dy}{dt}|_{t=0^+}$ is $-2$ $-1$ $0$ $1$
Consider the differential equation$\frac{d^2y(t)}{dt^2}+2\frac{dy(t)}{dt}+y(t)=\delta(t)$ with $y(t)|_{t=0^-}=-2$ and $\frac{dy}{dt}|_{t=0^-}=0$.The numerical value of $\...
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54
GATE2012-1
If $x=\sqrt{-1},$ then the value of $x^x$ is $e^{-\pi/2}$ $e^{\pi/2}$ $x$ $1$
If $x=\sqrt{-1},$ then the value of $x^x$ is $e^{-\pi/2}$$e^{\pi/2}$$x$$1$
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55
GATE2015-14
The magnitude of the directional derivative of the function $f(x,y)=x^2+3y^2$ in a direction normal to the circle $x^2+y^2=2,$ at the point $(1,1),$ is $4\sqrt{2}$ $5\sqrt{2}$ $7\sqrt{2}$ $9\sqrt{2}$
The magnitude of the directional derivative of the function $f(x,y)=x^2+3y^2$ in a direction normal to the circle $x^2+y^2=2,$ at the point $(1,1),$ is$4\sqrt{2}$$5\sqrt{...
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57
GATE2019 IN: 16
A 3 x 3 matrix has eigenvalues 1, 2 and 5. The determinant of the matrix is $\_\_\_\_$.
A 3 x 3 matrix has eigenvalues 1, 2 and 5. The determinant of the matrix is $\_\_\_\_$.
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58
GATE2016-4
The vector that is $NOT$ perpendicular to the vectors $(i+j+k)$ and $(i+2j+3k)$ is ______. $(i-2j+k)$ $(-i+2j-k)$ $(0i+0j+0k)$ $(4i+3j+5k)$
The vector that is $NOT$ perpendicular to the vectors $(i+j+k)$ and $(i+2j+3k)$ is ______.$(i-2j+k)$ $(-i+2j-k)$$(0i+0j+0k)$$(4i+3j+5k)$
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59
GATE2016-28
Consider the matrix $A= \begin{pmatrix} 2 & 1 & 1\\ 2& 3& 4\\ -1& -1 & -2 \end{pmatrix} $ whose eigenvalues are $1, -1$ and $3$. Then Trace of $(A^3-3A^2)$ is $\_\_\_\_\_\_.$
Consider the matrix $A= \begin{pmatrix} 2 & 1 & 1\\ 2& 3& 4\\ -1& -1 & -2 \end{pmatrix} $ whose eigenvalues are $1, -1$ and $3$. Then Trace of $(A^3-3A^2)$ is $\_\_\_\_\_...
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60
GATE2016-30
The value of the integral $\displaystyle{}\frac{1}{2\pi j}\int_c \frac{Z^2+1}{Z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in the complex plane is $\_\_\_\_\_\_\_\_.$
The value of the integral $\displaystyle{}\frac{1}{2\pi j}\int_c \frac{Z^2+1}{Z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in th...
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61
GATE2013-38
The Laplace Transform representation of the triangular pulse shown below is $\frac{1}{s^2}[1+e^{-2s}]$ $\frac{1}{s^2}[1-e^{-s}+e^{-2s}]$ $\frac{1}{s^2}[1-e^{-s}+2e^{-2s}]$ $\frac{1}{s^2}[1-2e^{-s}+e^{-2s}]$
The Laplace Transform representation of the triangular pulse shown below is $\frac{1}{s^2}[1+e^{-2s}]$$\frac{1}{s^2}[1-e^{-s}+e^{-2s}]$$\frac{1}{s^2}[1-e^{-s}+2e^{-2s}]$$...
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62
GATE2012-2
With initial condition $x(1)=0.5$, the solution of the differential equation, $t\frac{dx}{dt}+x=t$ is $x=t-\frac{1}{2}$ $x=t^2-\frac{1}{2}$ $x=\frac{t^2}{2}$ $x=\frac{t}{2}$
With initial condition $x(1)=0.5$, the solution of the differential equation,$t\frac{dx}{dt}+x=t$ is $x=t-\frac{1}{2}$$x=t^2-\frac{1}{2}$$x=\frac{t^2}{2}$$x=\frac{t}{2}$
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65
GATE2013-5
For a vector $E$, which one of the following statements is $\text{NOT TRUE}$? If $\Delta.E=0,\;E$ is called solenoidal If $\Delta \times E=0,\;E$ is called conservative If $\Delta\times E=0,\;E$ is called irrotational If $\Delta.E=0,\;E$ is called irrotational
For a vector $E$, which one of the following statements is $\text{NOT TRUE}$?If $\Delta.E=0,\;E$ is called solenoidalIf $\Delta \times E=0,\;E$ is called conservativeIf $...
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66
GATE2012-29
The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1, 6]$ is $21$ $25$ $41$ $46$
The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1, 6]$ is $21$$25$$41$$46$
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67
GATE2012-4
The unilateral Laplace transform of $f(t)$ is $\frac{1}{s^2+s+1}.$ The unilateral Laplace transform of $tf(t)$ is $-\frac{s}{(s^2+s+1)^2}$ $-\frac{2s+1}{(s^2+s+1)^2}$ $\frac{s}{(s^2+s+1)^2}$ $\frac{2s+1}{(s^2+s+1)^2}$
The unilateral Laplace transform of $f(t)$ is $\frac{1}{s^2+s+1}.$ The unilateral Laplace transform of $tf(t)$ is$-\frac{s}{(s^2+s+1)^2}$$-\frac{2s+1}{(s^2+s+1)^2}$$\frac...
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68
GATE2018IN: 3
X and Y are two independent random variables with variances 1 and 2, respectively. Let Z = X – Y. The variance of Z is 0 1 2 3
X and Y are two independent random variables with variances 1 and 2, respectively. Let Z = X – Y. The variance of Z is0123
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69
GATE2015-57
The fundamental period of the signal $x(t)=2\cos(\frac{2\pi t}{3})+\cos(\pi t)$, in seconds, is ________________ s.
The fundamental period of the signal $x(t)=2\cos(\frac{2\pi t}{3})+\cos(\pi t)$, in seconds, is ________________ s.
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70
GATE2015-11
Let $A$ be an $n\times n$ matrix with rank $r(0<r<n).$ Then $\text{Ax=0}$ has $p$ independent solutions, where $p$ is $r$ $n$ $n-r$ $n+r$
Let $A$ be an $n\times n$ matrix with rank $r(0<r<n).$ Then $\text{Ax=0}$ has $p$ independent solutions, where $p$ is$r$$n$$n-r$$n+r$
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71
GATE2014-3
The figure shows the plot of $y$ as a function of $x$ The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is : $\frac{d^2y}{dx^2}=1$ $\frac{dy}{dx}=x$ $\frac{dy}{dx}=-x$ $\frac{dy}{dx}=|x|$
The figure shows the plot of $y$ as a function of $x$The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is :$\fr...
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72
GATE2013-26
While numerically solving the differential equation $\frac{dy}{dx}+2xy^2=0,\; y(0)=1$ using Euler’s predictor-corrector (improved Euler-Cauchy) method with a step size of 0.2, the value of $y$ after the first step is $1.00$ $1.03$ $0.97$ $0.96$
While numerically solving the differential equation $\frac{dy}{dx}+2xy^2=0,\; y(0)=1$ using Euler’s predictor-corrector (improved Euler-Cauchy) method with a step size ...
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73
GATE2012-5
Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}.$ If $C$ is a counterclockwise path in the $z$-plane such that $|z+1|=1,$ the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is $-2$ $-1$ $1$ $2$
Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}.$ If $C$ is a counterclockwise path in the $z$-plane such that $|z+1|=1,$ the value of $\frac{1}{2\pi j}\oint_cf(z)dz$ is$-2$$-1$$...
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74
GATE2020 IN: 27
Consider the differential equation $\frac{dx}{dt}=\sin(x),$ with the initial condition $x(0)=0. $ The solution to this ordinary differential equation is __________ $x(t)=0$ $x(t)=\sin(t)$ $x(t)=\cos(t)$ $x(t)=\sin(t)-\cos(t)$
Consider the differential equation $\frac{dx}{dt}=\sin(x),$ with the initial condition $x(0)=0. $The solution to this ordinary differential equation is __________$x(t)=0$...
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75
GATE2017: 4
The eigenvalues of the matrix $A=\begin{bmatrix}1 &-1 &5\\0 &5 &6\\0 &-6 &5\end{bmatrix}$ are $-1,\;5,\;6$ $1,\;-5\pm j6$ $1,\;5\pm j6$ $1,\;5,\;5$
The eigenvalues of the matrix $A=\begin{bmatrix}1 &-1 &5\\0 &5 &6\\0 &-6 &5\end{bmatrix}$ are$-1,\;5,\;6$$1,\;-5\pm j6$$1,\;5\pm j6$$1,\;5,\;5$
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76
GATE2014-2
Given that $x$ is a random variable in the range $[0,\infty]$ with a probability density function $\frac{e \frac{-x}{2}}{K}$, the value of the constant $K$ is __________.
Given that $x$ is a random variable in the range $[0,\infty]$ with a probability density function $\frac{e \frac{-x}{2}}{K}$, the value of the constant $K$ is __________....
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78
GATE2017: 1
If $\text{v}$ is a non-zero vector of dimensions $3\times1$, then the matrix $A=VV^T$ has a rank = ____________.
If $\text{v}$ is a non-zero vector of dimensions $3\times1$, then the matrix $A=VV^T$ has a rank = ____________.
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79
GATE2013-13
The type of the partial differential equation $\frac{\partial f}{\partial t}=\frac{\partial^2 f}{\partial x^2}$ is Parabolic Elliptic Hyperbolic Nonlinear
The type of the partial differential equation $\frac{\partial f}{\partial t}=\frac{\partial^2 f}{\partial x^2}$ isParabolicEllipticHyperbolicNonlinear
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80
GATE2017: 28
The following table lists an $n^{th}$ order polynominal $f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ and the forward differences evaluated at equally spaced values of $x$. The order of the polynominal is $1$ $2$ $3$ $4$
The following table lists an $n^{th}$ order polynominal $f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ and the forward differences evaluated at equally spaced values of $x$. T...
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