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GATE CH 2021 | Question: 5
A three-dimensional velocity field is given by $V=5x^2y \: i + Cy \: j-10xyz \: k$, where $i,j,k$ are the unit vectors in ... fluid flow, the value of $C$ is $-1$ $0$ $1$ $5$
A three-dimensional velocity field is given by $V=5x^2y \: i + Cy \: j-10xyz \: k$, where $i,j,k$ are the unit vectors in $x,y,z$ directions, respectively, describing a c...
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GATE Chemical 2019 | Question: 1
A system of $n$ homogeneous linear equations containing $n$ unknowns will have non-trivial solutions if and only if the determinant of the coefficient matrix is$1$$-1$$0$...
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GATE Chemical 2019 | Question: 35
Two unbiased dice are thrown. Each dice can show any number between $1$ and $6$. The probability that the sum of the outcomes of the two dice is divisible by $4$ is _____...
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GATE Chemical 2019 | Question: 22
The product of the eigenvalues of the matrix $\begin{pmatrix} 2 &3 \\ 0& 7 \end{pmatrix}$ is ____________ (rounded off to one decimal place).
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GATE Chemical 2019 | Question: 27
The value of the complex number $i^{-1/2}$ (where $i=\sqrt{-1}$) is $\frac{1}{\sqrt{2}}\left ( 1-i \right )$$-\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}...
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GATE Chemical 2018 | Question: 47
For the matrix $A=\begin{bmatrix} \cos\theta &-\sin\theta \\ \sin \theta & \cos\theta \end{bmatrix}$ if $\det$ stands for the determinant and $A^{T}$ is the transpose of ...
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GATE CH 2021 | Question: 2
The function $\cos(x)$ is approximated using Taylor series around $x=0$ as $\cos(x) \approx 1 + ax + bx^2 + cx^3 + dx^4$. The values ... $a=-0.5, \: b=0,\: c=0.042, \: d=0$
The function $\cos(x)$ is approximated using Taylor series around $x=0$ as $\cos(x) \approx 1 + ax + bx^2 + cx^3 + dx^4$. The values of $a,b,c$ and $d$ are$a=1, \: b=-0....
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GATE Chemical 2020 | Question: 27
The maximum value of the function $f(x)=-\dfrac{5}{3} x^3 +10x^2-15x+16$ in the interval $(0.5,3.5)$ is$0$$8$$16$$48$
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GATE Chemical 2017 | Question: 4
Let $i$ and $j$ be the unit vectors in the $x$ and $y$ directions, respectively. For the function $F\left ( x,y \right )=x^{3}+y^{2}$ the gradient of the function. i.e.. ...
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GATE Chemical 2019 | Question: 26
The solution of the ordinary differential equation$\frac{dy}{dx}+3y=1$, subjects to the initial condition $y=1$ at $x=0$, is$\frac{1}{3}\left ( 1+2e^{-x/3} \right )$$\fra...
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GATE Chemical 2020 | Question: 1
Which one of the following methods requires specifying an initial interval containing the root (i.e., bracketing) to obtain the solution of $f(x) =0$, where $f(x)$ is a c...
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GATE Chemical 2019 | Question: 36
The Newton-Raphson method is used to determine the root of the equation $f(x)=e^{-x}-x$. If the initial guess for the root is 0, the estimate of the root after two iterat...
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GATE CH 2021 | Question: 1
An ordinary differential equation $\text{(ODE)}, \: \dfrac{dy}{dx}=2y$, with an initial condition $y(0)=1$, has the analytical ... $0.06$ $0.8$ $4.0$ $8.0$
An ordinary differential equation $\text{(ODE)}, \: \dfrac{dy}{dx}=2y$, with an initial condition $y(0)=1$, has the analytical solution $y=e^{2x}$.Using Runge-Kutta secon...
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GATE Chemical 2017 | Question: 5
The marks obtained by a set of students are: $38,\:84,\:45,\:70,\:75,\:60,\;48$.The mean and median marks, respectively, are$45$ and $75$$55$ and $48$$60$ and $60$$60$ an...
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GATE Chemical 2017 | Question: 1
The value of $\underset{x\rightarrow 0}{\lim}\:\frac{\tan\left ( x \right )}{x}$ is ________________.
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GATE Chemical 2017 | Question: 29
A box has $6$ red balls and $4$ white balls. A ball is picked at random and replaced in the box, after which a second ball is picked.The probability of both the balls bei...
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GATE Chemical 2018 | Question: 1
Consider the following two equation:$\frac{dx}{dt}+x+y=0$$\frac{dy}{dt}-x=0$The above set of equations is represented by$\frac{d^{2}y}{dt^{2}}-\frac{dy}{dt}-y=0$$\frac{d^...
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GATE Chemical 2019 | Question: 2
The value of the expression $\underset{x\rightarrow \frac{\pi }{2}}{\lim}\: \mid \frac{\tan\:x}{x} \mid $ is $\infty$$0$$1$$-1$
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GATE Chemical 2019 | Question: 34
If $x,y$ and $z$ are directions in a Cartesian coordinate system and $i$, $j$ and $k$ are the respective unit vectors, the directional derivative of the function $u\left ...
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GATE Chemical 2012 | Question: 27
If $a$ is a constant, then the value of the integral $a^{2}\int^\infty_0 xe^{-ax}dx$ is$1/a$$a$$1$$0$
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GATE Chemical 2020 | Question: 25
Sum of the eigenvalues of the matrix $\begin{bmatrix} 2 & 4 & 6 \\ 3 & 5 & 9 \\ 12 & 1 & 7 \end{bmatrix}$ is ______ (round off to nearest integer).
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GATE Chemical 2012 | Question: 28
The Newton – Raphson method is used to find the roots of the equation $f(x) = x- \cos\pi x$ $0\...
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GATE Chemical 2012 | Question: 29
If $i = \sqrt{-1}$, the value of the integral $$\oint_{c}\dfrac{7z + i}{z(z^2 + 1)}dz \quad\quad\quad\quad|z|<2$$,u...
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GATE Chemical 2020 | Question: 26
In a box, there are $5$ green balls and $10$ blue balls. A person picks $6$ balls randomly. The probability that the person has picked $4$ green balls and $2$ blue balls ...
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GATE Chemical 2018 | Question: 46
If $y=e^{-x^{2}}$ then the value of $\underset{x\rightarrow \infty }{\lim}\frac{1}{x}\frac{dy}{dx}$ is __________
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GATE CH 2021 | Question: 26
Let $A$ be a square matrix of size $n \times n (n>1)$. The elements of $A=\{a_{ij}\}$ ... $A$ is $0$ $1$ $n!$ $(n!)^2$
Let $A$ be a square matrix of size $n \times n (n>1)$. The elements of $A=\{a_{ij}\}$ are given by $$a_{ij} = \begin{cases} i \times j, & \text{if } i \geq j \\ 0, & \tex...
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GATE Chemical 2016 | Question: 2
The Laplace transform of $e^{at}\sin\left ( bt \right )$ is$\frac{b}{\left ( s-a \right )^{2}+b^{2}}$$\frac{s-a}{\left ( s-a \right )^{2}+b^{2}}$$\frac{s-a}{\left ( s-a \...
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GATE Chemical 2020 | Question: 4
Consider the following unit step function.The Laplace transform of this function is$\dfrac{e^{-3s}}{s} \\ $$\dfrac{e^{-3s}}{s^2} \\$$\dfrac{e^{-3s}}{3s} \\$$\dfrac{e^{-6s...
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GATE Chemical 2013 | Question: 4
Evaluate ${\displaystyle \int \frac{dx}{e^x – 1}}$(Note: C is a constant of integration)$\frac{e^x}{e^x -1}$ + C$\frac {In(e^x -1)}{e^x}$ + CIn$(\frac {e^x}{e^x -1})$ ...
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GATE Chemical 2017 | Question: 27
The Laplace transform of a function is $\frac{s+1}{s\left ( s+2 \right )}$The initial and final values, respectively, of the function are $0$ and $1$$1$ and $\frac{1}{2}$...
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GATE Chemical 2016 | Question: 29
The Lagrange mean-value theorem is satisfied for $f\left ( x \right )=x^{3}+5$, in the interval $\left ( 1,4 \right )$ at a value (rounded off to the second decimal place...
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GATE Chemical 2020 | Question: 55
Consider the following dataset.$$\begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 15 & 25 \\ \hline f(x) & 6 & 8 & 10 & 12 & 5 \\ \hline \end{array}$$The value of the ...
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GATE Chemical 2012 | Question: 1
Consider the following set of linear algebraic equations $x{_1} + 2x{_2} + 3x{_3} = 2$ ...
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GATE Chemical 2018 | Question: 31
The decay ratio for a system having complex conjugate poles as $\left ( -\frac{1}{10}+j\frac{2}{15} \right )$ and $\left ( -\frac{1}{10}-j\frac{2}{15} \right )$ is $7\ti...
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GATE Chemical 2014 | Question: 30
In rolling of two fair dice, the outcome of an experiments is considered to be the sum of the numbers appearing on the dice. The probability is highest for the outcome of...
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GATE Chemical 2017 | Question: 3
The number of positive roots of the function $f(x)$ shown below in the range $0<x<6$ is ___________________.
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GATE Chemical 2015 | Question: 12
For the matrix $\begin{pmatrix} 4 & 3\\ 3& 4 \end{pmatrix}$, if $\begin{pmatrix} 1\\ 1 \end{pmatrix}$ is an eigenvector, the corresponding eigenvalue is __________ .
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GATE Chemical 2014 | Question: 29
Consider the following two normal distributions$f_{1}\left ( x \right )=exp\left ( -\pi x^{2} \right )$$f_{2}\left ( x \right )=\frac{1}{2 \pi}exp\left \{ -\frac{1}{4\pi ...
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GATE Chemical 2017 | Question: 2
The real part of $6e^{i\pi /3}$ is _______________.
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GATE Chemical 2017 | Question: 26
For the initial value problem$$\frac{dx}{dt}=\sin\left ( t \right ),\:\:\:\:x\left ( 0 \right )=0$$the value of $x$ at $t=\pi /3$ , is _____________.
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