Recent questions and answers in Engineering Mathematics

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#GATE QUESTION BANK
The minimum number of equal length subintervals needed to approximate $\int_{1}^{2}xe^xdx$ to an accuracy of atleast (10^(-6))/3 using trapezoidal rule is __________.

asked Jul 15 in Numerical Methods by Gokulan K (120 points)

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GATE2019 ME-2: 3
The differential equation $\frac{dy}{dx}+4y=5$ is valid in the domain $0 \leq x \leq 1$ with $y(0)=2.25$. The solution of the differential equation is $y=e^{-4x}+5$ $y=e^{-4x}+1.25$ $y=e^{4x}+5$ $y=e^{4x}+1.25$

answered May 24, 2019 in Differential Equations by ankitgupta.1729 (410 points)

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GATE2019 ME-2: 4
An analytic function $f(z)$ of complex variable $z=x+iy$ may be written as $f(z)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ ...

answered May 24, 2019 in Calculus by ankitgupta.1729 (410 points)

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GATE2017 ME-2: 28
Consider the matrix $A=\begin{bmatrix} 50 &70 \\ 70 & 80 \end{bmatrix}$ whose eigenvectors corresponding to eigenvalues $\lambda _{1}$ and $\lambda _{2}$ are $x_{1}=\begin{bmatrix} 70 \\ \lambda_{1}-50 \end{bmatrix}$ and $x_{2}=\begin{bmatrix} \lambda _{2}-80\\ 70 \end{bmatrix}$, respectively. The value of $x^{T}_{1} x_{2}$ is _________.

answered May 24, 2019 in Linear Algebra by ankitgupta.1729 (410 points)

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GATE2019 ME-2: 1
In matrix equation $[A] \{X\}=\{R\}$, $[A] = \begin{bmatrix} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{bmatrix} \{X\} = \begin{Bmatrix} 2 \\ 1 \\ 4 \end{Bmatrix} \text{ and} \{ R \} = \begin{Bmatrix} 32 \\ 16 \\ 64 \end{Bmatrix}$ One of the eigen values of matrix $[A]$ is $4$ $8$ $15$ $16$

answered May 24, 2019 in Linear Algebra by ankitgupta.1729 (410 points)

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GATE2019 ME-1: 1
Consider the matrix $P=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ The number of distinct eigenvalues $0$ $1$ $2$ $3$

answered Feb 15, 2019 in Linear Algebra by aditya_kr (140 points)

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GATE2019 ME-2: 2
The directional derivative of the function $f(x,y)=x^2+y^2$ along a line directed from $(0,0)$ to $(1,1)$, evaluated at the point $x=1, y=1$ is $\sqrt{2}$ $2$ $2 \sqrt{2}$ $4 \sqrt{2}$

asked Feb 9, 2019 in Calculus by Arjun (21.2k points)

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GATE2012-64
An automobile plant contracted to buy shock absorbers from two suppliers $X$ and $Y.$ $X$ supplies $60\%$ and $Y$ supplies $40\%$ ... chosen shock absorber, which is found to be reliable, is made by $Y$ is $0.288$ $0.334$ $0.667$ $0.720$

asked Mar 20, 2018 in Probability and Statistics by Andrijana3306 (1.5k points)

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GATE2018-2-24
The arrival of customers over fixed time intervals in a bank follow a Poisson distribution with an average of $30$ customers/hour. The probability that the time between successive customer arrival is between $1$ and $3$ minutes is _____ (correct to two decimal places)

answered Feb 22, 2018 in Probability and Statistics by Balaji Jegan (1.2k points)

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GATE2018-2-27
Let $X_1$ and $X_2$ be two independent exponentially distributed random variables with means $0.5$ and $0.25$, respectively. Then $Y=\text{min}(X_1, X_2)$ is exponentially distributed with mean $1/6$ exponentially distributed with mean $2$ normally distributed with mean $3/4$ normally distributed with mean $1/6$

answered Feb 22, 2018 in Probability and Statistics by Balaji Jegan (1.2k points)

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GATE2018-2-19
If $A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 1 \end{bmatrix}$ then $\text{det}(A^{-1})$ is _______ (correct to two decimal palces).

answered Feb 22, 2018 in Linear Algebra by Balaji Jegan (1.2k points)

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GATE2018-2-36
Given the ordinary differential equation $\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$ with $y(0)=0$ and $\frac{dy}{dx}(0)=1$, the value of $y(1)$ is __________ (correct to two decimal places).

asked Feb 17, 2018 in Differential Equations by Arjun (21.2k points)

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GATE2018-2-35
The problem of maximizing $z=x_1-x_2$ subject to constraints $x_1+x_2 \leq 10, \: x_1 \geq 0, x_2 \geq 0$ and $x_2 \leq 5$ has no solution one solution two solutions more than two solutions

asked Feb 17, 2018 in Numerical Methods by Arjun (21.2k points)

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GATE2018-2-26
Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$, centered at origin, $\oint_C \frac{1}{5z-4} dz=A \pi i$, the value of $A$ is $2/5$ $1/2$ $2$ $4/5$

asked Feb 17, 2018 in Numerical Methods by Arjun (21.2k points)

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GATE2018-2-28
For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. Given a function $\phi =\text{ln} \mid \overrightarrow{r} \mid$, its ... $\frac{\overrightarrow{r}}{\overrightarrow{r} \cdot \overrightarrow{r} } \\ $ $\frac{\overrightarrow{r}}{\mid \overrightarrow{r} \mid^3} $

asked Feb 17, 2018 in Calculus by Arjun (21.2k points)

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GATE2018-2-3
Consider a function $u$ which depends on position $x$ and time $t$. The partial differential equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u }{\partial x^2}$ is known as the Wave equation Heat equation Laplace's equation Elasticity equation

asked Feb 17, 2018 in Differential Equations by Arjun (21.2k points)

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GATE2018-2-4
If $y$ is the solution of the differential equation $y^3 \frac{dy}{dx}+x^3 = 0, \: y(0)=1,$ the value of $y(-1)$ is $-2$ $-1$ $0$ $1$

asked Feb 17, 2018 in Differential Equations by Arjun (21.2k points)

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GATE2018-2-2
The divergence of the vector field $\overrightarrow{u}=e^x(\cos \: y\hat{i}+\sin \: y \hat{j})$ is $0$ $e^x \cos y + e^x \sin y$ $2e^x \cos y$ $2e^x \sin y$

asked Feb 17, 2018 in Calculus by Arjun (21.2k points)

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GATE2018-2-1
The Fourier cosine series for an even function $f(x)$ is given by $ f(x)=a_0 + \Sigma_{n=1}^\infty a_n \cos (nx).$ The value of the coefficient $a_2$ for the function $f(x)=\cos ^2 (x)$ in $[0, \pi]$ is $-0.5$ $0.0$ $0.5$ $1.0$

asked Feb 17, 2018 in Calculus by Arjun (21.2k points)

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GATE2018-1-51
The minimum value of $3x+5y$ such that $3x+5y \leq 15$ $4x+9y \leq 18$ $13x+2y \leq 2$ $x \geq 0, \: y \geq 0$ is ______

asked Feb 17, 2018 in Numerical Methods by Arjun (21.2k points)

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GATE2018-1-1
Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled out of the box at random one after another without replacement. The probability that all the three balls are red is $1/72$ $1/55$ $1/36$ $1/27$

asked Feb 17, 2018 in Probability and Statistics by Arjun (21.2k points)

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GATE2018-1-2
The rank of the matrix $\begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix}$ is $1$ $2$ $3$ $4$

asked Feb 17, 2018 in Linear Algebra by Arjun (21.2k points)

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GATE2018-1-3
According to the Mean Value Theorem, for a continuous function $f(x)$ in the interval $[a,b]$, there exists a value $\xi$ in this interval such that $\int_a^b f(x) dx = $ $f(\xi)(b-a)$ $f(b)(\xi-a)$ $f(a)(b-\xi)$ $0$

asked Feb 17, 2018 in Calculus by Arjun (21.2k points)

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GATE2018-1-4
$F(z)$ is a function of the complex variable $z=x+iy$ given by $F(z)+ i \: z + k \: Re(z) + i \: Im(z)$. For what value of $k$ will $F(z)$ satisfy the Cauchy-Riemann equations? $0$ $1$ $-1$ $y$

asked Feb 17, 2018 in Calculus by Arjun (21.2k points)

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GATE2017 ME-2: 3
The determinant of a $2 \times 2$ matrix is $50$. If one eigenvalue of the matrix is $10$, the other eigenvalue is _________.

answered Feb 14, 2018 in Linear Algebra by m2n037 (940 points)

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GATE2017 ME-2: 27
Consider the differential equation $3y" (x)+27 y (x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of $y$ at $x=1$ is ________.

asked Feb 27, 2017 in Differential Equations by Arjun (21.2k points)

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GATE2017 ME-2: 4
A sample of $15$ data is as follows: $17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3$. The mode of the data is $4$ $13$ $17$ $20$

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2017 ME-2: 5
The Laplace transform of $te^{t}$ is $\frac{s}{(s+1)^{2}}$ $\frac{1}{(s-1)^{2}}$ $\frac{1}{(s+1)^{2}}$ $\frac{s}{(s-1)}$

asked Feb 27, 2017 in Calculus by Arjun (21.2k points)

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GATE2017 ME-2: 2
The divergence of the vector $-yi+xj$ is ________.

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2017 ME-2: 1
Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is _______.

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2017 ME-1: 26
Consider the matrix $P=\begin{bmatrix} \frac{1}{\sqrt{2}} & 0 &\frac{1}{\sqrt{2}} \\ 0 & 1 & 0\\ -\frac{1}{\sqrt{2}} &0 & \frac{1}{\sqrt{2}} \end{bmatrix}$ Which one of the following statements about $P$ is INCORRECT ? Determinant of P is equal to $1$. $P$ is orthogonal. Inverse of $P$ is equal to its transpose. All eigenvalues of $P$ are real numbers.

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2017 ME-1: 2
The value of $\lim_{x \rightarrow 0}\frac{x^{3}-\sin(x)}{x}$ is $0$ $3$ $1$ $-1$

asked Feb 27, 2017 in Calculus by Arjun (21.2k points)

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GATE2017 ME-1: 1
The product of eigenvalues of the matrix $P$ is $P=\begin{bmatrix} 2 & 0 & 1\\ 4& -3 &3 \\ 0 & 2 & -1 \end{bmatrix}$ $-6$ $2$ $6$ $-2$

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2017 ME-1: 3
Consider the following partial differential equation for $u(x, y)$, with the constant $c > 1$: $\frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0$ Solution of this equation is $u(x, y) = f (x+cy)$ $u(x, y) = f (x-cy)$ $u(x, y) = f (cx+y)$ $u(x, y) = f (cx-y)$

asked Feb 27, 2017 in Linear Algebra by Arjun (21.2k points)

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GATE2013-1
The partial differential equation $\frac{\partial u }{\partial t}+u\frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2}$ is a linear equation of order $2$ non-linear equation of order $1$ linear equation of order $1$ non-linear equation of order $2$

asked Feb 19, 2017 in Differential Equations by piyag476 (1.4k points)

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GATE2013-3
Match the CORRECT pairs: Numerical Integration Scheme Order of Fitting Polynomial P. Simpson's 3/8 Rule 1. First Q. Trapezoidal Rule 2. Second R. Simpson's 1/3 Rule 3. Third $P-2; Q-1; R-3$ $P-3; Q-2; R-1$ $P-1; Q-2; R-3$ $P-3; Q-1; R-2$

asked Feb 19, 2017 in Numerical Methods by piyag476 (1.4k points)

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GATE2013-2
The eigen values of a symmetric matrix are all complex with non-zero positive imaginary part. complex with non-zero negative imaginary part. real. pure imaginary.

asked Feb 19, 2017 in Linear Algebra by piyag476 (1.4k points)

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