Engineering Sciences Q&A

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1

GATE XE 2023 | Question: 1
Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,$ and $3$. If $B=A^{2}+2 A+I$, where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ are $4,9,16$ $1,2,3$ $1,4,9$ $4,16,25$

Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,$ and $3$. If $B=A^{2}+2 A+I$, where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ ar...

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2

GATE XE 2023 | Question: 2
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by $f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{array}\right.$ Then which one of the following statement is TRUE? $f$ is NOT continuous at $(0,0)$. $\frac{\partial f}{\partial x}(0,0)=0$, and $\frac{\partial f}{\partial y}(0,0)=1$ $\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=0$ $\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=1$

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by $f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{...

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3

GATE XE 2023 | Question: 3
If the quadrature formula $\int_{-1}^{1} f(x) d x \approx \frac{1}{9}\left(c_{1} f(-1)+c_{2} f\left(\frac{1}{2}\right)+c_{3} f(1)\right)$ is exact for all polynomials of degree less than or equal to $2$, then $c_{1}+\frac{c_{2}}{4}+c_{3}=6$ $c_{1}+\frac{c_{2}}{3}+c_{3}=4$ $c_{1}+\frac{c_{2}}{2}+c_{3}=2$ $c_{1}+c_{2}+c_{3}=5$

If the quadrature formula $\int_{-1}^{1} f(x) d x \approx \frac{1}{9}\left(c_{1} f(-1)+c_{2} f\left(\frac{1}{2}\right)+c_{3} f(1)\right)$is exact for all polynomials of d...

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4

GATE XE 2023 | Question: 4
The second smallest eigenvalue of the eigenvalue problem $\frac{d^{2} y}{d x^{2}}+(\lambda-3) y=0, \quad y(0)=y(\pi)=0$, is $4$ $3$ $7$ $9$

The second smallest eigenvalue of the eigenvalue problem$\frac{d^{2} y}{d x^{2}}+(\lambda-3) y=0, \quad y(0)=y(\pi)=0$,is$4$$3$$7$$9$

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5

GATE XE 2023 | Question: 5
Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$? $f(z)=z|z|, \quad z \in \mathbb{C}$ $f(z)=\sin (z), \quad z \in \mathbb{C}$ $f(z)=\left\{\begin{array}{r}e^{\frac{1}{z}}, z \neq 0 \\ 0, z=0\end{array} \quad\right.$ for $\quad z \in \mathbb{C}$ $f(z)=e^{-z^{2}}, \quad z \in \mathbb{C}$

Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?$f(z)=z|z|, \quad z \in \mathb...

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6

GATE XE 2023 | Question: 7
The value of $m$ for which the vector field $\vec{F}(x, y)=\left(4 x^{m} y^{2}-2 x y^{m}\right) \hat{\imath}+\left(2 x^{4} y-3 x^{2} y^{2}\right) \hat{\jmath}$ is a conservative vector field, is __________$\text{(in integer)}$.

The value of $m$ for which the vector field$\vec{F}(x, y)=\left(4 x^{m} y^{2}-2 x y^{m}\right) \hat{\imath}+\left(2 x^{4} y-3 x^{2} y^{2}\right) \hat{\jmath}$is a conserv...

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7

GATE XE 2023 | Question: 6
If the polynomial $P(x)=a_{0}+a_{1} x+a_{2} x(x-1)+a_{3} x(x-1)(x-2)$ interpolates the points $(0,2),(1,3),(2,2)$, and $(3,5)$, then the value of $P\left(\frac{5}{2}\right)$ is _______$\text{(round off to 2 decimal places)}$.

If the polynomial $P(x)=a_{0}+a_{1} x+a_{2} x(x-1)+a_{3} x(x-1)(x-2)$interpolates the points $(0,2),(1,3),(2,2)$, and $(3,5)$, then the value of $P\left(\frac{5}{2}\right...

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8

GATE XE 2023 | Question: 8
Let $P=\left[\begin{array}{lll}4 & -2 & 2 \\ 6 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$, and $Q=\left[\begin{array}{lll}3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5\end{array}\right]$ The eigenvalues of both $P$ and $Q$ are $1, 1$, and $2$. Which one of the following statements is TRUE? Both $P$ and $Q$ are diagonalizable $P$ is diagonalizable but $Q$ is NOT diagonalizable $P$ is NOT diagonalizable but $Q$ is diagonalizable Both $P$ and $Q$ are NOT diagonalizable

Let $P=\left[\begin{array}{lll}4 & -2 & 2 \\ 6 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$, and $Q=\left[\begin{array}{lll}3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5\end{array}...

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9

GATE XE 2023 | Question: 10
The probability of a person telling the truth is $\frac{4}{6}$. An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in both the throws are same. Then the probability that actually the numbers appeared in both the throws are same is ________$\text{(round off to 2 decimal places)}$.

The probability of a person telling the truth is $\frac{4}{6}$. An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in bot...

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10

GATE XE 2023 | Question: 9
The surface area of the portion of the paraboloid $z=x^{2}+y^{2}$ that lies between the planes $z=0$ and $z=\frac{1}{4}$ is $\frac{\pi}{6}(2 \sqrt{2}-1)$ $\frac{\pi}{2}(2 \sqrt{2}-1)$ $\pi(2 \sqrt{2}-1)$ $\frac{\pi}{3}(2 \sqrt{2}-1)$

The surface area of the portion of the paraboloid $z=x^{2}+y^{2}$that lies between the planes $z=0$ and $z=\frac{1}{4}$ is$\frac{\pi}{6}(2 \sqrt{2}-1)$$\frac{\pi}{2}(2 \s...

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11

GATE XE 2023 | Question: 11
Let $u(x, t)$ be the solution of the initial boundary value problem $\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad x \in(0,2), t>0$ $u(x, 0)=\sin (\pi x), \quad x \in(0,2)$ $u(0, t)=u(2, t)=0$ Then the value of $e^{\pi^{2}}\left(u\left(\frac{1}{2}, 1\right)-u\left(\frac{3}{2}, 1\right)\right)$ is ________$\text{(in integer}$).

Let $u(x, t)$ be the solution of the initial boundary value problem$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad x \in(0,2), t>0$$u(x, 0)=...

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12

GATE XE 2023 | Question: 13
Among the following non-dimensional numbers, which one characterizes periodicity present in a transient flow? Froude number Strouhal number Peclet numbe Lewis number

Among the following non-dimensional numbers, which one characterizes periodicity present in a transient flow?Froude numberStrouhal numberPeclet numbeLewis number

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13

GATE XE 2023 | Question: 12
Match the following measuring instruments with the appropriate figures. I - Pitot probe II - Pitot-static probe III - Piezometer $\text{I - P; II - Q; III - R}$ $\text{I - R; II - Q; III - P}$ $\text{I - R; II - P; III - Q}$ $\text{I - Q; II - P; III - R}$

Match the following measuring instruments with the appropriate figures.I - Pitot probeII - Pitot-static probeIII - Piezometer$\text{I - P; II - Q; III - R}$$\text{I - R; ...

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14

GATE XE 2023 | Question: 15
Among the shear stress versus shear strain rate curves shown in the figure, which one corresponds to a shear thinning fluid ? $\mathrm{P}$ $\mathrm{Q}$ $\mathrm{R}$ $\mathrm{S}$

Among the shear stress versus shear strain rate curves shown in the figure, which one corresponds to a shear thinning fluid ? $\mathrm{P}$$\mathrm{Q}$$\mathrm{R}$$\mathrm...

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15

GATE XE 2023 | Question: 14
For an incompressible boundary layer flow over a flat plate shown in the figure, the momentum thickness is expressed as $\int_0^{\infty} \frac{u}{U_{\infty}} d y$ $\int_0^{\infty}\left(1-\frac{u}{U_{\infty}}\right) d y$ $\int_0^{\infty} \frac{u}{U_{\infty}}\left(1-\frac{u}{U_{\infty}}\right) d y$ $\int_0^{\infty}\left(1-\frac{u^2}{U_{\infty}^2}\right) d y$

For an incompressible boundary layer flow over a flat plate shown in the figure, the momentum thickness is expressed as$\int_0^{\infty} \frac{u}{U_{\infty}} d y$$\int_0^{...

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16

GATE XE 2023 | Question: 16
Consider steady incompressible flow over a flat plate, where the dashed line represents the edge of the boundary layer, as shown in the figure. Which one among the following statements is true? Bernoulli's equation can be applied in Region I between any two arbitrary points. Bernoulli's equation can be applied in Region I only along a streamline. Bernoulli's equation cannot be applied in Region II. Bernoulli's equation cannot be applied in Region I.

Consider steady incompressible flow over a flat plate, where the dashed line represents the edge of the boundary layer, as shown in the figure. Which one among the follow...

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17

GATE XE 2023 | Question: 17
An inviscid steady incompressible flow is formed by combining a uniform flow with velocity $U_{\infty}$ and a clockwise vortex of strength $K$ at the origin, as shown in the figure. Velocity potential $(\phi)$ for the combined flow in polar coordinate $(r, \theta)$ is $\phi=\frac{K \theta}{2 \pi}-U_{\infty} r \cos \theta$ $\phi=\frac{K \theta}{2 \pi}-U_{\infty} r \sin \theta$ $\phi=K \ln r+U_{\infty} r \cos \theta$ $\phi=-K \ln r+U_{\infty} r \sin \theta$

An inviscid steady incompressible flow is formed by combining a uniform flow with velocity $U_{\infty}$ and a clockwise vortex of strength $K$ at the origin, as shown in ...

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18

GATE XE 2023 | Question: 18
Which of the following statements are true? $\text{(i)}$ Conservation of mass for an unsteady incompressible flow can be represented as $\nabla \cdot \vec{V}=0$, where $\vec{V}$ denotes velocity vector. $\text{(ii)}$ Circulation is defined as the line integral of vorticity about a closed curve. $\text{(iii)}$ For some fluids, shear stress can be a nonlinear function of the shear strain rate. $\text{(iv)}$ Integration ... to the Euler's equation. $\text{(ii) and (iv)}$ only $\text{(i), (ii) and (iii)}$ only $\text{(i) and (iii)}$ only $\text{(ii) and (iv)}$ only

Which of the following statements are true?$\text{(i)}$ Conservation of mass for an unsteady incompressible flow can be represented as $\nabla \cdot \vec{V}=0$, where $\v...

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19

GATE XE 2023 | Question: 19
For a two-dimensional flow field given as $\vec{V}=-x \hat{\imath}+y \hat{\jmath}$, a streamline passes through points $(2,1)$ and $(5, p)$. The value of $p$ is $5$ $5 / 2$ $2 / 5$ $2$

For a two-dimensional flow field given as $\vec{V}=-x \hat{\imath}+y \hat{\jmath}$, a streamline passes through points $(2,1)$ and $(5, p)$. The value of $p$ is$5$$5 / 2$...

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20

GATE XE 2023 | Question: 20
A stationary object is fully submerged in a static fluid, as shown in the figure. Here, $\text{CG}$ and $\mathrm{CB}$ stand for center of gravity and center of buoyancy, respectively. Which one(s) among the following statements is/are true? The object is in stable equilibrium if $y_{C G}>y_{C B}$. The object is in stable equilibrium if $y_{C G}$ The object is in neutral equilibrium if $y_{C G}=y_{C B}$. The object is in unstable equilibrium if $y_{C G}=y_{C B}$.

A stationary object is fully submerged in a static fluid, as shown in the figure. Here, $\text{CG}$ and $\mathrm{CB}$ stand for center of gravity and center of buoyancy, ...

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