1
GATE ECE 2024
MCQ (More than One Correct Answer)
+1
-0

Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho }{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to $\frac{\partial \rho }{\partial t} + \nabla \cdot(\rho u) = 0$?

A
GATE ECE 2024 Engineering Mathematics - Vector Calculus Question 2 English Option 1
B
GATE ECE 2024 Engineering Mathematics - Vector Calculus Question 2 English Option 2
C

$\int\limits_{V} \frac{\partial \rho}{\partial t} \, dv = - \int\limits_{V} \nabla \cdot(\rho u) \, dv$

D

$\int\limits_{V} \frac{\partial \rho}{\partial t} \, dv = \int\limits_{V} \nabla \cdot(\rho u) \, dv$

2
GATE ECE 2024
Numerical
+1
-0

Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors

$$ \{ [2 \ -3 \ \alpha], \ [3 \ -1 \ 3], \ [1 \ -5 \ 7] \}$$

does not form a basis of $\mathbb{R}^3$ is _______.

Your input ____
3
GATE ECE 2024
Numerical
+1
-0

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______ .

Your input ____
4
GATE ECE 2024
MCQ (Single Correct Answer)
+2
-1.33

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is _______.

A

$\left( 2 - \sqrt{3} \right) \pi R^2$

B

$\frac{\left( \sqrt{2} - 1 \right) \pi R^2}{2}$

C

$\frac{2 \pi R^2}{3}$

D

$\frac{\left( 2 + \sqrt{3} \right) \pi R^2}{8 \sqrt{2}}$

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