1
GATE ECE 2024
MCQ (Single Correct Answer)
+1
-0.33

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is ____.

A

$\dfrac{d^2 y}{d t^2} + 4 \dfrac{d y}{d t} + 4 y = f(t)$

B

$\dfrac{d^2 y}{d t^2} + 4 y = f(t)$

C

$\dfrac{d^2 y}{d t^2} + 3 \dfrac{d y}{d t} + 2 y = f(t)$

D

$\dfrac{d^2 y}{d t^2} + 5 \dfrac{d y}{d t} + 6 y = f(t)$

2
GATE ECE 2024
MCQ (More than One Correct Answer)
+1
-0.33

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

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

3
GATE ECE 2024
Numerical
+1
-0.33

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 ____
4
GATE ECE 2024
Numerical
+1
-0.33

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 ____
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