1
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 ____
2
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 ____
3
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

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

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

Let $z$ be a complex variable. If $f(z)=\frac{\sin(\pi z)}{z^{7}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint\limits_{C} f(z)dz$ is _______.

A

$ \pi^2 j $

B

$ j\pi\left(\frac{1}{2}-\pi\right) $

C

$ j\pi\left(\frac{1}{2}+\pi\right) $

D

$-\pi^2 j$

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