$ \text{The integral } \displaystyle \int \left[ \left(\frac{x}{2}\right)^{x} + \left(\frac{2}{x}\right)^{x} \right] \ln!\left(\frac{e x}{2}\right), dx \text{ is equal to:} $

1

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by:

Topic: JEE Main 2023 (8 April Evening Shift)

2

Let $A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the…

Topic: JEE Main 2023 (8 April Evening Shift)

1

The probability that the random variable $X$ takes value $x$ is given by $P(X = x) = k(x + 1)3^{-x}, \; x = 0, 1, 2, …

Topic: JEE Main 2023 (8 April Evening Shift)

2

The set $A={1,2,3,4,5,6,7}$. The relation $R={(x,y)\in A\times A:\ x+y=7}$ is:

Topic: JEE Main 2023 (8 April Evening Shift)

3

The mean and variance of $12$ observations are $\dfrac{9}{2}$ and $4$ respectively. Later, it was observed that two obs…

Topic: JEE Main 2023 (8 April Evening Shift)

4

The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $\left(2x^{2}+\dfrac{1}{2x}\rig…

Topic: JEE Main 2023 (8 April Evening Shift)

5

The area of the quadrilateral $ABCD$ with vertices $A(2,1,1)$, $B(1,2,5)$, $C(-2,-3,5)$ and $D(1,-6,-7)$ is equal to:

Topic: JEE Main 2023 (8 April Evening Shift)

6

If the number of words (with or without meaning) that can be formed using all the letters of the word MATHEMATICS — in …

Topic: JEE Main 2023 (8 April Evening Shift)

7

The numbers $\alpha>\beta>0$ are the roots of the equation $a x^{2}+b x+1=0$, and $\displaystyle \lim_{x\to \frac{1}{\a…

Topic: JEE Main 2023 (8 April Evening Shift)

8

The value of $36,(4\cos^{2}9^\circ-1)(4\cos^{2}27^\circ-1)(4\cos^{2}81^\circ-1)(4\cos^{2}243^\circ-1)$ is:

Topic: JEE Main 2023 (8 April Evening Shift)

9

$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$ …

Topic: JEE Main 2023 (8 April Evening Shift)

10

The set $S$ is all values of $\theta\in[-\pi,\pi]$ for which the system $x+y+\sqrt{3},z=0,\quad -x+(\tan\theta),y+\sqrt…

Topic: JEE Main 2023 (8 April Evening Shift)