Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(E)$ is

1

In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to

Topic: JEE Main 2025 (24 January Evening Shift)

1

Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^2+ax+1$ is strictly increasing and …

Topic: JEE Main 2025 (24 January Evening Shift)

2

Let $A={,x\in(0,\pi)-{\tfrac{\pi}{2}}: \log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2,}$ and $B={,x\ge 0:\sqrt{x}(\sqr…

Topic: JEE Main 2025 (24 January Evening Shift)

3

For some $a, b,$ let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\fra…

Topic: JEE Main 2025 (24 January Evening Shift)

4

The area of the region enclosed by the curves $y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|$ and $y$-axis is :

Topic: JEE Main 2025 (24 January Evening Shift)

5

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^2\right)}{(\alpha-\beta)}\r…

Topic: JEE Main 2025 (24 January Evening Shift)

6

If the system of equations $x+2y-3z=2$, $2x+\lambda y+5z=5$, $14x+3y+\mu z=33$ has infinitely many solutions, then $\la…

Topic: JEE Main 2025 (24 January Evening Shift)

7

Suppose $A$ and $B$ are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial ex…

Topic: JEE Main 2025 (24 January Evening Shift)

8

Let the position vectors of three vertices of a triangle be $4\vec p+\vec q-3\vec r$, $-5\vec p+\vec q+2\vec r$ and $2\…

Topic: JEE Main 2025 (24 January Evening Shift)

9

If $7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha) + \frac{1}{7^3}(5+3\alpha) + \cdots$, then the value of $\…

Topic: JEE Main 2025 (24 January Evening Shift)

10

If the equation of the parabola with vertex $V!\left(\frac{3}{2},,3\right)$ and directrix $x+2y=0$ is $\alpha x^2+\beta…

Topic: JEE Main 2025 (24 January Evening Shift)