Let $A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$ and $P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]$. The sum of the prime factors of $\left|P^{-1} A P-2 I\right|$ is equal to

1

Let $x = \dfrac{m}{n}$ $(m, n$ are co-prime natural numbers$)$ be a solution of the equation $\cos(2\sin^{-1}x) = \dfra…

Topic: JEE Main 2025 (29 January Evening Shift)

2

The sum of the solutions $x \in \mathbb{R}$ of the equation $\dfrac{3\cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - …

Topic: JEE Main 2025 (29 January Evening Shift)

3

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z = 2 - i\left(2\tan\dfrac{5\pi}{…

Topic: JEE Main 2025 (29 January Evening Shift)

4

If each term of a geometric progression $a_1,a_2,a_3,\dots$ with $a_1=\dfrac{1}{8}$ and $a_2\neq a_1$ is the arithmetic…

Topic: JEE Main 2025 (29 January Evening Shift)

5

The function $f(x)=2x+3x^{\frac{1}{3}},; x\in\mathbb{R}$ has:

Topic: JEE Main 2025 (29 January Evening Shift)

6

Let $\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is …

Topic: JEE Main 2025 (29 January Evening Shift)

7

An integer is chosen at random from the integers $1,2,3,\dots,50$. The probability that the chosen integer is a multipl…

Topic: JEE Main 2025 (29 January Evening Shift)

8

If $\sin\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{x}$ is a solution of the differential equation $x\cos\left(\…

Topic: JEE Main 2025 (29 January Evening Shift)

9

If $\log_e a,;\log_e b,;\log_e c$ are in an A.P. and $\log_e a-\log_e(2b),;\log_e(2b)-\log_e(3c),;\log_e(3c)-\log_e a$ …

Topic: JEE Main 2025 (29 January Evening Shift)

10

Let $y=\log_e!\left(\dfrac{1-x^2}{1+x^2}\right)$, with $-1

Topic: JEE Main 2025 (29 January Evening Shift)