Let $A=\begin{bmatrix} 2&1&2\\ 6&2&11\\ 3&3&2 \end{bmatrix} \quad\text{and}\quad P=\begin{bmatrix} 1&2&0\\ 5&0&2\\ 7&1&5 \end{bmatrix}. $ The sum of the prime factors of $\left|\,P^{-1}AP-2I\,\right|$ is equal to:

1

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

Topic: JEE Main 2024 (29 January Evening Shift)

2

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\!\left(2\tan\frac{5\pi}{8}\…

Topic: JEE Main 2024 (29 January Evening Shift)

3

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

Topic: JEE Main 2024 (29 January Evening Shift)

4

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

Topic: JEE Main 2024 (29 January Evening Shift)

5

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

Topic: JEE Main 2024 (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 2024 (29 January Evening Shift)

7

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

Topic: JEE Main 2024 (29 January Evening Shift)

8

If $\sin\!\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$ is the solution of the differential equation $x\cos\!\…

Topic: JEE Main 2024 (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 2024 (29 January Evening Shift)

10

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

Topic: JEE Main 2024 (29 January Evening Shift)