Among the statements: (S1): $2023^{2022}-1999^{2022}$ is divisible by $8$. (S2): $13(13)^n-12n-13$ is divisible by $144$ for infinitely many $n\in\mathbb{N}$.

1

Let $P$ be a square matrix such that $P^{2}=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^{\alpha}+P^{…

Topic: JEE Main 2023 (6 April Evening Shift)

2

Let $a\ne b$ be two non-zero real numbers. Then the number of elements in the set $X=\{\, z\in\mathbb{C} : \operatorn…

Topic: JEE Main 2023 (6 April Evening Shift)

3

The area bounded by the curves $y=\lvert x-1\rvert+\lvert x-2\rvert$ and $y=3$ is equal to:

Topic: JEE Main 2023 (6 April Evening Shift)

4

If the coefficient of $x^{7}$ in $\left(a x^{2}+\dfrac{1}{2 b x}\right)^{11}$ and $x^{-7}$ in $\left(a x-\dfrac{1}{3 b …

Topic: JEE Main 2023 (6 April Evening Shift)

5

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary wi…

Topic: JEE Main 2023 (6 April Evening Shift)

1

If the solution curve $f(x,y)=0$ of the differential equation $(1+\log_e x)\frac{dx}{dy}-x\log_e x=e^y,\; x>0,$ pas…

Topic: JEE Main 2023 (6 April Evening Shift)

2

Let \(f(x)\) be a function satisfying \(f(x)+f(\pi-x)=\pi^{2}\), \(\forall x\in\mathbb{R}\). Then \(\displaystyle \int_…

Topic: JEE Main 2023 (6 April Evening Shift)

3

\[ \lim_{n\to\infty} \left\{ \left(2^{\tfrac12}-2^{\tfrac13}\right)\left(2^{\tfrac12}-2^{\tfrac15}\right)\cdots\left(2^…

Topic: JEE Main 2023 (6 April Evening Shift)

4

In a group of 100 persons, 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. I…

Topic: JEE Main 2023 (6 April Evening Shift)

5

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ an…

Topic: JEE Main 2023 (6 April Evening Shift)

6

For the system of equations \[ \begin{cases} x+y+z=6,\\ x+2y+\alpha z=10,\\ x+3y+5z=\beta, \end{cases} \] which one of …

Topic: JEE Main 2023 (6 April Evening Shift)

7

Let the sets A and B denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$,…

Topic: JEE Main 2023 (6 April Evening Shift)