For three positive integers $p, q, r$, $x^{p q^{2}} = y^{q r} = z^{p^{2} r}$ and $r = pq + 1$ such that $3,\ 3\log_{y}x,\ 3\log_{z}y,\ 7\log_{x}z$ are in A.P. with common difference $\dfrac{1}{2}$. Then $r - p - q$ is equal to:

1

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^{2}+B=A^{2}B$, then:

Topic: JEE Main 2023 (24 January Morning Shift)

2

The equation $x^{2}-4x+[x]+3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

Topic: JEE Main 2023 (24 January Morning Shift)

3

The area enclosed by the curves $y^2 + 4x = 4$ and $y - 2x = 2$ is:

Topic: JEE Main 2023 (24 January Morning Shift)

4

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event. Given below are two statements: (S1): If $…

Topic: JEE Main 2023 (24 January Morning Shift)

5

Let $y=y(x)$ be the solution of the differential equation $x^{3}\,dy+(xy-1)\,dx=0,\quad x>0,$ with $y\!\left(\dfrac{1…

Topic: JEE Main 2023 (24 January Morning Shift)

6

$\tan^{-1}\!\left(\dfrac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\!\left(\sqrt{\dfrac{8+4\sqrt{3}}{6+3\sqrt{3}}}\righ…

Topic: JEE Main 2023 (24 January Morning Shift)

7

$\displaystyle \lim_{t\to 0}\left(1^{\frac{1}{\sin^2 t}}+2^{\frac{1}{\sin^2 t}}+\cdots+n^{\frac{1}{\sin^2 t}}\right)^{\…

Topic: JEE Main 2023 (24 January Morning Shift)

8

The relation $\mathsf{R} = \{(a,b) : \gcd(a,b)=1,\ 2a \ne b,\ a,b \in \mathbb{Z}\}$ is:

Topic: JEE Main 2023 (24 January Morning Shift)

9

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations …

Topic: JEE Main 2023 (24 January Morning Shift)

10

Let $p, q \in \mathbb{R}$ and $(1 - \sqrt{3}i)^{200} = 2^{199}(p + iq),\ i = \sqrt{-1}$ Then $p + q + q^2$ and $p - q…

Topic: JEE Main 2023 (24 January Morning Shift)

1

Let $\triangle PQR$ be a triangle. The points $A, B,$ and $C$ are on the sides $QR, RP,$ and $PQ$ respectively such tha…

Topic: JEE Main 2023 (24 January Morning Shift)

2

Let $$ f(x)= \begin{cases} x^{2}\sin\!\left(\dfrac{1}{x}\right), & x\ne 0,\\[6pt] 0, & x=0 \end{cases} $$ Then at $x=0…

Topic: JEE Main 2023 (24 January Morning Shift)

3

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\displaystyl…

Topic: JEE Main 2023 (24 January Morning Shift)