JEE MAIN 2023 Previous Year Questions (PYQs) – Page 19 of 34

Let $z_{1}=2+3i$ and $z_{2}=3+4i$. The set $S=\left\{\,z\in\mathbb{C}:\ |z-z_{1}|^{2}-|z-z_{2}|^{2}=|z_{1}-z_{2}|^{2}\,\right\}$ represents a

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in \mathbb{R}\setminus\{0\}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

Let $\mathrm{P}\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9x^{2}+4y^{2}=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ be mutually perpendicular and pass through the origin. If $\dfrac{1}{(PQ)^{2}}+\dfrac{1}{(RS)^{2}}=\dfrac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to:

Let $y=y(x)$ be the solution curve of the differential equation $\displaystyle \frac{dy}{dx}=\frac{y}{x}\bigl(1+xy^{2}(1+\log_{e}x)\bigr),\ x>0,\ y(1)=3.$ Then $\displaystyle \frac{y^{2}(x)}{9}$ is equal to:

If the local maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\sin^{2}x},; x\in\left(0,\dfrac{\pi}{2}\right),$ is $\dfrac{k}{e},$ then $\left(\dfrac{k}{e}\right)^{8}+\dfrac{k^{8}}{e^{5}}+k^{8}$ is equal to:

$ \text{The area of the region enclosed by the curve } y=x^{3} \text{ and its tangent at the point } (-1,-1) \text{ is: } $

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^{2}=6x$. The locus of its circumcentre is:

Let $D$ be the domain of the function $f(x)=\sin^{-1}\!\left(\log_{3x}\!\left(\dfrac{6+2\log_{3}x}{-5x}\right)\right)$. If the range of the function $g: D \to \mathbb{R}$ defined by $g(x)=x-[x]$ (where $[x]$ is the greatest integer function) is $(\alpha,\beta)$, then $\alpha^{2}+\dfrac{5}{\beta}$ is equal to:

The foot of the perpendicular from the point $(2,0,5)$ on the line $\dfrac{x+1}{2}=\dfrac{y-1}{5}=\dfrac{z+1}{-1}$ is $(\alpha,\beta,\gamma)$. Then, which of the following is NOT correct?