JEE MAIN 2024 Previous Year Questions (PYQs) – Page 1 of 39

If the system of equations $x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0$ $x+(\cos\alpha)y+(\sin\alpha)z=0$ $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha\in\left(0,\frac{\pi}{2}\right)$ is equal to:

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\overline{z})^2 + |z| = 0,\; z \in \mathbb{C}$. Then $4(\alpha^2 + \beta^2)$ is equal to:

Let $f:\mathbb{R}\to\mathbb{R}$ be a function given by $ f(x)= \begin{cases} \dfrac{1-\cos 2x}{x^2}, & x<0,\\[6pt] \alpha, & x=0,\\[6pt] \dfrac{\beta\sqrt{\,1-\cos x\,}}{x}, & x>0, \end{cases} $ where $\alpha,\beta\in\mathbb{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to:

Consider a hyperbola $\text{H}$ having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola $\text{H}$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $\text{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of $\text{H}$ is:

If the coefficients of $x^4$, $x^5$, and $x^6$ in the expansion of $(1+x)^n$ are in arithmetic progression, then the maximum value of $n$ is:

The value of $\dfrac{1\times2^2+2\times3^2+\ldots+100\times(101)^2}{1\times3+2\times4+3\times5+\ldots+100\times101}$ is:

If the function $ f(x)= \begin{cases} \dfrac{7^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos^{2}x}}, & x\neq0,\\[6pt] a\log_{e}2\log_{e}3, & x=0 \end{cases} $ is continuous at $x=0$, then the value of $a^{2}$ is equal to:

Let $P$ be the point of intersection of the lines $\dfrac{x-2}{1}=\dfrac{y-4}{5}=\dfrac{z-2}{1}$ and $\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}$. Then, the shortest distance of $P$ from the line $4x=2y=z$ is:

Let $f(x)=3\sqrt{x-2}+\sqrt{4-x}$ be a real-valued function. If $\alpha$ and $\beta$ are respectively the minimum and maximum values of $f$, then $\alpha^2+2\beta^2$ is equal to:

Let $f(x)=\int_{0}^{x}\left(t+\sin(1-e^{t})\right)dt,\;x\in\mathbb{R}$. Then, $\displaystyle\lim_{x\to0}\dfrac{f(x)}{x^{3}}$ is equal to: