JEE MAIN 2024 Previous Year Questions (PYQs) – Page 5 of 39

Let the line $2x+3y-k=0,\ k>0$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. If the circle having $AB$ as a diameter is $x^{2}+y^{2}-3x-2y=0$ and the length of the latus rectum of the ellipse $x^{2}+9y^{2}=k^{2}$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $2m+n$ is equal to:

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and $I_1=\displaystyle\int_a^b x\sin(4x-x^2)\,dx,\ \ I_2=\displaystyle\int_a^b \sin(4x-x^2)\,dx,$ then $36\,\dfrac{I_1}{I_2}$ is equal to:

The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y},dy$ is:

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\dfrac{dx}{dt}+ax=0$ and $\dfrac{dy}{dt}+by=0$ respectively, $a,b\in\mathbb{R}$. Given that $x(0)=2$, $y(0)=1$ and $3y(1)=2x(1)$, the value of $t$ for which $x(t)=y(t)$ is:

Suppose $\theta\in[0,\tfrac{\pi}{4}]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:

The distance of the point $(7,-2,11)$ from the line $\dfrac{x-6}{1}=\dfrac{y-4}{0}=\dfrac{z-8}{3}$ along the line $\dfrac{x-5}{2}=\dfrac{y-1}{-3}=\dfrac{z-5}{6}$ is:

For $f(x)=\sin x+3x-\dfrac{2}{\pi}(x^{2}+x)$, where $x\in\left[0,\tfrac{\pi}{2}\right]$, consider: (I) $f$ is increasing in $\left(0,\tfrac{\pi}{2}\right)$. (II) $f'$ is decreasing in $\left(0,\tfrac{\pi}{2}\right)$.

The length of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose midpoint is $\left(1,\dfrac{9}{5}\right)$, is equal to:

Suppose $\theta\in\left[0,\tfrac{\pi}{4}\right]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:

Consider the function $ f(x)= \begin{cases} \dfrac{a\,(7x-12-x^{2})}{\,b\,\lfloor x^{2}-7x+12\rfloor\,}, & x<3,\\[6pt] \dfrac{\sin(x-3)}{2^{\,x-1}}, & x>3,\\[6pt] b, & x=3, \end{cases} $ where $\lfloor x\rfloor$ denotes the greatest integer $\le x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is: