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:
Previous 10 Questions — JEE Main 2024 (5 April Morning Shift)
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If the function $f(x)=\dfrac{\sin 3x+\alpha\sin x-\beta\cos 3x}{x^{3}},; x\in\mathbb{R},$ is continuous at $x=0$, then …
Topic: JEE Main 2024 (5 April Morning Shift)
Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\df…
Topic: JEE Main 2024 (5 April Morning Shift)
If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{…
Topic: JEE Main 2024 (5 April Morning Shift)
The integral $\displaystyle \int_{0}^{\pi/4}\frac{136\sin x}{3\sin x+5\cos x},dx$ is equal to:
Topic: JEE Main 2024 (5 April Morning Shift)
If the system
$11x+y+\lambda z=-5,\quad 2x+3y+5z=3,\quad 8x-19y-39z=\mu$
has infinitely many solutions, then $\lambda^{…
Topic: JEE Main 2024 (5 April Morning Shift)
The coefficients $a,b,c$ in the quadratic $ax^{2}+bx+c=0$ are chosen from the set ${1,2,3,4,5,6,7,8}$. The probability …
Topic: JEE Main 2024 (5 April Morning Shift)
Let $f(x)=x^{5}+2x^{3}+3x+1,; x\in\mathbb{R}$, and let $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbb{…
Topic: JEE Main 2024 (5 April Morning Shift)
Let a circle $C$ of radius $1$ and closer to the origin be such that the lines passing through the point $(3,2)$ and pa…
Topic: JEE Main 2024 (5 April Morning Shift)
Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then
$\left|,A^{T}A,( \operatorname{…
Topic: JEE Main 2024 (5 April Morning Shift)
Consider the following two statements:
Statement I: For any two non-zero complex numbers $z_1,z_2$,
$(|z_1|+|z_2|)\lef…
Topic: JEE Main 2024 (5 April Morning Shift)
Next 10 Questions — JEE Main 2024 (5 April Morning Shift)
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The value of $\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y},dy$ is:
Topic: JEE Main 2024 (5 April Morning Shift)
Suppose $\theta\in[0,\tfrac{\pi}{4}]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:
Topic: JEE Main 2024 (5 April Morning Shift)
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 increasin…
Topic: JEE Main 2024 (5 April Morning Shift)
Suppose $\theta\in\left[0,\tfrac{\pi}{4}\right]$ is a solution of $4\cos\theta-3\sin\theta=1$. Then $\cos\theta$ is:
Topic: JEE Main 2024 (5 April Morning Shift)
If $y=y(x)$ solves the differential equation $\dfrac{dy}{dx}+2y=\sin(2x)$ with $y(0)=\dfrac{3}{4}$, then $y!\left(\dfra…
Topic: JEE Main 2024 (5 April Morning Shift)
