Let $H:\dfrac{-x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be the hyperbola whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha,6)$, $\alpha>0$, lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha,6)$, then $\alpha^2+\beta$ is equal to:
Previous 10 Questions — JEE Main 2024 (8 April Morning Shift)
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Let $f(x)$ be a positive function such that the area bounded by $y=f(x)$, $y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^{2}…
Topic: JEE Main 2024 (8 April Morning Shift)
Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\dfrac{15}{2}\right)^2=r_2^2$ touch each …
Topic: JEE Main 2024 (8 April Morning Shift)
The set of all $\alpha$ for which the vectors $\vec a=\alpha t,\hat i+6,\hat j-3,\hat k$ and $\vec b=t,\hat i-2,\hat j-…
Topic: JEE Main 2024 (8 April Morning Shift)
Let $f(x)=4\cos^{3}x+3\sqrt{3}\cos^{2}x-10$. The number of points of local maxima of $f$ in the interval $(0,2\pi)$ is:
Topic: JEE Main 2024 (8 April Morning Shift)
The value of $k\in\mathbb{N}$ for which the integral $I_n=\displaystyle\int_{0}^{1}(1-x^{k})^{n},dx,\ n\in\mathbb{N}$, …
Topic: JEE Main 2024 (8 April Morning Shift)
Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$…
Topic: JEE Main 2024 (8 April Morning Shift)
Next 10 Questions — JEE Main 2024 (8 April Morning Shift)
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The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:
Topic: JEE Main 2024 (8 April Morning Shift)
Let $y=y(x)$ be the solution of the differential equation
$(1+y^{2})e^{\tan x},dx+\cos^{2}x,(1+e^{2\tan x}),dy=0$, $y(0…
Topic: JEE Main 2024 (8 April Morning Shift)
Let the sum of two positive integers be $24$. If the probability that their product is not less than $\dfrac{3}{4}$ tim…
Topic: JEE Main 2024 (8 April Morning Shift)
For the function $f(x)=\cos x - x + 1,; x\in\mathbb{R}$, consider the statements
(S1) $f(x)=0$ for only one value of $x…
Topic: JEE Main 2024 (8 April Morning Shift)
Let $P(x,y,z)$ be a point in the first octant whose projection on the $xy$–plane is $Q$. Let $OP=\gamma$; the angle bet…
Topic: JEE Main 2024 (8 April Morning Shift)
Let $z$ be a complex number such that $\lvert z+2\rvert=1$ and $\operatorname{Im}!\left(\dfrac{z+1}{z+2}\right)=\dfrac{…
Topic: JEE Main 2024 (8 April Morning Shift)
The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^x+48=0$ is:
Topic: JEE Main 2024 (8 April Morning Shift)
The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4x+y=14$ and $3x-2y=5$, respectively. The point $\lef…
Topic: JEE Main 2024 (8 April Morning Shift)
If $\sin x=-\frac{3}{5}$, where $\pi< x
Topic: JEE Main 2024 (8 April Morning Shift)
Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A …
Topic: JEE Main 2024 (8 April Morning Shift)
