Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1$ at the points $A$ and $B$. Let the line $y=x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is
Previous 10 Questions — JEE Main 2025 (4 April Evening Shift)
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If the sum of the first $20$ terms of the series $\dfrac{4\cdot1}{4+3\cdot1^{2}+1^{4}}+\dfrac{4\cdot2}{4+3\cdot2^{2}+2^…
Topic: JEE Main 2025 (4 April Evening Shift)
If a curve $y=y(x)$ passes through the point $\left(1,\dfrac{\pi}{2}\right)$ and satisfies the differential equation $(…
Topic: JEE Main 2025 (4 April Evening Shift)
Let $a>0$. If the function $f(x)=6x^3-45ax^2+108a^2x+1$ attains its local maximum and minimum values at the points $x_1…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the sum of the focal distances of the point $P(4,3)$ on the hyperbola $H:\ \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the mean and the standard deviation of the observation $2,3,3,3,4,5,7,a,b$ be $4$ and $\sqrt{2}$ respectively. Then…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the domains of the functions $f(x)=\log_{4}\big(\log_{3}\big(\log_{7}\big(8-\log_{2}(x^{2}+4x+5)\big)\big)\big)$ an…
Topic: JEE Main 2025 (4 April Evening Shift)
Let $A={-3,-2,-1,0,1,2,3}$ and $R$ be a relation on $A$ defined by $xRy$ iff $2x-y\in{0,1}$. Let $l$ be the number of e…
Topic: JEE Main 2025 (4 April Evening Shift)
Next 10 Questions — JEE Main 2025 (4 April Evening Shift)
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The centre of a circle $C$ is at the centre of the ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$. Let…
Topic: JEE Main 2025 (4 April Evening Shift)
A line passing through the point $A(-2,0)$ touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. Th…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\al…
Topic: JEE Main 2025 (4 April Evening Shift)
If $1^{2}\cdot{^{15}C_{1}}+2^{2}\cdot{^{15}C_{2}}+3^{2}\cdot{^{15}C_{3}}+\cdots+15^{2}\cdot{^{15}C_{15}}=2^{m}\cdot3^{n…
Topic: JEE Main 2025 (4 April Evening Shift)
Let $A$ be the point of intersection of the lines $L_{1}:\ \dfrac{x-7}{1}=\dfrac{y-5}{0}=\dfrac{z-3}{-1}$ and $L_{2}:\ …
Topic: JEE Main 2025 (4 April Evening Shift)
The sum of the infinite series $\cot^{-1}\left(\dfrac{7}{4}\right)+\cot^{-1}\left(\dfrac{19}{4}\right)+\cot^{-1}\left(\…
Topic: JEE Main 2025 (4 April Evening Shift)
The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and…
Topic: JEE Main 2025 (4 April Evening Shift)
Let $f(x)+2f!\left(\frac{1}{x}\right)=x^{2}+5$ and $2g(x)-3g!\left(\frac{1}{x}\right)=x$, $x>0$. If $\alpha=\displaysty…
Topic: JEE Main 2025 (4 April Evening Shift)
Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of $A$…
Topic: JEE Main 2025 (4 April Evening Shift)
Let the values of $p$, for which the shortest distance between the lines
$\dfrac{x+1}{3}=\dfrac{y}{4}=\dfrac{z}{5}$ and…
Topic: JEE Main 2025 (4 April Evening Shift)
