A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $A B$ internally in the ratio $2: 1$ is $9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0$, then $\alpha-\beta-\gamma$ is equal to :
Previous 10 Questions — JEE Main 2025 (23 January Evening Shift)
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If in the expansion of $(1+x)^p(1-x)^q$, the coefficients of $x$ and $x^2$ are $1$ and $-2$, respectively, then $p^2+q^…
Topic: JEE Main 2025 (23 January Evening Shift)
Let $\mathrm{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x+y| \geqslant 3\}$ and $\mathrm{B}=\{(x, y) \in \mathbf{R} …
Topic: JEE Main 2025 (23 January Evening Shift)
Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\…
Topic: JEE Main 2025 (23 January Evening Shift)
$\displaystyle \lim_{x\to\infty}\frac{(2x^{2}-3x+5),(3x-1)^{x/2}}{(3x^{2}+5x+4),\sqrt{(3x+2)^{x}}}$ is equal to:
Topic: JEE Main 2025 (23 January Evening Shift)
The distance of the line
$\displaystyle \frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$
from the point $(1,4,0)$ along the l…
Topic: JEE Main 2025 (23 January Evening Shift)
The system of equations
$\begin{cases}
x+y+z=6,\\
x+2y+5z=9,\\
x+5y+\lambda z=\mu
\end{cases}$
has no solution if:
Topic: JEE Main 2025 (23 January Evening Shift)
Let the range of the function
$f(x)=6+16\cos x\cdot \cos\!\left(\frac{\pi}{3}-x\right)\cdot \cos\!\left(\frac{\pi}{3}+…
Topic: JEE Main 2025 (23 January Evening Shift)
If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq \mathrm{a}+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \m…
Topic: JEE Main 2025 (23 January Evening Shift)
Let X = ℝ × ℝ. Define a relation R on X by
(a₁,b₁) R (a₂,b₂) ⇔ b₁ = b₂.
Statement I: R is an equivalence relation.
St…
Topic: JEE Main 2025 (23 January Evening Shift)
If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2…
Topic: JEE Main 2025 (23 January Evening Shift)
Next 10 Questions — JEE Main 2025 (23 January Evening Shift)
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Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\r…
Topic: JEE Main 2025 (23 January Evening Shift)
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream…
Topic: JEE Main 2025 (23 January Evening Shift)
Let the shortest distance from $(a,0)$, $a>0$, to the parabola $y^{2}=4x$ be $4$.
Then the equation of the circle passi…
Topic: JEE Main 2025 (23 January Evening Shift)
Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin …
Topic: JEE Main 2025 (23 January Evening Shift)
The value of
$ \cot^{-1}\left(\dfrac{\sqrt{1+\tan^2(2)}-1}{\tan(2)}\right) - \cot^{-1}\left(\dfrac{\sqrt{1+\tan^2\left(…
Topic: JEE Main 2025 (23 January Evening Shift)
