Let $A$ be a $3\times3$ real matrix such that \[ A\!\begin{pmatrix}1\\0\\1\end{pmatrix} =2\!\begin{pmatrix}1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}-1\\0\\1\end{pmatrix} =4\!\begin{pmatrix}-1\\0\\1\end{pmatrix},\qquad A\!\begin{pmatrix}0\\1\\0\end{pmatrix} =2\!\begin{pmatrix}0\\1\\0\end{pmatrix}. \] Then, the system $(A-3I)\!\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ has:
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Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line
\[
\frac{x-1}{2}=\frac{y-2}{3}=\frac…
Topic: JEE Main 2024 (31 January Evening Shift)
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at le…
Topic: JEE Main 2024 (31 January Evening Shift)
If $a=\sin^{-1}(\sin 5)$ and $b=\cos^{-1}(\cos 5)$, then $a^{2}+b^{2}$ is equal to:
Topic: JEE Main 2024 (31 January Evening Shift)
Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$.
Let an ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}…
Topic: JEE Main 2024 (31 January Evening Shift)
The number of solutions of the equation $e^{\sin x}-2e^{-\sin x}=2$ is:
Topic: JEE Main 2024 (31 January Evening Shift)
The shortest distance, between lines $L_1$ and $L_2$, where $L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$…
Topic: JEE Main 2024 (31 January Evening Shift)
The area of the region enclosed by the parabolas
$y=4x-x^{2}$ and $3y=(x-4)^{2}$ is equal to:
Topic: JEE Main 2024 (31 January Evening Shift)
Let $f,g:(0,\infty)\to\mathbb{R}$ be defined by
$f(x)=\int_{-x}^{x}\big(|t|-t^{2}\big)e^{-t^{2}}\,dt,\qquad
g(x)=\int_{…
Topic: JEE Main 2024 (31 January Evening Shift)
Let $f:\mathbb{R}\to(0,\infty)$ be a strictly increasing function such that
$\displaystyle \lim_{x\to\infty}\frac{f(7x)…
Topic: JEE Main 2024 (31 January Evening Shift)
The temperature $T(t)$ of a body at time $t=0$ is $160^\circ\!F$ and it decreases
continuously as per the differential …
Topic: JEE Main 2024 (31 January Evening Shift)
