Consider the lines $L_{1}$ and $L_{2}$ given by $L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2}$ $L_{2}:\ \dfrac{x-2}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}$ A line $L_{3}$ having direction ratios $1,-1,-2$ intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $PQ$ is:
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Let $y(x)=(1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$.
Then $y' - y''$ at $x=-1$ is equal to:
Topic: JEE Main 2023 (25 January Morning Shift)
The distance of the point $P(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with di…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $f(x)=\displaystyle \int \frac{2x}{(x^{2}+1)(x^{2}+3)}\,dx$.
If $f(3)=\dfrac{1}{2}(\log_{e}5-\log_{e}6)$, then $f…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $M$ be the maximum value of the product of two positive integers when their sum is $66$.
Let the sample space $S=…
Topic: JEE Main 2023 (25 January Morning Shift)
The value of $\displaystyle \lim_{n\to\infty}
\frac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n}
{\sqrt{2n^{4}+4n+3}-\sqrt{n^{…
Topic: JEE Main 2023 (25 January Morning Shift)
The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of on…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $x=2$ be a local minima of the function $f(x)=2x^{4}-18x^{2}+8x+12,\ x\in(-4,4)$.
If $M$ is the local maximum val…
Topic: JEE Main 2023 (25 January Morning Shift)
The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its wa…
Topic: JEE Main 2023 (25 January Morning Shift)
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The points of intersection of the line $ax+by=0,\ (a\ne b)$ and the circle $x^{2}+y^{2}-2x=0$ are $A(\alpha,0)$ and $B(…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $z_{1}=2+3i$ and $z_{2}=3+4i$. The set $S=\left\{\,z\in\mathbb{C}:\ |z-z_{1}|^{2}-|z-z_{2}|^{2}=|z_{1}-z_{2}|^{2}\,…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in \mathbb{R}\setminus\{0\}$ for which the system of linear …
Topic: JEE Main 2023 (25 January Morning Shift)
Let $y=y(x)$ be the solution curve of the differential equation
$\displaystyle \frac{dy}{dx}=\frac{y}{x}\bigl(1+xy^{2}…
Topic: JEE Main 2023 (25 January Morning Shift)
Let $f:(0,1)\to \mathbb{R}$ be a function defined by $f(x)=\dfrac{1}{1-e^{-x}}$, and $g(x)=\bigl(f(-x)-f(x)\bigr)$.&nbs…
Topic: JEE Main 2023 (25 January Morning Shift)
