Line $L_1$ passes through the point $(1, 2, 3)$ and is parallel to the $z$-axis. Line $L_2$ passes through the point $(\lambda, 5, 6)$ and is parallel to the $y$-axis. Let for $\lambda = \lambda_1, \lambda_2,$ $\lambda_2 < \lambda_1,$ the shortest distance between the two lines be $3$. Then the square of the distance of the point $(\lambda_1, \lambda_2, 7)$ from the line $L_1$ is
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A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse
$\dfrac{x^2}{36} + \dfrac{y^2}{25} = 1$…
Topic: JEE Main 2025 (3 April Morning Shift)
Let $\quad f(x)= \begin{cases}(1+a x)^{1 / x} & , x0\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to:
Topic: JEE Main 2025 (3 April Morning Shift)
Let a line passing through the point $(4,1,0)$ intersect the line $\mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}…
Topic: JEE Main 2025 (3 April Morning Shift)
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3…
Topic: JEE Main 2025 (3 April Morning Shift)
Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$.
If $\int_0^{b - a} [x^2] , dx …
Topic: JEE Main 2025 (3 April Morning Shift)
Let $f(x) = \int x^3 \sqrt{3 - x^2} , dx.$ If $5f(\sqrt{2}) = -4$, then $f(1)$ is equal to
Topic: JEE Main 2025 (3 April Morning Shift)
The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is
Topic: JEE Main 2025 (3 April Morning Shift)
Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $0 \leq x^2+…
Topic: JEE Main 2025 (3 April Morning Shift)
$
\text { The number of solutions of the equation } 2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \…
Topic: JEE Main 2025 (3 April Morning Shift)
The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ up to $20$ terms is equal to:
Topic: JEE Main 2025 (3 April Morning Shift)
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Let $g$ be a differentiable function such that
$\displaystyle \int_0^x g(t),dt = x - \int_0^x t g(t),dt,; x \ge 0$
and …
Topic: JEE Main 2025 (3 April Morning Shift)
The sum of all rational terms in the expansion of $(2 + \sqrt{3})^8$ is:
Topic: JEE Main 2025 (3 April Morning Shift)
If $\displaystyle \sum_{r=1}^{9} \left(\dfrac{r + 3}{2^r}\right) \cdot {^9C_r} = \alpha \left(\dfrac{3}{2}\right)^9 - \…
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Let $z \in \mathbb{C}$ be such that $\dfrac{z^2+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^2$ is:
Topic: JEE Main 2025 (3 April Morning Shift)
$
\text { If } y(x)=\left|\begin{array}{ccc}
\sin x & \cos x & \sin x+\cos x+1 \\
27 & 28 & 27 \\
1 & 1 & 1
\end{array}…
Topic: JEE Main 2025 (3 April Morning Shift)
A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1…
Topic: JEE Main 2025 (3 April Morning Shift)
Let $A$ be a matrix of order $3\times 3$ and $|A|=5$. If $\left|,2,\operatorname{adj}\left(3A,\operatorname{adj}(2A)\ri…
Topic: JEE Main 2025 (3 April Morning Shift)
