Let $L_1:\ \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $L_2:\ \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ be two lines. Which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

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Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac{\pi}{4}$, $\arg(z_2)=0$ …

Topic: JEE Main 2025 (22 January Morning Shift)

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Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its l…

Topic: JEE Main 2025 (22 January Morning Shift)

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The area of the region, inside the circle $ (x - 2\sqrt{3})^2 + y^2 = 12 $ and outside the parabola $ y^2 = 2\sqrt{3}x …

Topic: JEE Main 2025 (22 January Morning Shift)

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Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m< n\right.$ and $\left.\operatorname{gcd}(m, n)=1…

Topic: JEE Main 2025 (22 January Morning Shift)

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Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the…

Topic: JEE Main 2025 (22 January Morning Shift)

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The product of all solutions of the equation $e^{5(\log_e x)^2+3}=x^8,\ x>0$, is:

Topic: JEE Main 2025 (22 January Morning Shift)

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From all the English alphabets, five letters are chosen and arranged in alphabetical order. The total number of ways in…

Topic: JEE Main 2025 (22 January Morning Shift)

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A circle $C$ of radius $2$ lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a…

Topic: JEE Main 2025 (22 January Morning Shift)

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Let $x=x(y)$ be the solution of the differential equation $y^2\,dx+\left(x-\dfrac{1}{y}\right)dy=0$. If $x(1)=1$, then …

Topic: JEE Main 2025 (22 January Morning Shift)

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A coin is tossed three times. Let X denote the number of times a tail follows a head. If \mu and \sigma^2 denote the me…

Topic: JEE Main 2025 (22 January Morning Shift)

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Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for …

Topic: JEE Main 2025 (22 January Morning Shift)

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Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the…

Topic: JEE Main 2025 (22 January Morning Shift)

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Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms. If $a_1a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal t…

Topic: JEE Main 2025 (22 January Morning Shift)