Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac{\pi}{4}$, $\arg(z_2)=0$ and $\arg(z_3)=\frac{\pi}{4}$. If $\left|\,z_1\overline{z_2}+z_2\overline{z_3}+z_3\overline{z_1}\,\right|^2=\alpha+\beta\sqrt{2}$, $\alpha,\beta\in\mathbb{Z}$, then the value of $\alpha^2+\beta^2$ is:
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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…
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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 …
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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 tw…
Topic: JEE Main 2025 (22 January Morning Shift)
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)
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)
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)
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)
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)
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)
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)
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)
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)
