Let X = ℝ × ℝ. Define a relation R on X by (a₁,b₁) R (a₂,b₂) ⇔ b₁ = b₂. Statement I: R is an equivalence relation. Statement II: For some (a,b) ∈ X, the set S = { (x,y) ∈ X : (x,y) R (a,b) } represents a line parallel to y = x.

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If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2…

Topic: JEE Main 2025 (23 January Evening Shift)

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Let the point $A$ divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r:1\…

Topic: JEE Main 2025 (23 January Evening Shift)

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The length of the chord of the ellipse $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{2}=1$ whose midpoint is $\left(1,\dfrac{1}{2}\ri…

Topic: JEE Main 2025 (23 January Evening Shift)

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The number of complex numbers $z$ satisfying $|z|=1$ and $\left|\dfrac{z}{\overline{z}}+\dfrac{\overline{z}}{z}\right|=…

Topic: JEE Main 2025 (23 January Evening Shift)

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If $I=\displaystyle\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$, then $\displaystyle\int_{0}^{2I}\f…

Topic: JEE Main 2025 (23 January Evening Shift)

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If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq \mathrm{a}+\mathrm{e}^{|x|}-\mathrm{e}^{-x}, \m…

Topic: JEE Main 2025 (23 January Evening Shift)

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Let the range of the function $f(x)=6+16\cos x\cdot \cos\!\left(\frac{\pi}{3}-x\right)\cdot \cos\!\left(\frac{\pi}{3}+…

Topic: JEE Main 2025 (23 January Evening Shift)

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The system of equations $\begin{cases} x+y+z=6,\\ x+2y+5z=9,\\ x+5y+\lambda z=\mu \end{cases}$ has no solution if:

Topic: JEE Main 2025 (23 January Evening Shift)

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The distance of the line $\displaystyle \frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the l…

Topic: JEE Main 2025 (23 January Evening Shift)

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$\displaystyle \lim_{x\to\infty}\frac{(2x^{2}-3x+5),(3x-1)^{x/2}}{(3x^{2}+5x+4),\sqrt{(3x+2)^{x}}}$ is equal to:

Topic: JEE Main 2025 (23 January Evening Shift)

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Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\…

Topic: JEE Main 2025 (23 January Evening Shift)

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Let $\mathrm{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x+y| \geqslant 3\}$ and $\mathrm{B}=\{(x, y) \in \mathbf{R} …

Topic: JEE Main 2025 (23 January Evening Shift)

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If in the expansion of $(1+x)^p(1-x)^q$, the coefficients of $x$ and $x^2$ are $1$ and $-2$, respectively, then $p^2+q^…

Topic: JEE Main 2025 (23 January Evening Shift)

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A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respect…

Topic: JEE Main 2025 (23 January Evening Shift)

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Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\r…

Topic: JEE Main 2025 (23 January Evening Shift)