JEE Main 2025 Jan 23 Shift 2 — Mathematics

Let $A = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geq 3\}$ and $B = \{(x, y) \in \mathbf{R} \times \mathbf{R} : |x| + |y| \leq 3\}$. If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$, then $\sum_{(x, y) \in C} |x+y|$ is :

Let $X = \mathbf{R} \times \mathbf{R}$. Define a relation R on X as :

$(a_1, b_1) \text{ R } (a_2, b_2) \Leftrightarrow b_1 = b_2$.

Statement I : R is an equivalence relation.

Statement II : For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y) \text{ R } (a, b)\}$ represents a line parallel to $y = x$.

In the light of the above statements, choose the correct answer from the options given below :

The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\dfrac{z}{\bar{z}} + \dfrac{\bar{z}}{z}\right| = 1$, is :

The system of equations $$x + y + z = 6,$$ $$x + 2y + 5z = 9,$$ $$x + 5y + \lambda z = \mu,$$ has no solution if

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, then $a_{23}$ equals :

If in the expansion of $(1+x)^{\mathrm{p}} (1-x)^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and $-2$, respectively, then $\mathrm{p}^2 + \mathrm{q}^2$ is equal to :

A board has 16 squares as shown in the figure :

a 4 × 4 grid of 16 squares

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

A rod of length eight units moves such that its ends A and B always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$, respectively. If the locus of the point P, that divides the rod AB internally in the ratio $2 : 1$ is $9(x^2 + \alpha y^2 + \beta xy + \gamma x + 28y) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to :

The length of the chord of the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1$, whose mid-point is $\left(1, \dfrac{1}{2}\right)$, is :

Let the shortest distance from $(a, 0)$, $a > 0$, to the parabola $y^2 = 4x$ be 4. Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

Let the range of the function $f(x) = 6 + 16\cos x \cdot \cos\left(\dfrac{\pi}{3} - x\right) \cdot \cos\left(\dfrac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is :

The distance of the line $\dfrac{x-2}{2} = \dfrac{y-6}{3} = \dfrac{z-3}{4}$ from the point $(1, 4, 0)$ along the line $\dfrac{x}{1} = \dfrac{y-2}{2} = \dfrac{z+3}{3}$ is :

If the square of the shortest distance between the lines $\dfrac{x-2}{1} = \dfrac{y-1}{2} = \dfrac{z+3}{-3}$ and $\dfrac{x+1}{2} = \dfrac{y+3}{4} = \dfrac{z+5}{-5}$ is $\dfrac{\mathrm{m}}{\mathrm{n}}$, where m, n are coprime numbers, then $\mathrm{m} + \mathrm{n}$ is equal to :

$\lim\limits_{x \to \infty} \dfrac{(2x^2 - 3x + 5)(3x - 1)^{\frac{x}{2}}}{(3x^2 + 5x + 4)\sqrt{(3x + 2)^x}}$ is equal to :

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81\ \mathrm{cm}^3/\text{min}$ and the thickness of the ice-cream layer decreases at the rate of $\dfrac{1}{4\pi}$ cm/min. The surface area (in $\mathrm{cm}^2$) of the chocolate ball (without the ice-cream layer) is :

If the area of the region $\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq \mathrm{a} + \mathrm{e}^{|x|} - \mathrm{e}^{-x}, \mathrm{a} > 0\}$ is $\dfrac{\mathrm{e}^2 + 8\mathrm{e} + 1}{\mathrm{e}}$, then the value of a is :

Let $\int x^3 \sin x\, \mathrm{d}x = g(x) + C$, where C is the constant of integration. If $8\left(g\left(\dfrac{\pi}{2}\right) + g'\left(\dfrac{\pi}{2}\right)\right) = \alpha\pi^3 + \beta\pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbf{Z}$, then $\alpha + \beta - \gamma$ equals :

If $\mathrm{I} = \displaystyle\int_0^{\frac{\pi}{2}} \dfrac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}\, \mathrm{d}x$, then $\displaystyle\int_0^{2\mathrm{I}} \dfrac{x \sin x \cos x}{\sin^4 x + \cos^4 x}\, \mathrm{d}x$ equals :

Let $x = x(y)$ be the solution of the differential equation $y = \left(x - y \dfrac{\mathrm{d}x}{\mathrm{d}y}\right) \sin\left(\dfrac{x}{y}\right)$, $y > 0$ and $x(1) = \dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to :

Let the point A divide the line segment joining the points $P(-1, -1, 2)$ and $Q(5, 5, 10)$ internally in the ratio $r : 1$ $(r > 0)$. If O is the origin and $\left(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}}\right) - \dfrac{1}{5}\left|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}\right|^2 = 10$, then the value of $r$ is :

The roots of the quadratic equation $3x^2 - \mathrm{p}x + \mathrm{q} = 0$ are $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with common difference $\dfrac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $\mathrm{q} - 2\mathrm{p}$ is equal to _________.

Let $\alpha, \beta$ be the roots of the equation $x^2 - \mathrm{a}x - \mathrm{b} = 0$ with $\text{Im}(\alpha) < \text{Im}(\beta)$. Let $\mathrm{P_n} = \alpha^{\mathrm{n}} - \beta^{\mathrm{n}}$. If $\mathrm{P_3} = -5\sqrt{7}\, i$, $\mathrm{P_4} = -3\sqrt{7}\, i$, $\mathrm{P_5} = 11\sqrt{7}\, i$ and $\mathrm{P_6} = 45\sqrt{7}\, i$, then $|\alpha^4 + \beta^4|$ is equal to _________.

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is _________.

The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is _________.

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle C of radius 5. If the values of $\lambda$, for which C passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$, are $\lambda_1$ and $\lambda_2$, $\lambda_1 < \lambda_2$, then $12\lambda_1 + 29\lambda_2$ is equal to _________.