JEE Main 2024 Apr 5 Shift 2 — Mathematics

Let the set $S = \{2, 4, 8, 16, \ldots, 512\}$ be partitioned into 3 sets $A$, $B$, $C$ with equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = \phi$. The maximum number of such possible partitions of $S$ is equal to :

Let $f, g : \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$f(x) = |x - 1|$ and $g(x) = \begin{cases} \mathrm{e}^{x}, & x \geq 0 \\ x + 1, & x \leq 0. \end{cases}$

Then the function $f(g(x))$ is

Let $S_1 = \{z \in \mathbf{C} : |z| \leq 5\}$, $S_2 = \left\{z \in \mathbf{C} : \operatorname{Im}\left(\dfrac{z + 1 - \sqrt{3}\,i}{1 - \sqrt{3}\,i}\right) \geq 0\right\}$ and $S_3 = \{z \in \mathbf{C} : \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :

The values of $m$, $n$, for which the system of equations

$x + y + z = 4$,

$2x + 5y + 5z = 17$,

$x + 2y + mz = n$

has infinitely many solutions, satisfy the equation :

Let $\alpha\beta \neq 0$ and $A = \begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}$. If $B = \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}$ is the matrix of cofactors of the elements of $A$, then $\det(AB)$ is equal to :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text{th}}$ word is :

The coefficients $a$, $b$, $c$ in the quadratic equation $ax^2 + bx + c = 0$ are from the set $\{1, 2, 3, 4, 5, 6\}$. If the probability of this equation having one real root bigger than the other is $p$, then $216p$ equals :

For $x \geq 0$, the least value of $K$, for which $4^{1+x} + 4^{1-x}$, $\dfrac{K}{2}$, $16^x + 16^{-x}$ are three consecutive terms of an A.P., is equal to :

If the constant term in the expansion of $\left(\dfrac{\sqrt[5]{3}}{x} + \dfrac{2x}{\sqrt[3]{5}}\right)^{12}$, $x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25\alpha$ is equal to :

Let $f : [-1, 2] \rightarrow \mathbf{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

Let $\beta(m, n) = \displaystyle\int_0^1 x^{m-1}(1-x)^{n-1}\,\mathrm{d}x$, $m, n > 0$. If $\displaystyle\int_0^1 (1 - x^{10})^{20}\,\mathrm{d}x = a \times \beta(b, c)$, then $100(a + b + c)$ equals ________.

The area enclosed between the curves $y = x|x|$ and $y = x - |x|$ is :

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius $r$ of the circle passing through the point F and touching the line segments BC and CD satisfies :

Let the circle $C_1 : x^2 + y^2 - 2(x + y) + 1 = 0$ and $C_2$ be a circle having centre at $(-1, 0)$ and radius 2. If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point P, then the square of the distance of P from the centre of $C_1$ is :

Let $A(-1, 1)$ and $B(2, 3)$ be two points and P be a variable point above the line AB such that the area of $\triangle PAB$ is 10. If the locus of P is $ax + by = 15$, then $5a + 2b$ is :

Let $(\alpha, \beta, \gamma)$ be the image of the point $(8, 5, 7)$ in the line $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-2}{5}$. Then $\alpha + \beta + \gamma$ is equal to :

Let $\vec{a} = 2\hat{i} + 5\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i})$. If $\vec{a} \cdot \vec{c} = -29$, then $\vec{c} \cdot (-2\hat{i} + \hat{j} + \hat{k})$ is equal to :

Consider three vectors $\vec{a}, \vec{b}, \vec{c}$. Let $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $\vec{a} = \vec{b} \times \vec{c}$. If $\alpha \in \left[0, \dfrac{\pi}{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c} - \vec{a}|^2$ is equal to :

If $y(\theta) = \dfrac{2\cos\theta + \cos 2\theta}{\cos 3\theta + 4\cos 2\theta + 5\cos\theta + 2}$, then at $\theta = \dfrac{\pi}{2}$, $y'' + y' + y$ is equal to :

The number of solutions of $\sin^2 x + (2 + 2x - x^2)\sin x - 3(x - 1)^2 = 0$, where $-\pi \leq x \leq \pi$, is ________.

If $1 + \dfrac{\sqrt{3} - \sqrt{2}}{2\sqrt{3}} + \dfrac{5 - 2\sqrt{6}}{18} + \dfrac{9\sqrt{3} - 11\sqrt{2}}{36\sqrt{3}} + \dfrac{49 - 20\sqrt{6}}{180} + \ldots \text{upto } \infty = 2 + \left(\sqrt{\dfrac{b}{a}} + 1\right)\log_{\mathrm{e}}\left(\dfrac{a}{b}\right)$, where $a$ and $b$ are integers with $\gcd(a, b) = 1$, then $11a + 18b$ is equal to ________.

Let $a > 0$ be a root of the equation $2x^2 + x - 2 = 0$. If $\lim\limits_{x \to \frac{1}{a}} \dfrac{16\left(1 - \cos(2 + x - 2x^2)\right)}{(1 - ax)^2} = \alpha + \beta\sqrt{17}$, where $\alpha, \beta \in \mathbb{Z}$, then $\alpha + \beta$ is equal to ________.

Let the maximum and minimum values of $\left(\sqrt{8x - x^2 - 12} - 4\right)^2 + (x - 7)^2$, $x \in \mathbf{R}$ be $M$ and $m$, respectively. Then $M^2 - m^2$ is equal to ________.

If $f(t) = \displaystyle\int_0^{\pi} \dfrac{2x\,\mathrm{d}x}{1 - \cos^2 t \sin^2 x}$, $0 < t < \pi$, then the value of $\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{\pi^2\,\mathrm{d}t}{f(t)}$ equals ________.

Let $y = y(x)$ be the solution of the differential equation

$$\dfrac{\mathrm{d}y}{\mathrm{d}x} + \dfrac{2x}{(1 + x^2)^2}\,y = x\,\mathrm{e}^{\frac{1}{(1 + x^2)}};\ y(0) = 0.$$

Then the area enclosed by the curve $f(x) = y(x)\,\mathrm{e}^{-\frac{1}{(1 + x^2)}}$ and the line $y - x = 4$ is ________.

Let a line perpendicular to the line $2x - y = 10$ touch the parabola $y^2 = 4(x - 9)$ at the point P. The distance of the point P from the centre of the circle $x^2 + y^2 - 14x - 8y + 56 = 0$ is ________.

Let the point $(-1, \alpha, \beta)$ lie on the line of the shortest distance between the lines $\dfrac{x + 2}{-3} = \dfrac{y - 2}{4} = \dfrac{z - 5}{2}$ and $\dfrac{x + 2}{-1} = \dfrac{y + 6}{2} = \dfrac{z - 1}{0}$. Then $(\alpha - \beta)^2$ is equal to ________.

Let the mean and the standard deviation of the probability distribution

$X$	$\alpha$	$1$	$0$	$-3$
$P(X)$	$\dfrac{1}{3}$	$K$	$\dfrac{1}{6}$	$\dfrac{1}{4}$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to ________.

The number of real solutions of the equation $x|x + 5| + 2|x + 7| - 2 = 0$ is ________.