JEE Main 2024 Apr 4 Shift 2 — Mathematics

Let a relation R on $\mathbb{N} \times \mathbb{N}$ be defined as: $(x_1, y_1)$ R $(x_2, y_2)$ if and only if $x_1 \le x_2$ or $y_1 \le y_2$. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?

The area (in sq. units) of the region $S = \{z \in \mathbb{C} : |z-1| \le 2; (z+\bar{z}) + i(z-\bar{z}) \le 2, \operatorname{Im}(z) \ge 0\}$ is

Let $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = I + \operatorname{adj}(A) + (\operatorname{adj} A)^2 + \ldots + (\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

If the coefficients of $x^4$, $x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$ is

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a$, $b$ and $c$ is $8$, then the cube of the geometric mean of $a$, $b$ and $c$ is

Let $f(x) = 3\sqrt{x-2} + \sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2 + 2\beta^2$ is equal to

If the function $$f(x) = \begin{cases} \dfrac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}} & , \ x \ne 0 \\ a \log_e 2 \log_e 3 & , \ x = 0 \end{cases}$$ is continuous at $x = 0$, then the value of $a^2$ is equal to

Let $f(x) = \int_0^x \left(t + \sin\left(1 - e^t\right)\right) dt, x \in \mathbb{R}$. Then, $\lim\limits_{x \to 0} \dfrac{f(x)}{x^3}$ is equal to

If the value of the integral $\int_{-1}^{1} \dfrac{\cos \alpha x}{1 + 3^x} dx$ is $\dfrac{2}{\pi}$. Then, a value of $\alpha$ is

The area (in sq. units) of the region described by $\{(x, y) : y^2 \le 2x, \text{ and } y \ge 4x - 1\}$ is

Let $y = y(x)$ be the solution of the differential equation $(x^2 + 4)^2 dy + (2x^3y + 8xy - 2)dx = 0$. If $y(0) = 0$, then $y(2)$ is equal to

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope $-1$. Then, a distance (in units) between the chord PQ and the chord MN is

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola H and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of H is

Let PQ be a chord of the parabola $y^2 = 12x$ and the midpoint of PQ be at $(4, 1)$. Then, which of the following point lies on the line passing through the points P and Q?

Let P be the point of intersection of the lines $\dfrac{x-2}{1} = \dfrac{y-4}{5} = \dfrac{z-2}{1}$ and $\dfrac{x-3}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{2}$. Then, the shortest distance of P from the line $4x = 2y = z$ is

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, x \in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b} + \vec{c}$ such that $\vec{a} \cdot \vec{d} = 1$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

For $\lambda > 0$, let $\theta$ be the angle between the vectors $\vec{a} = \hat{i} + \lambda\hat{j} - 3\hat{k}$ and $\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$. If the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are mutually perpendicular, then the value of $(14 \cos\theta)^2$ is equal to

If the mean of the following probability distribution of a radam variable X:

X	0	2	4	6	8
P(X)	$a$	$2a$	$a+b$	$2b$	$3b$

is $\dfrac{46}{9}$, then the variance of the distribution is

Given that the inverse trigonometric function assumes principal values only. Let $x$, $y$ be any two real numbers in $[-1, 1]$ such that $\cos^{-1} x - \sin^{-1} y = \alpha$, $\dfrac{-\pi}{2} \le \alpha \le \pi$. Then, the minimum value of $x^2 + y^2 + 2xy \sin\alpha$ is

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{2x}{\sqrt{1 + 9x^2}}$. If the composition of $f$, $\underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text{ times}}(x) = \dfrac{2^{10} x}{\sqrt{1 + 9\alpha x^2}}$, then the value of $\sqrt{3\alpha + 1}$ is equal to ______

Let $S = \{\sin^2 2\theta : (\sin^4\theta + \cos^4\theta)x^2 + (\sin 2\theta)x + (\sin^6\theta + \cos^6\theta) = 0$ has real roots$\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set S, respectively, then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals _________

Let $A$ be a $2 \times 2$ symmetric matrix such that $A \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$ and the determinant of $A$ be 1. If $A^{-1} = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals _________

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ___________

Let $f : \mathbb{R} \to \mathbb{R}$ be a thrice differentiable function such that $f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2$ and $f(4) = -2$. Then, the minimum number of zeros of $(3f'f'' + ff''')(x)$ is ___________

If $\int \operatorname{cosec}^5 x \, dx = \alpha \cot x \operatorname{cosec} x \left(\operatorname{coesc}^2 x + \dfrac{3}{2}\right) + \beta \log_e \left|\tan \dfrac{x}{2}\right| + C$ where $\alpha, \beta \in \mathbb{R}$ and C is the constant of integration, then the value of $8(\alpha + \beta)$ equals ________

Let $y = y(x)$ be the solution of the differential equation $(x + y + 2)^2 dx = dy$, $y(0) = -2$. Let the maximum and minimum values of the function $y = y(x)$ in $\left[0, \dfrac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma + \delta$ equals ________

Consider a triangle ABC having the vertices $A(1, 2)$, $B(\alpha, \beta)$ and $C(\gamma, \delta)$ and angles $\angle ABC = \dfrac{\pi}{6}$ and $\angle BAC = \dfrac{2\pi}{3}$. If the points B and C lie on the line $y = x + 4$, then $\alpha^2 + \gamma^2$ is equal to _____.

Consider a line L passing through the points $P(1, 2, 1)$ and $Q(2, 1, -1)$. If the mirror image of the point $A(2, 2, 2)$ in the line L is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + 6\gamma$ is equal to _______.

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\dfrac{1}{3}$ and $\dfrac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|x - y| \le 2)$ is $p$, then $3^9 p$ equals ___________.