JEE Main 2024 Apr 8 Shift 2 — Mathematics

Let $A = \{2, 3, 6, 8, 9, 11\}$ and $B = \{1, 4, 5, 10, 15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)R(c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is

Let $f(x) = \begin{cases} -a & \text{if } -a \le x \le 0 \\ x + a & \text{if } 0 < x \le a \end{cases}$ where $a > 0$ and $g(x) = (f(|x|) - |f(x)|)/2$.

Then the function $g : [-a, a] \to [-a, a]$ is

If the system of equations $x + 4y - z = \lambda$, $7x + 9y + \mu z = -3$, $5x + y + 2z = -1$ has infinitely many solutions, then $(2\mu + 3\lambda)$ is equal to :

If $\alpha \ne a$, $\beta \ne b$, $\gamma \ne c$ and $\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0$, then $\dfrac{a}{\alpha - a} + \dfrac{b}{\beta - b} + \dfrac{\gamma}{\gamma - c}$ is equal to :

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

If the term independent of $x$ in the expansion of $\left( \sqrt{a}x^2 + \dfrac{1}{2x^3} \right)^{10}$ is 105, then $a^2$ is equal to :

The sum of all possible values of $\theta \in [-\pi, 2\pi]$, for which $\dfrac{1 + i\cos\theta}{1 - 2i\cos\theta}$ is purely imaginary, is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\dfrac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{\text{th}}$, $6^{\text{th}}$ and $8^{\text{th}}$ terms is equal to :

For $a, b > 0$, let $f(x) = \begin{cases} \dfrac{\tan((a+1)x) + b\tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \dfrac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a}\, x\sqrt{x}}, & x > 0 \end{cases}$

be a continous function at $x = 0$. Then $\dfrac{b}{a}$ is equal to :

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$, $a > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :

Let $\displaystyle\int_{\alpha}^{\log_e 4} \dfrac{\mathrm{d}x}{\sqrt{\mathrm{e}^x - 1}} = \dfrac{\pi}{6}$. Then $\mathrm{e}^{\alpha}$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :

The area of the region in the first quadrant inside the circle $x^2 + y^2 = 8$ and outside the parabola $y^2 = 2x$ is equal to :

Let $y = y(x)$ be the solution curve of the differential equation $\sec y \dfrac{\mathrm{d}y}{\mathrm{d}x} + 2x\sin y = x^3\cos y$, $y(1) = 0$. Then $y(\sqrt{3})$ is equal to :

If the line segment joining the points $(5, 2)$ and $(2, a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

If the image of the point $(-4, 5)$ in the line $x + 2y = 2$ lies on the circle $(x+4)^2 + (y-3)^2 = r^2$, then $r$ is equal to :

If the shortest distance between the lines $\dfrac{x - \lambda}{2} = \dfrac{y - 4}{3} = \dfrac{z - 3}{4}$ and $\dfrac{x - 2}{4} = \dfrac{y - 4}{6} = \dfrac{z - 7}{8}$ is $\dfrac{13}{\sqrt{29}}$, then a value of $\lambda$ is :

Let $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = 11\hat{i} - \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (-2\vec{a} + 3\vec{b})$. If $(2\vec{a} + 3\vec{b}) \cdot \vec{c} = 1670$, then $|\vec{c}|^2$ is equal to :

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$ and $\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b} + \vec{c}$. If $\vec{r} \cdot \vec{a} = 3$, then $3\lambda$ is equal to :

There are three bags X, Y and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :

If the value of $\dfrac{3\cos 36^\circ + 5\sin 18^\circ}{5\cos 36^\circ - 3\sin 18^\circ}$ is $\dfrac{a\sqrt{5} - b}{c}$, where $a$, $b$, $c$ are natural numbers and $\gcd(a, c) = 1$, then $a + b + c$ is equal to :

The number of distinct real roots of the equation $|x+1|\,|x+3| - 4|x+2| + 5 = 0$, is ________

Let a ray of light passing through the point $(3, 10)$ reflects on the line $2x + y = 6$ and the reflected ray passes through the point $(7, 2)$. If the equation of the incident ray is $ax + by + 1 = 0$, then $a^2 + b^2 + 3ab$ is equal to ________.

An arithmetic progression is written in the following way

$$\begin{array}{ccccccc} & & & 2 & & & \\ & & 5 & & 8 & & \\ & 11 & & 14 & & 17 & \\ 20 & & 23 & & 26 & & 29 \end{array}$$ $$- - - - - - - - - - - - - -$$ $$- - - - - - - - - - - - - - - -$$

The sum of all the terms of the $10^{\text{th}}$ row is ________.

Let A be the region enclosed by the parabola $y^2 = 2x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is ________.

If $\alpha = \displaystyle\lim_{x \to 0^+} \left( \dfrac{\mathrm{e}^{\sqrt{\tan x}} - \mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right)$ and $\beta = \displaystyle\lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $ax^2 + bx - \sqrt{\mathrm{e}} = 0$, then $12\log_e(a + b)$ is equal to ________.

If $\displaystyle\int \dfrac{1}{\sqrt[5]{(x-1)^4 (x+3)^6}}\, \mathrm{d}x = A\left( \dfrac{\alpha x - 1}{\beta x + 3} \right)^{B} + C$, where C is the constant of integration, then the value of $\alpha + \beta + 20AB$ is ________.

Let $\alpha|x| = |y|\mathrm{e}^{xy - \beta}$, $\alpha, \beta \in \mathbb{N}$ be the solution of the differential equation $x\mathrm{d}y - y\mathrm{d}x + xy(x\mathrm{d}y + y\mathrm{d}x) = 0$, $y(1) = 2$. Then $\alpha + \beta$ is equal to ________

Let S be the focus of the hyperbola $\dfrac{x^2}{3} - \dfrac{y^2}{5} = 1$, on the positive $x$-axis. Let C be the circle with its centre at $\mathrm{A}\left(\sqrt{6}, \sqrt{5}\right)$ and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to ________

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(1, 6, 4)$ in the line $\dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-2}{3}$. Then $2\alpha + \beta + \gamma$ is equal to ________

Let $a, b, c \in \mathbb{N}$ and $a < b < c$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9, 25, a, b, c$ be 18, 4 and $\dfrac{136}{5}$, respectively. Then $2a + b - c$ is equal to ________