JEE Main 2024 Apr 6 Shift 1 — Mathematics

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, ...., 20\}$ be given by $R_1 = \{(x, y): 2x - 3y = 2\}$ and $R_2 = \{(x, y): -5x + 4y = 0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

The function $f(x) = \dfrac{x^2 + 2x - 15}{x^2 - 4x + 9}$, $x \in \mathbb{R}$ is

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $\alpha$, $\beta$ be the distinct roots of the equation $x^2 - (t^2 - 5t + 6)x + 1 = 0$, $t \in \mathbb{R}$ and $a_n = \alpha^n + \beta^n$. Then the minimum value of $\dfrac{a_{2023} + a_{2025}}{a_{2024}}$ is

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n-1)}{2} \end{vmatrix}$. Then $2A_{10} - A_8$ is

Let $f : (-\infty, \infty) - \{0\} \to \mathbb{R}$ be a differentiable function such that $f'(1) = \lim\limits_{a \to \infty} a^2 f\left(\dfrac{1}{a}\right)$. Then $\lim\limits_{a \to \infty} \dfrac{a(a+1)}{2} \tan^{-1}\left(\dfrac{1}{a}\right) + a^2 - 2\log_e a$ is equal to

Let $A = \{n \in [100, 700] \cap \mathbb{N}: n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

If $f(x) = \begin{cases} x^3 \sin\left(\dfrac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$, then

The interval in which the function $f(x) = x^x$, $x > 0$, is strictly increasing is

Let a variable line of slope $m > 0$ passing through the point $(4, -9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

$\displaystyle\int_0^{\pi/4} \dfrac{\cos^2 x \, \sin^2 x}{\left(\cos^3 x + \sin^3 x\right)^2} \, dx$ is equal to

Let the area of the region enclosed by the curves $y = 3x$, $2y = 27 - 3x$ and $y = 3x - x\sqrt{x}$ be $A$. Then $10A$ is equal to

Let $y = y(x)$ be the solution of the differential equation $\left(1 + x^2\right) \dfrac{dy}{dx} + y = e^{\tan^{-1} x}$, $y(1) = 0$. Then $y(0)$ is

Let $y = y(x)$ be the solution of the differential equation $(2x \log_e x) \dfrac{dy}{dx} + 2y = \dfrac{3}{x} \log_e x$, $x > 0$ and $y(e^{-1}) = 0$. Then, $y(e)$ is equal to

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n^2$ is equal to

Let $C$ be the circle of minimum area touching the parabola $y = 6 - x^2$ and the lines $y = \sqrt{3}\,|x|$. Then, which one of the following points lies on the circle $C$?

The shortest distance between the lines $\dfrac{x-3}{2} = \dfrac{y+15}{-7} = \dfrac{z-9}{5}$ and $\dfrac{x+1}{2} = \dfrac{y-1}{1} = \dfrac{z-9}{-3}$ is

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

If $A(3, 1, -1)$, $B\left(\dfrac{5}{3}, \dfrac{7}{3}, \dfrac{1}{3}\right)$, $C(2, 2, 1)$ and $D\left(\dfrac{10}{3}, \dfrac{2}{3}, \dfrac{-1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is

A company has two plants $A$ and $B$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. 80% of the motorcycles manufactured at plant $A$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126p$ is

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$ and $\left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \dfrac{125}{16}m$. Then the value of $m$ is ________

Let $\alpha\beta\gamma = 45$; $\alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$ for some $x, y, z \in \mathbb{R}$, $xyz \neq 0$, then $6\alpha + 4\beta + \gamma$ is equal to ________

If the second, third and fourth terms in the expansion of $(x + y)^n$ are 135, 30 and $\dfrac{10}{3}$, respectively, then $6(n^3 + x^2 + y)$ is equal to ________

Let the first term of a series be $T_1 = 6$ and its $r^{\text{th}}$ term $T_r = 3T_{r-1} + 6^r$, $r = 2, 3, ......, n$. If the sum of the first $n$ terms of this series is $\dfrac{1}{5}\left(n^2 - 12n + 39\right)\left(4 \cdot 6^n - 5 \cdot 3^n + 1\right)$, then $n$ is equal to ________.

Let $r_k = \dfrac{\int_0^1 \left(1 - x^7\right)^k \, dx}{\int_0^1 \left(1 - x^7\right)^{k+1} \, dx}$, $k \in \mathbb{N}$. Then the value of $\displaystyle\sum_{k=1}^{10} \dfrac{1}{7\left(r_k - 1\right)}$ is equal to ________

Let $L_1$, $L_2$ be the lines passing through the point $P(0, 1)$ and touching the parabola $9x^2 + 12x + 18y - 14 = 0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\triangle PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$, then $16\left(m_1^2 + m_2^2\right)$ is equal to ________.

Let a conic $C$ pass through the point $(4, -2)$ and $P(x, y)$, $x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3, -5)$. If the focal distance of the point $(7, 1)$ on $C$ is $d$, then $12d$ equals ________.

Let $P$ be the point $(10, -2, -1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1, 7, 6)$ on the line passing through the points $(2, -5, 11)$ and $(-6, 7, -5)$. Then the length of the line segment $PQ$ is equal to ________.

Let $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a} \times \left(\vec{b} + \vec{c}\right) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}$. If $\vec{a} \cdot \vec{c} = 13$, then $\left(24 - \vec{b} \cdot \vec{c}\right)$ is equal to ________.

For $n \in \mathbb{N}$, if $\cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \dfrac{\pi}{4}$, then $n$ is equal to ________.