JEE Main 2024 Apr 4 Shift 1 — Mathematics

If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)+\log_{e}\left(\frac{3x^{2}-8x+5}{x^{2}-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha+10\beta$ is equal to :

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0$, $z \in \mathbf{C}$. Then $4(\alpha^{2}+\beta^{2})$ is equal to :

If 2 and 6 are the roots of the equation $ax^{2}+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is :

If the system of equations

$$x+(\sqrt{2}\sin\alpha)y+(\sqrt{2}\cos\alpha)z=0$$ $$x+(\cos\alpha)y+(\sin\alpha)z=0$$ $$x+(\sin\alpha)y-(\cos\alpha)z=0$$

has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to :

Let $\alpha \in (0, \infty)$ and $A=\begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\det(\mathrm{adj}(2A-A^{T})\cdot\mathrm{adj}(A-2A^{T}))=2^{8}$, then $(\det(A))^{2}$ is equal to :

Let $f: \mathbf{R} \to \mathbf{R}$ be a function given by

$$f(x)=\begin{cases} \frac{1-\cos 2x}{x^{2}}, & x<0 \\ \alpha, & x=0, \\ \frac{\beta\sqrt{1-\cos x}}{x}, & x>0 \end{cases}$$

where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^{2}+\beta^{2}$ is equal to :

There are 5 points $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $P_{6}, P_{7}, \ldots, P_{11}$ on the side BC and 7 points $P_{12}, P_{13}, \ldots, P_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $P_{1}, P_{2}, \ldots, P_{18}$ as vertices, is :

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

Let the first three terms 2, $p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text{th}}$, $8^{\text{th}}$ and $13^{\text{th}}$ terms of an A.P. If the $5^{\text{th}}$ term of the G.P. is the $n^{\text{th}}$ term of the A.P., then $n$ is equal to :

Let $f(x)=\begin{cases} -2, & -2 \leq x \leq 0 \\ x-2, & 0<x \leq 2 \end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\int\limits_{-2}^{2} h(x)\, dx$ is equal to :

Let $f(x)=x^{5}+2e^{x/4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8g'(2)$ is :

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2x^{2}-3x+8}{2x^{2}+3x+8}$ be $\frac{m}{n}$, where $\gcd(m, n)=1$. Then $m+n$ is equal to :

One of the points of intersection of the curves $y=1+3x-2x^{2}$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l\sqrt{5}+m)-n\log_{e}(1+\sqrt{5})$, where $l$, $m$, $n \in \mathbf{N}$. Then $l+m+n$ is equal to

If the solution $y=y(x)$ of the differential equation $(x^{4}+2x^{3}+3x^{2}+2x+2)dy-(2x^{2}+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :

A square is inscribed in the circle $x^{2}+y^{2}-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_{i}, y_{i})$ are the vertices of the square, then $\Sigma(x_{i}^{2}+y_{i}^{2})$ is equal to :

The vertices of a triangle are $A(-1, 3)$, $B(-2, 2)$ and $C(3, -1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Let the point, on the line passing through the points $P(1, -2, 3)$ and $Q(5, -4, 7)$, farther from the origin and at a distance of 9 units from the point P, be $(\alpha, \beta, \gamma)$. Then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to :

Let a unit vector which makes an angle of $60°$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45°$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}+\left(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is :

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3, 4, 7, -6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m+n$ is equal to ________.

Let A be a $3 \times 3$ matrix of non-negative real elements such that $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=3\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$. Then the maximum value of $\det(A)$ is ________.

Let $a=1+\frac{^{2}C_{2}}{3!}+\frac{^{3}C_{2}}{4!}+\frac{^{4}C_{2}}{5!}+\ldots$,

$b=1+\frac{^{1}C_{0}+{}^{1}C_{1}}{1!}+\frac{^{2}C_{0}+{}^{2}C_{1}+{}^{2}C_{2}}{2!}+\frac{^{3}C_{0}+{}^{3}C_{1}+{}^{3}C_{2}+{}^{3}C_{3}}{3!}+\ldots$

Then $\frac{2b}{a^{2}}$ is equal to ________.

If $\lim\limits_{x \to 1} \frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m, n)=1$, then $8m+12n$ is equal to ________.

If $\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin^{2}x}{1+\sin x \cos x}\, dx=\frac{1}{a}\log_{e}\left(\frac{a}{3}\right)+\frac{\pi}{b\sqrt{3}}$, where $a, b \in \mathbf{N}$, then $a+b$ is equal to ________.

Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ________.

Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x, y)$ satisfying $A^{2}+xA+y\,I=O$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^{4}+l^{4}$ is equal to ________.

Let the length of the focal chord PQ of the parabola $y^{2}=12x$ be 15 units. If the distance of PQ from the origin is $p$, then $10p^{2}$ is equal to ________.

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\int\limits_{0}^{k}[x^{2}]\, dx=\alpha-\sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6\alpha^{3}$ is equal to ________.

Let ABC be a triangle of area $15\sqrt{2}$ and the vectors $\vec{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\vec{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\vec{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of the triangle ABC is ________.