JEE Main 2024 Apr 8 Shift 1 — Mathematics

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

Let $z$ be a complex number such that $|z+2|=1$ and $\mathrm{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $\left|\mathrm{Re}\left(\overline{z+2}\right)\right|$ is

The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^{x}+48=0$ is :

Let $A=\begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times 3$, then $2a+3b$ is equal to

The equations of two sides AB and AC of a triangle ABC are $4x+y=14$ and $3x-2y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side BC internally in the ratio 2:1. the equation of the side BC is

If the set $R=\{(a,b): a+5b=42, a,b\in\mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}\left(1-i^{n!}\right)=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

For the function $f(x)=(\cos x)-x+1$, $x\in\mathbb{R}$, between the following two statements

(S1) $f(x)=0$ for only one value of $x$ in $[0,\pi]$.

(S2) $f(x)$ is decreasing in $\left[0,\frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2},\pi\right]$.

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is

Let $f(x)=4\cos^3 x+3\sqrt{3}\cos^2 x-10$. The number of points of local maxima of $f$ in interval $(0, 2\pi)$ is

Let $I(x)=\int \frac{6}{\sin^2 x\,(1-\cot x)^2}\,dx$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to

The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n dx$, $n\in\mathbb{N}$, satisfies $147\, I_{20}=148\, I_{21}$ is

Let $f(x)$ be a positive function such that the area bounded by $y=f(x)$, $y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

Let $y=y(x)$ be the solution of the differential equation $(1+y^2)e^{\tan x}\,dx+\cos^2 x\,(1+e^{2\tan x})\,dy=0$, $y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

If the shortest distance between the lines $$L_1 : \vec{r}=(2+\lambda)\hat{i}+(1-3\lambda)\hat{j}+(3+4\lambda)\hat{k}, \quad \lambda\in\mathbb{R}$$ $$L_2 : \vec{r}=2(1+\mu)\hat{i}+3(1+\mu)\hat{j}+(5+\mu)\hat{k}, \quad \mu\in\mathbb{R}$$ is $\frac{m}{\sqrt{n}}$, where $\gcd(m,n)=1$, then the value of $m+n$ equals

Let the circles $C_1 : (x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2 : (x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio 2:1, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

Let $H : \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$, $\alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to

Let $P(x, y, z)$ be a point in the first octant, whose projection in the $xy$-plane is the point $Q$. Let $OP=\gamma$; the angle between $OQ$ and the positive $x$-axis be $\theta$; and the angle between $OP$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\,\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\,\hat{i}-2\hat{j}-2\alpha t\,\hat{k}$ are inclined at an obtuse angle for all $t\in\mathbb{R}$, is

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $n-m$ equals

If $\sin x=-\frac{3}{5}$, where $\pi<x<\frac{3\pi}{2}$, then $80(\tan^2 x-\cos x)$ is equal to

If the range of $f(\theta)=\frac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, $\theta\in\mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to __________.

Let $A=\begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to __________.

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to __________.

Let $\alpha=\sum_{r=0}^{n}\left(4r^2+2r+1\right){}^{n}C_r$ and $\beta=\left(\sum_{r=0}^{n}\frac{{}^{n}C_r}{r+1}\right)+\frac{1}{n+1}$. If $140<\frac{2\alpha}{\beta}<281$, then the value of $n$ is __________.

Let the positive integers be written in the form :

triangular arrangement of positive integers — row 1: 1; row 2: 2, 3; row 3: 4, 5, 6; row 4: 7, 8, 9, 10; and so on

If the $k^{\text{th}}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is __________.

The value of $\lim_{x\to 0} 2\left(\frac{1-\cos x\sqrt{\cos 2x}\,\sqrt[3]{\cos 3x}\,......\,\sqrt[10]{\cos 10x}}{x^2}\right)$ is __________.

Let the area of the region enclosed by the curve $y=\min\{\sin x, \cos x\}$ and the $x$-axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^2$ is equal to __________.

Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k}$, $\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\vec{c}=17\hat{i}-2\hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times\vec{a}=(\vec{b}+\vec{c})\times\vec{a}$ and $\vec{r}\cdot(\vec{b}-\vec{c})=0$, then $\frac{|593\vec{r}+67\vec{a}|^2}{(593)^2}$ is equal to _____.

If the orthocentre of the triangle formed by the lines $2x+3y-1=0$, $x+2y-1=0$ and $ax+by-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (−6, −8), then the value of $|a-b|$ is __________.

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7\bar{X}+4\bar{Y}$ is equal to __________.