JEE Main 2024 Jan 30 Shift 1 — Mathematics

If the domain of the function $f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\{\log_e(3-x)\}^{-1}$ is $[-\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to :

If $z=x+\mathrm{i}y$, $xy\neq 0$, satisfies the equation $z^2+\mathrm{i}\bar{z}=0$, then $|z^2|$ is equal to :

Consider the system of linear equations $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^2 z=\mu^2+15$, where $\lambda, \mu \in \mathbf{R}$. Which one of the following statements is NOT correct ?

If $f(x)=\begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3+\sin^2 2x \\ 3+2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3+2\sin^4 x & \sin^2 2x \end{vmatrix}$,

then $\frac{1}{5}f'(0)=$ is equal to :

Two integers $x$ and $y$ are chosen with replacement from the set $\{0, 1, 2, 3, ..., 10\}$. Then the probability that $|x-y|>5$, is :

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-S_5$ is :

The maximum area of a triangle whose one vertex is at $(0, 0)$ and the other two vertices lie on the curve $y=-2x^2+54$ at points $(x, y)$ and $(-x, y)$, where $y > 0$, is :

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\to \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $\lim\limits_{x\to 0} \frac{x\int_0^x f(t)\,\mathrm{d}t}{e^{x^2}-1}=\alpha$, then $8\alpha^2$ is equal to :

Let $g:\mathbf{R}\to\mathbf{R}$ be a non constant twice differentiable function such that $g'\left(\frac{1}{2}\right)=g'\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

The value of $\lim\limits_{n\to\infty} \sum\limits_{k=1}^{n} \frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$, is :

Let $y=y(x)$ be the solution of the differential equation $\sec x\,\mathrm{d}y+\{2(1-x)\tan x+x(2-x)\}\,\mathrm{d}x=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

A line passing through the point $\mathrm{A}(9, 0)$ makes an angle of $30°$ with the positive direction of $x$-axis. If this line is rotated about A through an angle of $15°$ in the clockwise direction, then its equation in the new position is :

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let $(\alpha, \beta, \gamma)$ be the foot of perpendicular from the point $(1, 2, 3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. Then $19(\alpha+\beta+\gamma)$ is equal to :

Let $\mathrm{A}(2, 3, 5)$ and $\mathrm{C}(-3, 4, -2)$ be opposite vertices of a parallelogram ABCD. If the diagonal $\overrightarrow{\mathrm{BD}}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to :

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$, $\vec{a}\cdot\vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times\vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to :

Let M denote the median of the following frequency distribution

Class	0 – 4	4 – 8	8 – 12	12 – 16	16 – 20
Frequency	3	9	10	8	6

Then $20\mathrm{M}$ is equal to :

If $2\sin^3 x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{n\pi}{2}\right]$, $n\in\mathbf{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to :

Let $\mathrm{A}=\{1, 2, 3, ...., 7\}$ and let $\mathrm{P(A)}$ denote the power set of A. If the number of functions $f:\mathrm{A}\to\mathrm{P(A)}$ such that $a\in f(a)$, $\forall\ a\in\mathrm{A}$ is $m^n$, $m$ and $n\in\mathbf{N}$ and $m$ is least, then $m+n$ is equal to ________.

Let $\alpha, \beta \in\mathbf{N}$ be roots of the equation $x^2-70x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3}\notin\mathbf{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :

Number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$ is equal to ________.

Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+...$ upto 10 terms and $\beta=\sum\limits_{n=1}^{10} n^4$. If $4\alpha-\beta=55k+40$, then $k$ is equal to ________.

The value of $9\int_0^9 \left[\sqrt{\frac{10x}{x+1}}\right]\mathrm{d}x$, where $[t]$ denotes the greatest integer less than or equal to $t$, is ________.

Let $y=y(x)$ be the solution of the differential equation $(1-x^2)\,\mathrm{d}y=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]\mathrm{d}x$, $-1<x<1$, $y(0)=0$. If $y\left(\frac{1}{2}\right)=\frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m+n$ is equal to ________.

Let the latus ractum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to ________.

If $\mathrm{d}_1$ is the shortest distance between the lines $x+1=2y=-12z$, $x=y+2=6z-6$ and $\mathrm{d}_2$ is the shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$, $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$, then the value of $\frac{32\sqrt{3}\,\mathrm{d}_1}{\mathrm{d}_2}$ is :

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, atmost 11 students passed in both Mathematics and Physics, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.

If the function $$f(x)=\begin{cases} \frac{1}{|x|}, & |x|\geq 2 \\ ax^2+2b, & |x|<2 \end{cases}$$ is differentiable on $\mathbf{R}$, then $48(a+b)$ is equal to ________.