JEE Main 2024 Jan 30 Shift 2 — Mathematics

If the domain of the function $f(x)=\log_e\left(\frac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5\beta-4\alpha$ is equal to

If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2z^2+2z+1=0$, is equal to

Let $R=\begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3\times 3$ matrix, where $x\sin\theta = y\sin\left(\theta+\frac{2\pi}{3}\right) = z\sin\left(\theta+\frac{4\pi}{3}\right) \neq 0,\ \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) Trace $(R)=0$

(II) If trace $(\mathrm{adj}(\mathrm{adj}(R)))=0$, then $R$ has exactly one non-zero entry.

Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^2 z=9$, $x+3y+\lambda z=\mu$, where $\lambda, \mu \in \mathbb{R}$. Then, which of the following statement is NOT correct?

Suppose $2-p,\ p,\ 2-\alpha,\ \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6\alpha+2p$ equals

Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

Let $f: \mathbb{R}-\{0\} \to \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y,\ f(y) \neq 0$. If $f'(1)=2024$, then

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases} x^2+3x+a & ,\ x \leq 1 \\ bx+2 & ,\ x > 1 \end{cases}$ be differentiable on $\mathbb{R}$. Then, the value of $\int_{-2}^{2} f(x)\,dx$ equals

Let $f(x)=(x+3)^2(x-2)^3,\ x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M-m$ is

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x)=ae^{2x}+be^x+cx$. If $f(0)=-1$, $f'(\log_e 2)=21$ and $\int_0^{\log_e 4}(f(x)-cx)\,dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals

For $\alpha, \beta \in (0, \pi/2)$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such that $\tan\alpha = k\tan\beta$. Then, the value of $k$ is equal to

Let $y=f(x)$ be a thrice differentiable function in $(-5, 5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\pi/6$ and $\pi/4$, respectively with positive $x$-axis. If $27\int_1^3 \left(\left(f'(t)\right)^2+1\right)f''(t)\,dt = \alpha+\beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha+\beta$ equals

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^4\right)^{1/4}}$, and $g(x)=f(f(f(f(x))))$. Then, $18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x)\,dx$ is equal to

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to,

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9}-\frac{y^2}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is

If $x^2-y^2+2hxy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$, then the value of $g+c+h-f$ equals

Let $L_1: \vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda(\hat{i}-\hat{j}+2\hat{k}),\ \lambda \in \mathbb{R}$,

$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3\hat{i}+\hat{j}+p\hat{k}),\ \mu \in \mathbb{R}$, and $L_3: \vec{r}=\delta(\ell\hat{i}+m\hat{j}+n\hat{k}),\ \delta \in \mathbb{R}$

be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then, the point which lies on $L_3$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times\vec{a}|=2$. Then $\left|(\vec{b}\times\vec{a})-\vec{b}\right|^2$ is equal to

Let $\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k},\ \alpha, \beta \in \mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2=6$. If $\vec{a}\cdot\vec{b}=3\sqrt{2}$, then the value of $(\alpha^2+\beta^2)\left|\vec{a}\times\vec{b}\right|^2$ is equal to

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is ______.

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is ________.

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $A$, $B$ and $C$. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section $A$ has 8 questions, section $B$ has 6 questions and section $C$ has 6 questions, then the total number of ways a student can select 15 questions is ______.

Let $\alpha=\sum_{k=0}^{n}\left(\frac{\left({}^{n}C_{k}\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{}^{n}C_{k}\ {}^{n}C_{k+1}}{k+2}\right)$. If $5\alpha=6\beta$, then $n$ equals ______.

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3, 7, 11, \ldots$ If $40<\left(\frac{6}{n(n+1)}\sum_{k=1}^{n} S_k\right)<42$, then $n$ equals ______.

The area of the region enclosed by the parabola $(y-2)^2=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is _______.

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y=Y'(x)(X-x)$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)}+1,\ Y'(x)\neq 0$. If $Y(1)=1$, then $12Y(2)$ equals _______.

Consider two circles $C_1: x^2+y^2=25$ and $C_2: (x-\alpha)^2+y^2=16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals _______.

Let a line passing through the point $(-1, 2, 3)$ intersect the lines $L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$ at $N(a, b, c)$. Then, the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals ________.

The variance $\sigma^2$ of the data

$x_i$	0	1	5	6	10	12	17
$f_i$	3	2	3	2	6	3	3

is ______.