JEE Main 2024 Jan 31 Shift 2 — Mathematics

If the function $f:(-\infty,-1] \rightarrow(a, b]$ defined by $f(x)=e^{x^{3}-3 x+1}$ is one - one and onto, then the distance of the point $P(2 b+4, a+2)$ from the line $x+e^{-3} y=4$ is :

The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$, is :

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1}+z_{2}=5$ and $z_{1}^{3}+z_{2}^{3}=20+15 i$. Then, $\left|z_{1}^{4}+z_{2}^{4}\right|$ equals -

Let $A$ be a $3 \times 3$ real matrix such that

$$A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}=2\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix},\ A\begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}=4\begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix},\ A\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}=2\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}.$$

Then, the system $(A-3 I)\begin{pmatrix}x \\ y \\ z\end{pmatrix}=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}$ has

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some $m, n$; ${ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2}>{ }^{8} C_{3}$ and ${ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8$, then ${ }^{n} P_{m+1}+{ }^{\mathrm{n}+1} C_{m}$ is equal to

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$ terms of a non-constant A. P. be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

Consider the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=e^{-\left|\log _{e} x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

Let $f: \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of $\lim _{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to

Let a variable line passing through the centre of the circle $x^{2}+y^{2}-16 x-4 y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to

The area of the region enclosed by the parabolas $y=4 x-x^{2}$ and $3 y=(x-4)^{2}$ is equal to

Let $f, g:(0, \infty) \rightarrow \mathbb{R}$ be two functions defined by $f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t$ and $g(x)=\int_{0}^{x^{2}} t^{1 / 2} e^{-t} d t$. Then, the value of $9\left(f\left(\sqrt{\log _{e} 9}\right)+g\left(\sqrt{\log _{e} 9}\right)\right)$ is equal to

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential equation $\frac{d T}{d t}=-K(T-80)$, where $K$ is a positive constant. If $T(15)=120^{\circ} \mathrm{F}$, then $T(45)$ is equal to

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2 x+3 y-4=0$ measured parallel to the line $x-2 y-1=0$ is

Let $(\alpha, \beta, \gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2 \alpha+3 \beta+4 \gamma$ is equal to

The shortest distance, between lines $L_{1}$ and $L_{2}$, where $L_{1}: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_{2}$ is the line, passing through the points $\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)$ and perpendicular to the line $\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}$, is

Let the mean and the variance of 6 observations $a, b, 68,44,48,60$ be 55 and 194, respectively. If $a>b$, then $a+3 b$ is

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^{2}+b^{2}$ is equal to

Let $A=\{1,2,3, \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_{1}$ be a symmetric relation on $A$ such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is n. Then, the minimum value of n is _________.

Let A be a $3 \times 3$ matrix and $\det(A)=2$. If $n=\det (\underbrace{a d j\left(a d j\left(\ldots \ldots(a d j\ A)\right)\right)}_{2024-\text{times}})$, then the remainder when $n$ is divided by 9 is equal to ________.

Let the coefficient of $x^{r}$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+\ldots \ldots \ldots+(x+2)^{n-1}$ be $\alpha_{r}$. If $\sum_{r=0}^{n} \alpha_{r}=\beta^{n}-\gamma^{n}$, $\beta, \gamma \in \mathbb{N}$, then the value of $\beta^{2}+\gamma^{2}$ equals ________.

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^{2}+\beta^{2}\right)$ is equal to ______.

If $\lim _{x \rightarrow 0} \frac{a x^{2} e^{x}-b \log _{e}(1+x)+c x e^{-x}}{x^{2} \sin x}=1$, then $16\left(a^{2}+b^{2}+c^{2}\right)$ is equal to ______.

$\left|\frac{120}{\pi^{3}} \int_{0}^{\pi} \frac{x^{2} \sin x \cos x}{\sin ^{4} x+\cos ^{4} x}\ d x\right|$ is equal to ________.

Let $y=y(x)$ be the solution of the differential equation $\sec ^{2} x\ d x+\left(e^{2 y} \tan ^{2} x+\tan x\right) d y=0$, $0<x<\frac{\pi}{2}$, $y\left(\pi / 4\right)=0$. If $y\left(\pi / 6\right)=\alpha$, then $e^{8 \alpha}$ is equal to _______.

Let $A(-2,-1), B(1,0), C(\alpha, \beta)$ and $D(\gamma, \delta)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2 x-y=5$ and the point $D$ lies on $3 x-2 y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to ________.

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a, b, c)$, where $a, b, c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a, b, c)$ and $Q(4,-12,3)$ is equal to __________.

Let $\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}$, $\vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}$ and $(\vec{a}-\vec{b}+\hat{i}) \cdot \vec{c}=-3$. Then $|\vec{c}|^{2}$ is equal to ______.