JEE Main 2024 Jan 31 Shift 1 — Mathematics

If $f(x)=\frac{4x+3}{6x-4}, x \neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(gogog)(4)$ is equal to

Let S be the set of postive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in S is :

If $f(x)=\begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in \mathbb{R}$, then $2f(0)+f'(0)$ is equal to

If the system of linear equations

$x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$

has infinitely many solutions, then $12\alpha+13\beta$ is equal to

The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$ up to 10-terms is

$\lim_{x \to 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b=\lim_{x \to 0}\left(\frac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root, where $c, d, e \in \mathbb{R}$, then $\mathrm{d}:\mathrm{c}:\mathrm{e}$ equals

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

(I) If $\alpha \in (-1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$

(II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$

Let $g(x)$ be a linear function and $f(x)=\begin{cases} g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0 \end{cases}$, is continuous at $x=0$. If $f'(1)=f(-1)$, then the value $g(3)$ is

The area of the region $\left\{(x, y): y^2 \leq 4x, x<4, \frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)=2$. Then, $y\left(\frac{\pi}{3}\right)$ is

The solution curve of the differential equation $y\frac{dx}{dy}=x(\log_e x-\log_e y+1), x>0, y>0$ passing through the point $(e, 1)$ is

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle C, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle C is :

Let $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ and let $A(\alpha, \beta)$, $B(1,0)$, $C(\gamma, \delta)$ and $D(1,2)$ be the vertices of a parallelogram ABCD. If $AB=\sqrt{10}$ and the points A and C lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to

The distance of the point Q(0,2,−2) form the line passing through the point P(5, −4, 3) and perpendicular to the lines $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}+5\hat{k})$, $\lambda \in \mathbb{R}$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu(-\hat{i}+3\hat{j}+2\hat{k})$, $\mu \in \mathbb{R}$ is :

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vectors $\vec{p}$ satisfies $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{p} \cdot \vec{a}=0$, then $\vec{p} \cdot (\hat{i}-\hat{j}-\hat{k})$ is equal to

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

For $\alpha, \beta, \gamma \neq 0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals

Let $A=\{1, 2, 3, 4\}$ and $R=\{(1, 2), (2,3), (1,4)\}$ be a relation on A. Let S be the equivalence relation on A such that $R \subset S$ and the number of elements in S is n. Then, the minimum value of n is ______

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi}\,i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}\,i}\right]$, $i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x-3y=7$ is _____

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _____

In the expansion of $(1+x)(1-x^2)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficients of $x^3$ and $x^{-13}$ is equal to _____

Let $S=(-1, \infty)$ and $f: S \rightarrow \mathbb{R}$ be defined as

$$f(x)=\int_{-1}^{x}\left(e^t-1\right)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61}\,dt,$$

Let p = Sum of squares of the values of $x$, where $f(x)$ attains local maxima on S, and q = Sum of the values of $x$, where $f(x)$ attains local minima on S. Then, the value of $\mathrm{p}^2+2\mathrm{q}$ is ________

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M=\int_{f(a)}^{f(1-a)} x\sin^4(x(1-x))\,dx$, $N=\int_{f(a)}^{f(1-a)} \sin^4(x(1-x))\,dx$; $a \neq \frac{1}{2}$. If $\alpha M=\beta N, \alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2+\beta^2$ is equal to ______

If the integral $525\int_{0}^{\frac{\pi}{2}} \sin 2x \cos^{\frac{11}{2}}x\left(1+Cos^{\frac{5}{2}}x\right)^{\frac{1}{2}} dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to _______

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $(\pm 5, 0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2b^2}=1$ equals

Let Q and R be the feet of perpendiculars from the point P($a$, $a$, $a$) on the lines $x=y, z=1$ and $x=-y, z=-1$ respectively. If $\angle$QPR is a right angle, then $12a^2$ is equal to _____

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1, |\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2\vec{a} \times \vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192\sin^2\alpha$ is equal to ______