JEE Main 2024 Jan 29 Shift 1 — Mathematics

If $f(x)=\begin{cases} 2+2x, & -1 \le x < 0 \\ 1-\dfrac{x}{3}, & 0 \le x \le 3 \end{cases}$ ; $g(x)=\begin{cases} -x, & -3 \le x \le 0 \\ x, & 0 < x \le 1 \end{cases}$, then range of $(fog)(x)$ is

Let R be a relation on $\mathbb{Z}\times\mathbb{Z}$ defined by (a, b) R (c, d) if and only if $ad - bc$ is divisible by 5. Then R is

If $z=\dfrac{1}{2}-2i$ is such that $|z+1| = \alpha z + \beta(1+i)$, $i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$, then $\alpha + \beta$ is equal to

Let A be a square matrix such that $AA^T = I$. Then $\dfrac{1}{2} A\left[\left(A+A^T\right)^2 + \left(A-A^T\right)^2\right]$ is equal to

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3 = 2^{21}$ where $\alpha, \beta \in \mathbb{Z}$. Then a value of $\alpha$ is

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_5$ is the greatest, then the common difference of the A.P. is equal to

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

$\lim\limits_{x \to \frac{\pi}{2}}\left(\dfrac{1}{\left(x-\frac{\pi}{2}\right)^2}\int\limits_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos\left(t^{\frac{1}{3}}\right)dt\right)$ is equal to

Suppose $f(x)=\dfrac{\left(2^x + 2^{-x}\right)\tan x\sqrt{\tan^{-1}\left(x^2 - x + 1\right)}}{\left(7x^2 + 3x + 1\right)^3}$. Then the value of $f'(0)$ is equal to

Consider the function $f:\left[\dfrac{1}{2},1\right] \to \mathbb{R}$ defined by $f(x)=4\sqrt{2}x^3 - 3\sqrt{2}x - 1$.

Consider the statements

(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.

(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos\dfrac{\pi}{12}$.

Then

If the value of the integral $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+\pi^x} + \dfrac{1+\sin^2 x}{1+e^{\sin x^{2023}}}\right)dx = \dfrac{\pi}{4}(\pi + a) - 2$, then the value of $a$ is

For $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\int \dfrac{\operatorname{cosec} x + \sin x}{\operatorname{cosec} x \sec x + \tan x \sin^2 x}\,dx$, and $\lim\limits_{x \to \left(\frac{\pi}{2}\right)^-} y(x) = 0$ then $y\left(\dfrac{\pi}{4}\right)$ is equal to

A function $y=f(x)$ satisfies $f(x)\sin 2x + \sin x - \left(1+\cos^2 x\right)f'(x) = 0$ with condition $f(0)=0$. Then, $f\left(\dfrac{\pi}{2}\right)$ is equal to

Let $\left(5,\dfrac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\dfrac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is

In a $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x - y = 2$. If $2AB = BC$ and the points $A$ and $B$ are respectively $(4, 6)$ and $(\alpha, \beta)$, then $\alpha + 2\beta$ is equal to

Let PQR be a triangle with R $(-1, 4, 2)$. Suppose M $(2, 1, 2)$ is the mid point of PQ. The distance of the centroid of $\triangle PQR$ from the point of intersection of the lines $\dfrac{x-2}{0}=\dfrac{y}{2}=\dfrac{z+3}{-1}$ and $\dfrac{x-1}{1}=\dfrac{y+3}{-3}=\dfrac{z+1}{1}$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha\vec{b}+\beta\vec{c}=\vec{0}$, then $\alpha + \beta$ is equal to

Let O be the origin and the position vectors of A and B be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line AB at C, then the length of OC is

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

If $\alpha, -\dfrac{\pi}{2} < \alpha < \dfrac{\pi}{2}$ is the solution of $4\cos\theta + 5\sin\theta = 1$, then the value of $\tan\alpha$ is

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + 2 = 0$ with $Im(\alpha) > Im(\beta)$.

Then $\alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2$ is equal to __________.

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is __________.

If $\dfrac{{}^{11}C_1}{2} + \dfrac{{}^{11}C_2}{3} + \ldots + \dfrac{{}^{11}C_9}{10} = \dfrac{n}{m}$ with $\gcd(n,m)=1$, then $n + m$ is equal to __________.

Let $f(x) = 2^x - x^2$, $x \in \mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the $x$-axis, then the value of $m + n$ is __________.

The area (in sq. units) of the part of the circle $x^2 + y^2 = 169$ which is below the line $5x - y = 13$ is $\dfrac{\pi\alpha}{2\beta} - \dfrac{65}{2} + \dfrac{\alpha}{\beta}\sin^{-1}\left(\dfrac{12}{13}\right)$, where $\alpha, \beta$ are coprime numbers. Then $\alpha + \beta$ is equal to _______

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)\,dx + x\,dy = 0$, $x > 0$ passes through the point $(1,1)$ and $y(e) = \dfrac{\alpha - \tan\left(\dfrac{3}{2}\right)}{\beta + \tan\left(\dfrac{3}{2}\right)}$, then $\alpha + 2\beta$ is __________.

Equations of two diameters of a circle are $2x - 3y = 5$ and $3x - 4y = 7$. The line joining the points $\left(-\dfrac{22}{7}, -4\right)$ and $\left(-\dfrac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then, $17\beta - \alpha$ is equal to _______.

If the points of intersection of two distinct conics $x^2 + y^2 = 4b$ and $\dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1$ lie on the curve $y^2 = 3x^2$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is __________.

A line with direction ratios 2, 1, 2 meets the lines $x = y + 2 = z$ and $x + 2 = 2y = 2z$ respectively at the points P and Q. If the length of the perpendicular from the point $(1, 2, 12)$ to the line PQ is $l$, then $l^2$ is __________.

If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, $\alpha$, $\beta$, 60 where $\alpha > \beta$, are 56 and 66.2 respectively, then $\alpha^2 + \beta^2$ is equal to ________.