JEE MAIN 2023 Previous Year Questions (PYQs) – Page 30 of 34

Let $a_1=1,\,a_2,\,a_3,\,a_4,\ldots$ be consecutive natural numbers. Then $\tan^{-1}\!\left(\dfrac{1}{1+a_1a_2}\right)+\tan^{-1}\!\left(\dfrac{1}{1+a_2a_3}\right)+\cdots+\tan^{-1}\!\left(\dfrac{1}{1+a_{2021}a_{2022}}\right)$ is equal to:

For $\alpha,\beta\in\mathbb{R}$, suppose the system of linear equations $\begin{aligned} x-y+z&=5,\\ 2x+2y+\alpha z&=8,\\ 3x-y+4z&=\beta \end{aligned}$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

The number of ways of selecting two numbers $a$ and $b$, $a\in\{2,4,6,\ldots,100\}$ and $b\in\{1,3,5,\ldots,99\}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is:

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive integers $a_1,a_2,a_3,\ldots,a_{100}$ is $25$. Then $S$ is:

Let $a,b,c>1$, $a^{3},b^{3}$ and $c^{3}$ be in A.P., and $\log_{a} b,\ \log_{c} a$ and $\log_{b} c$ be in G.P. If the sum of first $20$ terms of an A.P., whose first term is $\dfrac{a+4b+c}{3}$ and the common difference is $\dfrac{a-8b+c}{10}$, is $-444$, then $abc$ is equal to:

Let $f,g,h$ be the real valued functions defined on $\mathbb{R}$ as \[ f(x)= \begin{cases} \dfrac{x}{|x|}, & x\neq 0,\\[6pt] 1, & x=0, \end{cases} \qquad g(x)= \begin{cases} \dfrac{\sin(x+1)}{x+1}, & x\neq -1,\\[6pt] 1, & x=-1, \end{cases} \] and $h(x)=2\lfloor x\rfloor - f(x)$, where $\lfloor x\rfloor$ is the greatest integer $\le x$. Then the value of $\displaystyle \lim_{x\to 1} g\!\big(h(x-1)\big)$ is:

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^{2}-px+\dfrac{5}{4}p=0$ are rational. Then the area of the region $\left\{(x,y): 0\le y\le (x-q)^{2},\ 0\le x\le q\right\}$ is:

If the functions $f(x)=\dfrac{x^{3}}{3}+2bx+\dfrac{a x^{2}}{2}$ and $g(x)=\dfrac{x^{3}}{3}+a x+b x^{2},\ a\ne 2b$ have a common extreme point, then $a+2b+7$ is equal to:

The parabolas: $a x^{2}+2 b x+c y=0$ and $d x^{2}+2 e x+f y=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in G.P., then:

Let $\vec a$ and $\vec b$ be two vectors. Let $|\vec a|=1$, $|\vec b|=4$ and $\vec a\cdot\vec b=2$. If $\vec c=(2\,\vec a\times\vec b)-3\vec b$, then the value of $\vec b\cdot\vec c$ is: (A) $-48$