JEE Main - 24 January 2023 Shift 2

If the domain of the function $f\left(x\right)=\frac{\left[x\right]}{1+x^2}$, where $\left[x\right]$ is greatest integer $\le x$, is $[2,6)$, then its range is

Let $f:R\to R$ be a function such that $f\left(x\right)=\frac{x^2+2x+1}{x^2+1}$. Then

Let $f$ and $g$ be twice differentiable functions on $R$ such that

$f^{"}\left(x\right)=g^{"}\left(x\right)+6x$

$f^{\text{'}}\left(1\right)=4g^{\text{'}}\left(1\right)-3=9$

$f\left(2\right)=3 g\left(2\right)=12$

Then which of the following is NOT true ?

Consider a function $\mathrm{f}:\mathrm{\mathbb{N}}\to \mathrm{ℝ}$, satisfying $f\left(1\right)+2f\left(2\right)+3f\left(3\right)+\dots +xf\left(x\right)=x\left(x+1\right)f\left(x\right) ;x\ge 2$ with $f\left(1\right)=1$. Then $\frac{1}{f\left(2022\right)}+\frac{{\displaystyle 1}}{{\displaystyle f\left(2028\right)}}$ is equal to

Suppose $f:R\to \left(0,\infty \right)$ be a differentiable function such that $5f\left(x+y\right)=f\left(x\right)·f\left(y\right), \forall x, y\in R$, If $f\left(3\right)=320$, then $\sum _{n=0}^{5}f\left(n\right)$ is equal to:

Let $f^1\left(x\right)=\frac{3x+2}{2x+3},x\in R-\left\{-\frac{3}{2}\right\}$. For $n\ge 2$, define $f^n\left(x\right)=f^{1}o{f}^{n-1}\left(x\right)$. If $f^5\left(x\right)=\frac{ax+b}{bx+a}, \gcd \left(a,b\right)=1$, then $a+b$ is equal to ________

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f:A\to A$ such that $f\left(m·n\right)=f\left(m\right)·f\left(n\right)$ for every $m,n\in A$ with $m·n\in A$ is equal to

Let $f:R-\left\{0,1\right\}\to R$ be a function such that $f\left(x\right)+f\left(\frac{1}{1-x}\right)=1+x$. Then $f\left(2\right)$ is equal to :