JEE Main 2025 (Online) 3rd April Morning ShiftQuestion 1
Let the domain of the function $f(x)=\log_2\log_4\log_6\left(3+4x-x^{2}\right)$ be $(a,b)$. If $\int_{0}^{b-a}\left[x^{2}\right]dx=p-\sqrt{q}-\sqrt{r}$, $p,q,r\in\mathbb{N}$, $\gcd(p,q,r)=1$, where $[\cdot]$ is the greatest integer function, then $p+q+r$ is equal to
Functions
Solution
Need $\log_4\log_6(3+4x-x^2)>0\Rightarrow\log_6(\cdot)>1\Rightarrow x^2-4x+3<0$, so $(a,b)=(1,3)$ and $b-a=2$. $\int_0^2[x^2]dx=(\sqrt2-1)+2(\sqrt3-\sqrt2)+3(2-\sqrt3)=5-\sqrt2-\sqrt3$, so $p+q+r=10$.