Other Maths · JEE Advanced previous year questions

Let $a_0,a_1,\dots ,a_{23}$ be real numbers such that $${\left(1+\frac{2}{5}x\right)}^{23}=\sum _{i=0}^{23}{a}_i{x}^i$$ for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0\le j\le 23$. Then the value of $r$ is ____.

Let $a=3\sqrt{2}$ and $b=\frac{1}{5^{1/6}\sqrt{6}}$. If $x,y\in \mathbb{R}$ are such that $$3x+2y={\log }_a(18{)}^{\frac{5}{4}}\text{and}$$ $$2x-y={\log }_b(\sqrt{1080}),$$ then $4x+5y$ is equal to _____.