JEE Advanced 2024 Paper 1Question 1
Let $a=3\sqrt{2}$ and $b=\frac{1}{5^{1/6}\sqrt{6}}$. If $x,y\in\mathbb{R}$ are such that $$3x+2y=\log_a\left(18\right)^{\frac{5}{4}}$$ and $$2x-y=\log_b\left(\sqrt{1080}\right),$$ then $4x+5y$ is equal to ________.
Sets, Relations and Function
Answer$8$
Solution
$18=(3\sqrt2)^2$ gives $\log_a18^{5/4}=\frac52$; $\sqrt{1080}=6^{3/2}5^{1/2}=b^{-3}$ gives $\log_b\sqrt{1080}=-3$. Solving $3x+2y=\frac52$, $2x-y=-3$ gives $x=-\frac12,y=2$, so $4x+5y=8$.