CAT 2023 Slot 1 QA Question & Solution
AlgebraMedium
Question
If $\sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2})$, then $\sqrt{10x+9}$ is equal to
Options
$3\sqrt{31}$
$4\sqrt{5}$
$3\sqrt{7}$
$2\sqrt{7}$
Solution
Given, $\sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2})$
=> $\sqrt{5x+9} + \sqrt{5x-9} = 6 + 3\sqrt{2}$
=> $\sqrt{5x+9} + \sqrt{5x-9} = \sqrt{36} + \sqrt{18}$
Comparing the L.H.S. and R.H.S.
=> $5x+9=36$ => $5x=27$ => $x=\dfrac{27}{5}$ (can be verified using the second term as well).
=> $\sqrt{10x+9} = \sqrt{\left(10 \times \dfrac{27}{5}\right)+9} = \sqrt{63} = 3\sqrt{7}$