CAT 2023 Slot 1 QA Q3 Logarithms & Exponents

Question 3 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

If \(\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})\), then \(\sqrt{10 x+9}\) is equal to

A
\(4 \sqrt{5}\)
B
\(2 \sqrt{7}\)
C
\(3 \sqrt{31}\)
D
\(3 \sqrt{7}\)

Answer: (D) 3 √(7) Logarithms & Exponents MCQ

Solution

Let \(a=\sqrt{5x+9}\), \(b=\sqrt{5x-9}\). Then \(a^2-b^2 = 18\) and \(a+b = 3(2+\sqrt2)\).
\(a-b = \frac{18}{3(2+\sqrt2)} = \frac{6}{2+\sqrt2} = 3(2-\sqrt2)\).
Adding, \(2a = 12\), so \(a=6\): \(5x+9 = 36\), \(x = \frac{27}{5}\).
\(10x + 9 = 54 + 9 = 63\), so \(\sqrt{63} = 3\sqrt7\).

Shortcut. \(10x+9 = 2(5x+9) - 9 = 2\cdot 36 - 9 = 63\) once \(a=6\).