CAT 2024 Slot 3 QA Q3 Logarithms & Exponents

Question 3 of the Quantitative Ability (QA) section, CAT 2024 Slot 3, with CAT's official answer and a worked solution.

If \((a+b \sqrt{3})^2=52+30 \sqrt{3}\), where \(a\) and \(b\) are natural numbers, then \(a+b\) equals
  1. A
    7
  2. B
    10
  3. C
    8
  4. D
    9

Answer: (C) 8 Logarithms & Exponents MCQ

Solution

  1. \((a + b\sqrt3)^2 = a^2 + 3b^2 + 2ab\sqrt3\), so \(a^2 + 3b^2 = 52\) and \(ab = 15\).
  2. \((a, b) = (5, 3)\): \(25 + 27 = 52\). So \(a + b = 8\).

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