CAT 2024 Question Paper | Quant Slot 3

CAT Questions

CAT 2024 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 3 If \((a+b \sqrt{3})^2=52+30 \sqrt{3}\), where \(a\) and \(b\) are natural numbers, then \(a+b\) equals

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Video Explanation

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Explanatory Answer

\[ \begin{array}{l} (a+b \sqrt{3})^2=52+30 \sqrt{3} \\ a^2+3 b^2+2 a b \sqrt{3}=52+30 \sqrt{3} \end{array} \] Comparing the terms, we get: \[ \begin{array}{l} a^2+3 b^2=52 \\ a b=15 \end{array} \] So, \((a, b)=(1,15),(15,1),(3,5)\) and \((5,3)\)
The only possibility for the above system of equations is, \(a=5, b=3\)
Hence, \(a+b=5+3=8\)

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The question is "If \((a+b \sqrt{3})^2=52+30 \sqrt{3}\), where \(a\) and \(b\) are natural numbers, then \(a+b\) equals"

Hence, the answer is '8'

Choice C is the correct answer.

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