CAT 2023 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 1 : If \(x\) and \(y\) are positive real numbers such that \(\log _x\left(x^2+12\right)=4\) and \(3 \log _y x=1\), then \(x+y\) equals

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\( \left( x ^ { 2 } + 12 \right) = 4 \)

This means, \( x ^ { 2 } + 12 = x ^ { 4 } \)

Let k = x^{2}

k + 12 = k^{2}

k^{2} – k – 12 = 0

k^{2} – 4k + 3k – 12 = 0

k(k – 4) + 3(k – 4) = 0

k = 4 or k = -3

But since k = x^{2}, k is always non-negative.

∴ k = 4

∴ x^{2} = 4

Since x is the base of the log function, it can should always be positive.

∴ x = 2

\( 3 x = 1 \)

\( x ^ { 3 } = 1 \)

\( x ^ { 3 } = y ^ { 1 } \)

y = x^{3} = 2^{3} = 8

∴ x + y = 2 + 8 = 10

The question is **" If \(x\) and \(y\) are positive real numbers such that \(\log _x\left(x^2+12\right)=4\) and \(3 \log _y x=1\), then \(x+y\) equals " **

Choice D is the correct answer.

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