CAT 2024 Surds and Indices Question (Hard)

Question

If is the positive square root of , where and are integers, and is a natural number, then the maximum possible value of is.

Accepted Answer

Step 1 Expand the LHS a b 2 a 2 2ab b 2 n Compare with RHS 29 - 12 Rightarrow 2ab -12 Rightarrow 2ab -12 n 5 Rightarrow ab -6 Now, rational part a 2 b 2 n 29 Rightarrow a 2 5b 2 29 --- Step 2 Try integer values of , check if becomes a perfect square. Try Check a 2 5b 2 36 5 41 ne 29 Try a 2 5b 2 9 2

CAT 2024

Quantitative

Surds and Indices

Hard

If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$ , where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is.

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