CAT 2024 Slot 1 QA Question 18: If $a + b\sqrt{n}$ is a positive square root of $29 - 12\sqrt{5}$, where $a$ and $b$ are integers, and $n$ is a natural number, the maximum possible value of $a + b + n$ is

Question

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

Accepted Answer

We need $a + b\sqrt{n}$ to be the positive square root of $29 - 12\sqrt{5}$.
Since $29 - 12\sqrt{5} = (2\sqrt{5} - 3)^2$, the positive square root is $2\sqrt{5} - 3$.
$a + b\sqrt{n} = -3 + 2\sqrt{5}$
Now $2\sqrt{5}$ can be written in two valid ways with integer $b$ and natural $n$:
$2\sqrt{5} = 2\sq

If a+bna + b\sqrt{n}a+bn​ is a positive square root of 29−12529 - 12\sqrt{5}29−125​, where aaa and bbb are integers, and nnn is a natural number, the maximum possible value of a+b+na + b + na+b+n is

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