CAT 2019 Slot 2 QA Question & Solution
AlgebraMedium
Question
What is the largest positive integer n such that $\frac{n^2 + 7n + 12}{n^2 - n - 12}$ is also a positive integer?
Options
6
16
8
12
Solution
$\ \frac{\ n^2+3n+4n+12}{n^2-4n+3n-12}$
=$\ \frac{\ n^{ }\left(n+3\right)+4\left(n+3\right)}{n^{ }\left(n-4\right)+3\left(n-4\right)}$
=$\ \frac{\left(\ n+4\right)\left(n+3\right)}{\left(n-4\right)\left(n+3\right)}$
=$\ \frac{\left(\ n+4\right)}{\left(n-4\right)}$
=$\ \frac{\left(\ n-4\right)+8}{\left(n-4\right)}$
=$\ 1+\frac{8}{\left(n-4\right)}$ which will be maximum when n-4 =8
n=12
D is the correct answer.