CAT 2021 Slot 3 QA Q18 Inequalities & Modulus

Question 18 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

The number of distinct pairs of integers (m, n) satisfying |1+mn| < |m+n| < 5 is

TITA: type in the answer. There are no options and no negative marking.

Answer: 12 Inequalities & Modulus TITA

Solution

\((m+n)^2 - (1+mn)^2 = m^2 + n^2 - 1 - m^2n^2 = -(m^2 - 1)(n^2 - 1)\).
So \(|1 + mn| \lt |m + n|\) needs \((m^2 - 1)(n^2 - 1) \lt 0\): for integers, one of \(m, n\) is 0 and the other has absolute value at least 2.
With \(m = 0\): \(1 \lt |n| \lt 5\) gives \(n = \pm2, \pm3, \pm4\): 6 pairs. Symmetrically 6 with \(n = 0\).
Total 12 pairs.