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

CAT 2021 Question Paper | CAT Verbal Ability Slot 3

Question :

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

Started 3 months ago by Shashank in

Answer : The answer is '12'

Explanatory Answer

|1 + mn| < |m + n| < 5
For two numbers ‘a’ and ‘b’,
|a| < |b| is equivalent to a² < b²

So, we can say that:
(1 + mn)² < (m + n)²
1 + 2mn +m²n² < m² + n² + 2mn
1 - n² - m² + m²n² < 0
(1 - n²) - m²(1 - n²) < 0
(1 - m²)(1 - n²) < 0

For the product to be negative, either one of the two terms has to be negative.
But they cannot simultaneously be 0.
The only possibility for either of the two terms to be positive is when
n = 0 and |m| > 1, or |n| > 1 and m = 0

Now for the case when m = 0 and |n| > 1
|m + n| < 5
|0 + n| < 5
So n can be $\pm$ 2, $\pm$ 3, $\pm$ 4
Which are 6 cases

Similarly for the case when n = 1 and |m| > 1
|m + n| < 5
|0 + m| < 5
So m can be $\pm$ 2, $\pm$ 3, $\pm$ 4
Again we have 6 cases.
Hence the answer is 12.