CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

The question is about maximum possible value. Range of two integers are given and we need to maximize the given expression. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 12: Consider integers p, q such that – 3 < p < 4, – 8 < q < 7, what is the maximum possible value of p² + pq + q²?

1. 60
2. 67
3. 93
4. 84

Video Explanation

Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : Are the integers positive or negative or both?

Trial and error is the best approach for this question. We just need to be scientific about this.
p² and q² are both positive and depend on |p| and |q|. If p, q are large negative or large positive numbers, p² and q² will be high.
pq will be positive if p, q have the same sign, and negative if they have opposite signs.

So, for p² + pq + q² to be maximum, best scenarios would be if both p & q are positive or both are negative.

Let us try two possibilities.
p = – 2, q = – 7: p² + pq + q² = 4 + 14 + 49 = 67
p = 3, q = 6: p² + pq + q² = 9 + 15 + 36 = 60
Whenever we have an expression with multiple terms, there are two key points to note.


The equation will be most sensitive to the highest power.
The equation will be more sensitive to the term with the greater value.

In the case, q.

In this question, we have a trade–off between higher value for p² and q². For q², the choice is between 6² and (–7)². This impact will overshadow the choice for p (where we are choosing between –2 and 3).


So, the maximum value for the expression would be 67.

The question is "what is the maximum possible value of p² + pq + q²?"

Hence the answer is "67"

Choice B is the correct answer.