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 p^{2} + pq + q^{2}?

- 60
- 67
- 93
- 84

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Trial and error is the best approach for this question. We just need to be scientific about this.

p^{2} and q^{2} are both positive and depend on |p| and |q|. If p, q are large negative or large positive numbers, p^{2} and q^{2} will be high.

pq will be positive if p, q have the same sign, and negative if they have opposite signs.

So, for p^{2} + pq + q^{2} 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: p^{2} + pq + q^{2} = 4 + 14 + 49 = 67

p = 3, q = 6: p^{2} + pq + q^{2} = 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 p^{2} and q^{2}. For q^{2}, the choice is between 6^{2} 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 p ^{2} + pq + q^{2}?"**

Choice B is the correct answer.

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