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.
Answer choice (B)
Correct Answer: 67
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