CAT Practice : Inequalities

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This is an excellent question to understand how a polynomial behaves when the variable 'x' takes values between roots. Think about how you can make the polynomial take positive and negative values. What values of 'x' should you substitute in the polynomial?

Inequalities - Maximum possible value

    Q.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

 

  • Correct Answer
    Choice B. 67

Explanatory Answer

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Detailed Solution

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.
Answer choice (B)

Correct Answer: 67



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Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.