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?

Quadratic Inequalities

    Q.9: a and b are roots of the equation x2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?

 

  • Correct Answer
    P ≥ 8 3  or P ≤ -8 3 

Detailed Solution

The roots are a and b:

a + b = p and
ab = 12

(a + b)2 = p2
(a - b)2 = (a + b)2 - 4ab
=> (a - b)2 = p2 - 12 * 4 = p2 - 48

If |a - b| ≥ 12 { Difference between the roots is at least 12}
then, (a - b)2 ≥ 144

p2 - 48 ≥ 144
p2 ≥ 192
P ≥ 8 3  or P ≤ -8 3 

Correct Answer: P ≥ 8 3  or P ≤ -8 3 

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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.