# CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

###### CAT Questions | Algebra | Inequalities - Roots

The question is about maximum possible value. We need to maximize a expression which is related to the roots of the polynomial. 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 22: x4 โ 4x3 + ax2 โ bx = 1 = 0 has positive real roots. What is the maximum possible value of a + b?

1. 20
2. 12
3. 8
4. 10

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##### Method of solving this CAT Question from Algebra - Inequalities : When AM = GM, how roots are related to each other?

Fabulous question.
If we have an equation of the form ax4 + bx3 + cx2 + dx + e = 0 with Roots p, q, r and s.

We can say sum of the roots p + q + r + s = $$frac{-b}{a}\\$ Sum of the products taken two at a time pq + pr + ps + qr + qs + rs = ๐??$\frac{c}{a}\\$ . Sum of the products taken three at a time pqr + pqs + prs + qrs = $\frac{-d}{a}\\$ Product of the roots pqrs = ๐?$\frac{e}{a}\\$ . Note 1: We alternate between - and + Note 2: Start with the immediate lower power for sum of roots, stick a negative symbol and then alternate. So, if p, q, r, s were roots of this equation p + q + r + s = 4, pqrs = 1 Or, Arithmetic mean of p,q, r, s = 1 and geometric mean of pqrs = 1. P, q, r s are positive real numbers. AM = GM. What does this mean? This means that all 4 numbers are equal. Or, this expression is$x-1)4
a = pq + pr + ps + qr + qs + rs = 6
-(-b) = pqr + pqs + prs + qrs = 4
a = 6, b = 4. a + b = 10

The question is "What is the maximum possible value of a + b?"

##### Hence the answer is "10"

Choice D is the correct answer.

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