CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

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 – 4x³ + ax² – bx = 1 = 0 has positive real roots. What is the maximum possible value of a + b?

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

Explanatory Answer

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 = ð?‘?\\frac{c}{a}\\) .
Sum of the products taken three at a time pqr + pqs + prs + qrs = \\frac{-d}{a}\\)
Product of the roots pqrs = ð?‘’\\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.