CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

CAT Questions | Algebra | Quadratic Equations - Integer Roots

The question is about finding possible values of the constant in a Quadratic Equation. We need to find out the no. of values 'p' can take when the given quadratic equations has integer roots. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 6: x2 - 11x + |p| = 0 has integer roots. How many integer values can 'p' take?

  1. 6
  2. 4
  3. 8
  4. More than 8

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Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Quadratic Equations: Counting question with modulus also thrown in. Worry about all those negative possibilities as well.

To start with the discriminant should be a perfect square. Let the dicriminant be 'D'.
From the quadratic formula: \\frac{-11 ± \sqrt{ D }}{2}\\), we see that the numerator has to be an even number for the roots to be integers.
This implies that the discriminant should be a perfect square and be square of an odd number. (Only then we will have odd + odd = even in the numerator)

D = 121 – 4|p| = 121 – 4|p|
4|p| cannot be negative => D can take values 121, 81, 49, 25, 9, 1
|p| can be 0, 10, 18, 24, 28, 30
p can take 0, ±10, ± 18, ± 24, ± 28, ± 30

The question is "x2 - 11x + |p| = 0 has integer roots. How many integer values can 'p' take?"

Hence the answer is "More than 8"

Choice D is the correct answer.


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