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

CAT Questions | Algebra | Cubic Equations - AP and roots of an equation

The question invloves concepts from Progressions and Equations. A cubic equation in which the roots are in AP is given and we need to verify the given statements. 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 14: Let x3- x2 + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true

  1. b=2,c=2
  2. b=1,c=1
  3. b= -1,c = 1
  4. b= -1,c= -1

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Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Cubic Equations: Roots are in AP. What is the average of roots?

Given the roots are in A.P. so let a-d, a, a+d be the roots
From equation, sum of roots = 1
Sum of two roots taken at a time = +b
Product of two roots = -c
∴ (a-d)+ (a)+ (a+d) = 1
=> 3a = 1
=> a = \\frac {1}{3}\\)
Also, (a-d)a+ a(a+d)+ (a-d)(a+d) = b
=> a2 – ad + a2 + ad + a2 – d2 = b
=> 3a2 – d2 = b
=> 3 * \\frac {1}{9}\\) – d2 = b
=> d2 = b - \\frac {1}{3}\\)
Now, since d is a real number,
\\frac {1}{3}\\) - b > 0 => b < \\frac {1}{3}\\)
Also,
(a-d)(a)(a+d) = -c
=> a(a2 – d2) = -c
=> \\frac {1}{3}\\) * \\frac {1}{9}\\) - d2 = -c
=>c = \\frac {d^2}{3}\\) - \\frac {1}{27}\\)
=>c = \\frac {9d^2 - 1}{27}\\)
=> d2 = 3c + \\frac {1}{9}\\)
Again, 3c + \\frac {1}{9}\\) > 0
c > \\frac {-1}{27}\\) > 0

The question is "Let x3- x2 + bx + c = 0 has 3 real roots which are in A.P. which of the given could be true"

Hence the answer is "b=1,c=1"

Choice B is the correct answer.

 

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