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

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

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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##### 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. ###### 2IIM's Online CAT CoachingGet CAT Last Mile Prep Course for 799 /- Signup Now! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2020Enroll at 44,000/- Next Weekend Batch Starts Sat, Nov 2nd, 2019 ###### Best CAT Coaching in ChennaiRegister Online, get Rs 4000/- off Attend a Demo Class ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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