# 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. ###### Prepare for CAT 2022 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2022 Classroom Batches Starting Now! @Gopalapuram ###### Best CAT Coaching in Chennai Introductory offer of 5000/- Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ##### 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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