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 x^{3}- x^{2} + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true

- b=2,c=2
- b=1,c=1
- b= -1,c = 1
- b= -1,c= -1

b=1,c=1

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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

=> a^{2} – ad + a^{2} + ad + a^{2} – d^{2} = b

=> 3a^{2} – d^{2} = b

=> 3 * \\frac {1}{9}\\) – d^{2} = b

=> d^{2} = 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(a^{2} – d^{2}) = -c

=> \\frac {1}{3}\\) * \\frac {1}{9}\\) - d^{2} = -c

=>c = \\frac {d^2}{3}\\) - \\frac {1}{27}\\)

=>c = \\frac {9d^2 - 1}{27}\\)

=> d^{2} = 3c + \\frac {1}{9}\\)

Again, 3c + \\frac {1}{9}\\) > 0

c > \\frac {-1}{27}\\) > 0

The question is **"Let x ^{3}- x^{2} + bx + c = 0 has 3 real roots which are in A.P. which of the given could be true"**

Choice B is the correct answer.

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