Question

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in A.P., then the intervals in which β and γ lie, are

• A. $\left( - \infty,\frac{1}{3} \right\rbrack$, $\left\lbrack - \frac{1}{27},\infty \right)$
• B. $\left( - \frac{1}{27},\infty \right)$, $\left( - \infty,\frac{1}{3} \right)$
• C. (0, ∞), (–∞, 0)
• D. None of these

Answer 1

Answer: (A)

$\left( - \infty,\frac{1}{3} \right\rbrack$, $\left\lbrack - \frac{1}{27},\infty \right)$

Answer 2

Taking x₁ , x₂, x₃ as a – d, a and a + d, we have

a – d + a + a + d = 1 ... (i)

(a – d)a + a(a + d) + (a – d) (a + d) = β ... (ii)

(a – d) a (a + d) = –γ ... (iii)

From equation (i), a = 1/3 and putting a = $\frac{1}{3}$in equation (ii),

we get

3a² – d² = β ⇒ d² = $\frac{1}{3}$– β

⇒ $\frac{1}{3}$ – β ≥ 0 ⇒ β ≤ $\frac{1}{3}$

Again from (iii), a (a² – d²) = –γ

⇒ γ = $\frac{d^{2}}{3}$– $\frac{1}{27}$≥ – $\frac{1}{27}$

⇒ γ ∈ $\left\lbrack - \frac{1}{27},\infty \right)$