Let α, β, γ, δ are zeroes of P(x)=5 x4+p x3+q x2+r x+ s(p, q, r, s ∈ R) and α, γ, δ are zeroes of Q(x)=x3-9 x2 +a x-24(α

Question

Let α, β, &gamma;, δ are zeroes of P(x)=5 x4+p x3+q x2+r x+ s(p, q, r, s ∈ R) and α, &gamma;, δ are zeroes of Q(x)=x3-9 x2 +a x-24(α < β < &gamma; < δ) . If α, &gamma;, δ (taken in that order) are in arithmetic progression and α, β, &gamma;, δ (taken in that order) are in harmonic progression, then find the value of |(P(1)/Q(1))|. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

2 &gamma;=α+δ and α+β+&gamma;=9 ⇒ &gamma;=3 α &gamma; δ=24 ⇒ &gamma;=3, δ=4, α=2 ∵ (1/α), (1/β), (1/&gamma;), (1/δ) are in A.P. ⇒ β=(12/5) |(P(x)/Q(x))|=|5(x-β)|=7

2γ=α+δ and α+β+γ=9 ⇒γ=3 αγδ=24⇒γ=3,δ=4,α=2 ∵α1​,β1​,γ1​,δ1​ are in A.P. ⇒β=512​ ∣∣​Q(x)P(x)​∣∣​=∣5(x−β)∣=7

$2 γ = α + δ$ and $α + β + γ = 9$
$\Rightarrow γ = 3$
$α γ δ = 24 \Rightarrow γ = 3, δ = 4, α = 2$
$∵ \frac{1}{α}, \frac{1}{β}, \frac{1}{γ}, \frac{1}{δ}$ are in A.P.
$\Rightarrow β = \frac{12}{5}$
$∣ ∣ \frac{P ( x )}{Q ( x )} ∣ ∣ = ∣5 (x - β) ∣ = 7$