Progression (Sequence and Series)Medium
Question
Let α, β be the roots of x2 - x + p = 0 and γ, δ be the roots of x2 - 4x + q = 0 If α, β, γ, δ are in GP, then the integer values of p and q respectively are
Options
A.- 2, - 32
B.- 2, 3
C.- 6, 3
D.- 6, - 32
Solution
and 
Let r be the common ratio.
Since, α, β, γ and δ are in GP. Therefore
β = αr, γ = αr2
and δ = αr2
Then, α + αr = 1 ⇒ (α1 + r) = 1 ......(i)
and αr2 + αr3 = 4 ⇒ αr2 (1 + r) = 4 nbsp; ......(ii)
From Eqs. (i) and (ii), r2 = 4 ⇒ r =
2 Now, α.αr = p
and αr2. αr3 = q
On putting r = - 2, we get
α = - 1, p = - 2 and q = - 32
Again putting r = 2, we get α =- 1/3, and p = -
Since, q, p is an integer.
Therefore, we take p = - 2 and q = - 32
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