ProbabilityHard
Question
A quadratic equation is chosen from the set of all quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots is -
Options
A.1/2
B.1/3
C.1/4
D.2/3
Solution
Let the roots of the quadratic equation be α, β
After squaring α2, β2
αβ = (αβ)2 ⇒ αβ(αβ - 1) = 0
⇒ αβ = 0 .... (1)
⇒ αβ = 1 ... (2)
Now α2 + β2 = α + β
(α+ β)2 - 2αβ = (α + β)
(α + β) {(α + β) - 1 } = 0 (∴ αβ = 0)
α + β = 0 ... (3)
α + β = 1 ... (4)
Solving (1) & (3)
α = 0, β = 0
solving (1) & (4)
α(1 - α) = 0 ⇒ α = 0, 1
⇒ β = 1, 0
solving (2) & (4)
α + 1/α = 1
α2 - α + 1 = 0
(α, β) ∈ (ω, ω2)
Hence sample space → (0, 0) (1, 1) (0, 1) (ω, ω2)
∴ P(A) =
After squaring α2, β2
αβ = (αβ)2 ⇒ αβ(αβ - 1) = 0
⇒ αβ = 0 .... (1)
⇒ αβ = 1 ... (2)
Now α2 + β2 = α + β
(α+ β)2 - 2αβ = (α + β)
(α + β) {(α + β) - 1 } = 0 (∴ αβ = 0)
α + β = 0 ... (3)
α + β = 1 ... (4)
Solving (1) & (3)
α = 0, β = 0
solving (1) & (4)
α(1 - α) = 0 ⇒ α = 0, 1
⇒ β = 1, 0
solving (2) & (4)
α + 1/α = 1
α2 - α + 1 = 0
(α, β) ∈ (ω, ω2)
Hence sample space → (0, 0) (1, 1) (0, 1) (ω, ω2)
∴ P(A) =
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