We have, α is common root of the equations x2−x+2λ=0 and 3x2−10x+27λ=0. Now, common root of these equations is (3α2−10α+27λ)−(3α2−3α+6λ)=0⇒−7α+21λ=0 ⇒α=3λ Again, α is root of x2−x+2λ=0 ∴α2−α+2λ=0 ⇒(3λ)2−3λ+2λ=0 ⇒9λ2−λ=0 ⇒λ(9λ−1)=0 ⇒λ=0,
1
9
⇒λ=
1
9
[∵λ≠0] ∴α=3λ=3×
1
9
=
1
3
Again, α and β are roots of the equation x2−x+2λ=0
ll∴α+β=
−(−1)
1
=1 ⇒β=1−α=1−
1
3
=
2
3
And α and γ are the roots of the equation 3x2−10x+27λ=0 ∴α+γ=