Let λ ≠ 0 be in R. If α and β are the roots of the equation x² - x + 2λ = 0, and α and γ are the roots of equation 3x² - 10x + 27λ = 0, then (β γ)/λ is equal to ____________.

Solution

Maths sir · 4 steps

Both equations share α, so eliminate α² between them to find α in terms of λ, then fix λ.

1. Eliminate the square. The common root α satisfies α² - α + 2λ = 0 and 3α² - 10α + 27λ = 0. Multiply the first by 3 and subtract it from the second so that α² cancels.

   (3α² - 10α + 27λ) - (3α² - 3α + 6λ) = -7α + 21λ = 0 ⇒ α = 3λ

2. Find λ. Put α = 3λ into the first equation: 9λ² - 3λ + 2λ = 0, so λ(9λ - 1) = 0. Since λ ≠ 0, λ = 1/9 and α = 1/3.
3. Other roots. From the sum of roots of the first equation, β = 1 - 1/3 = 2/3. From the second, γ = 10/3 - 1/3 = 3. Check: αβ = 2/9 = 2λ and αγ = 1 = 9λ.
4. So. βγ = 2/3 × 3 = 2, and βγ/λ = 2 × 9 = 18.

The common slip: 58. Taking the sum of roots of 3x² - 10x + 27λ = 0 as 10 instead of 10/3 makes γ = 29/3 and gives 58, not 18.

The idea to keep. For a common root, scale one equation so the squared terms match and subtract. That gives the root linearly, and Vieta's sums then hand you the other roots.

Answer from the NTA final answer key. Solution by PrepWiser.