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
the equation, 3x² - 10x + 27λ = 0, then (β γ)/λ is equal to:

Solution

Maths sir · 4 steps

Get the common root first: subtract to kill α² and α falls out as 3λ.

1. Eliminate the square. Since α satisfies both equations, α² - α + 2λ = 0 and 3α² - 10α + 27λ = 0. Multiply the first by 3 and subtract it from the second. This gives -7α + 21λ = 0, so α = 3λ.
2. Find lambda. Put α = 3λ into the first equation: 9λ² - 3λ + 2λ = 0, so 9λ² = λ. Since λ ≠ 0, λ = 1/9 and α = 1/3.
3. Other two roots. From the sum of roots of the first equation, β = 1 - 1/3 = 2/3. From the sum of roots of the second, γ = 10/3 - 1/3 = 3. Check: αβ = 2/9 = 2λ and αγ = 1 = 9λ.

   βγ/λ = (2/3 × 3)/(1/9) = 2 × 9 = 18

4. So. βγ/λ = 18, which is option (d).

The idea to keep. For a common root, write both equations in that root and eliminate the square term. That gives the common root linearly, and then sum and product of roots give the rest.

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