Let α and β be the roots of x² - 3x + p=0 and γ and δ be the roots of x² - 6x + q = 0. If α, β, γ, δ form a geometric progression.Then ratio (2q + p) : (2q - p) is:

Solution

Maths sir · 4 steps

Write the roots as a, ar, ar², ar³; the sums give r² = 2, so q = r⁴ p = 4p.

1. Set up the GP. Take α = a, β = ar, γ = ar², δ = ar³. The first two belong to the first equation, the last two to the second.
2. Divide the sums. Sum of roots gives a(1 + r) = 3 and ar²(1 + r) = 6. Dividing the second by the first gives r² = 2.

   q = γδ = a² r⁵ = (a² r) r⁴ = p × r⁴ = 4p

3. Form the ratio. Product of roots gives p = αβ = a² r, so q = 4p. Then 2q + p = 9p and 2q - p = 7p.
4. So. (2q + p) : (2q - p) = 9 : 7, which is option (a). This holds whether r = √2 or r = -√2, since only r⁴ = 4 is used.

The idea to keep. When roots of two quadratics form one GP, write them as a, ar, ar², ar³. Dividing matching relations, sum by sum or product by product, cancels a and leaves only powers of r.

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