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Question

Let α and β be the roots of x2-3x+p=0 and 1γ and 1δ be the roots of x2-6x+q=0. If α,β,γ,δ form a geometric progression. Then ratio (2q+p):(2q-p) is :


A

33:31

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B

9:7

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C

3:1

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D

5:3

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Solution

The correct option is B

9:7


Explanation for the correct option:

Step 1: Use relation between roots and coefficients of the quadratic equation

The given quadratic equation is x2-3x+p=0, α&β are the roots.

Therefore,

Sum of roots, α+β=3

Product of roots, αβ=p

Also the in given quadratic equation x2-6x+q=0, γ&δ are roots.

therefore,

Sum of roots γ+δ=6

Product of roots γδ=q

Since α,β,γ,δ form a geometric progression

Step 2: Find the required ratio

Therefore, consider α=a,β=ar,γ=ar2,δ=ar3

a(1+r)=3....(i)α+β=3ar2(1+r)=6..(ii)γ+δ=6

Dividing (ii) by (i)

ar2(1+r)a(1+r)=63r2=2α.β=p=a2rγ.δ=q=a2r5(2q+p)(2q-p)=(2r4+1)(2r4-1)=(2×22+1)(2×22-1)=97

Thus, (2q+p):(2q-p)=9:7

Hence, the correct option is (B)


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