If - -are the roots of the equation x2-4x- -0 and - - are the roots of the equation x2-64x- -0and - - - - are given to be in increasing G-P- The value of - be 2 k- Find k -

Let nbsp; α = a, β = ar, γ = ar^2, δ = ar^3 According to the given conditions, nbsp; α + β = a(1 + r) = 4, a^2r = λ γ + δ = ar^2(1 + r) = 64, a ...

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Standard XII

Mathematics

Sum of Infinite Terms

If α ,βare ...

Question

If $α, β$ are the roots of the equation $x^{2} - 4 x + λ = 0$ and $γ, δ$ are the roots of the equation $x^{2} - 64 x + μ = 0$ and $α, β, γ, δ$ are given to be in increasing G.P. The value of $\frac{μ}{λ}$ be $2^{k}$ . Find $k$ ?

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Solution

Let $α = a, β = a r, γ = a r^{2}, δ = a r^{3}$
According to the given conditions, $α + β = a (1 + r) = 4, a^{2} r = λ$
$γ + δ = a r^{2} (1 + r) = 64, a^{2} r^{5} = μ$
Dividing the two conditions, we get $r^{2} = 16$ i.e. $r = 4$
Thus, $a \times 5 = 4$ or $a = \frac{4}{5}$
$\frac{λ}{μ} = \frac{1}{r^{4}} = \frac{1}{2^{8}}$
$k = - 8$

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If α,βare the roots of the equation x2−4x+λ=0 and γ,δ are the roots of the equation x2−64x+μ=0and α,β,γ,δ are given to be in increasing G.P. The value of μλ be 2k. Find k ?

Let α=a,β=ar,γ=ar2,δ=ar3According to the given conditions, α+β=a(1+r)=4,a2r=λγ+δ=ar2(1+r)=64,a2r5=μDividing the two conditions, we get r2=16 i.e. r=4Thus, a×5=4 or a=45λμ=1r4=128k=−8

If $α, β$ are the roots of the equation $x^{2} - 4 x + λ = 0$ and $γ, δ$ are the roots of the equation $x^{2} - 64 x + μ = 0$ and $α, β, γ, δ$ are given to be in increasing G.P. The value of $\frac{μ}{λ}$ be $2^{k}$ . Find $k$ ?