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Question

If α,β and γ are the roots of x3+4x+1=0, then the equation, whose roots are α2(β+γ),β2(α+γ),γ2(α+β), is


A

x34x1=0

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B

x34x+1=0

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C

x3+4x1=0

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D

x3+4x+1=0

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Solution

The correct option is C

x3+4x1=0


Explanation for the correct option:

Step 1. Find the required quadratic equation:

Given, α,β and γ are the roots of x3+4x+1=0

α=0,αβ=4 and αβγ=1

Step 2. Find the Sum of the roots:

α2β+γ+β2γ+α+γ2α+β=α2α+β2β+γ2γ=(α+β+γ)=0

Step 3. Find the sum of Product of the roots:

α2β2β+γγ+α+β2γ2γ+αα+β+γ2α2β+γα+β=αβ+βγ+γα=4

Step 4. Find the Product of the roots:

α2β2γ2β+γγ+αα+β=αβγ=1

The required equation is x3+4x1=0.

Hence, Option ‘C’ is Correct.


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