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Question

Let αβ, α2+3=5α and β2=5β3. The quadratic equation whose roots are αβ and βα will be:


A

3x219x+3=0

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B

3x2+19x+3=0

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C

3x219x3=0

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D

3x23x+1=0

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Solution

The correct option is A

3x219x+3=0


We have,
α25α+3=0 (1)
β25β+3=0 (2)

Subtracting (2) from (1), we get
α25α+3β2+5β+3=0
α2β25(αβ)=0
(αβ)((α+β5))=0
But αβ α+β=5 (3)

Adding (1) and (2), we get
α2+β25(α+β)+6=0
α2+β25(5)+6=0 [From(3)]
α2+β2=19 (4)

Now, (α+β)2=α2+β2+2αβ
25=19+2αβ
2αβ=6αβ=3 (5)

The roots of the quadratic equation are αβ and βα
Sum of roots =αβ+βα=α2+β2αβ=193
and product of roots=αβ×βα=1

Required quadratic equation is
x2193x+1=0
3x219x+3=0


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