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Question

The number of quadratic equations which are unchanged by squaring their roots, is:

A
2
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B
4
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C
6
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D
None of these
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Solution

The correct option is C 4
α2+β2=α+β ...(1)
α2β2=αβ ...(2)
αβ(αβ1)=0
α=0 or β=0 or αβ=1
For α=0
Substituting this in (1)
β2β=0
β=0,1
Hence roots are 0,0 and 0,1
Similarly for β=0
We get roots are 0,0 and 0,1
And for αβ=1
From (1)
(α+β)22αβ=α+β
substitute t=α+β
t2t2=0(t2)(t+1)=0
α+β=2,1
From α+β=2 and αβ=1
We get root 1,1
And from α+β=1 and αβ=1
We get root ω,ω2
Hence total 4 quadratic equations are possible

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