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Question

If α and β are the zeroes of the quadratic polynomial f(x)=2x25x+7,then a polynomial whose zeroes are 2α+3β and 3α+2β is

A
k(x2252x41)
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B
k(x2+252x+41)
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C
k(x2252x+41)
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D
k(x2252x+41)
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Solution

The correct option is C k(x2252x+41)
Since α and β are the roots of f(x)=2x25x+7
So, α+β=(52)=52 and αβ=72
Let S and P denote respectively the sum and product of zeroes of the required polynomial .Then,polynomial is
p(x)=k(x2Sx+P)
Now, S=(2α+3β)+(3α+2β)
=5(α+β)
=5×52
=252
and P=(2α+3β)(3α+2β)
=6(α2+β2)+13αβ
=6α2+6β2+12αβ+αβ
=6×(52)2+72
=752+72
=41
Hence,the required polynomial is given by
p(x)=k(x2Sx+P)
=>p(x)=k(x2252x+41)

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