Question

# If α and β are the zeroes of the quadratic polynomial f(x)=2x2−5x+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(x2−252x+41)Since α and β are the roots of f(x)=2x2−5x+7So, α+β=−(−52)=52 and αβ=72Let S and P denote respectively the sum and product of zeroes of the required polynomial .Then,polynomial is p(x)=k(x2−Sx+P)Now, S=(2α+3β)+(3α+2β) =5(α+β) =5×52 =252and P=(2α+3β)(3α+2β) =6(α2+β2)+13αβ =6α2+6β2+12αβ+αβ =6×(52)2+72 =752+72 =41Hence,the required polynomial is given by p(x)=k(x2−Sx+P)=>p(x)=k(x2−252x+41)

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