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Question

# Consider the following statements:S1: if the roots fo x2−bx+c=0 are two consecutive integers, then value of b2−4c is equal to 1S2: If α,β are roots of x2−x+3=0 then value of α4+β4 is equal 7.S3: If α,β,γ are roots of x3−7x2+16x−12=0 then value of α2+β2+γ2 is equal 17.State, in order whether S1,S2,S2 are true or false.

A
True
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B
False
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Solution

## The correct option is A TrueS1: If the roots of x2−bx+c=0 are two consecutive integersi.e., |α−β|=∣∣∣√b2−4aca∣∣∣=1⟹∣∣ ∣∣√(−b)2−4c1∣∣ ∣∣=1⟹√b2−4c=1∴b2−4c=1S2:α,β are the roots of x2−x+3=0 α+β=−ba=1 and αβ=ca=3 α4+β4=(α2+β2)2−2α2β2=[(α+β)2−2αβ]2−2(αβ)2=[(1)2−2(3)]2−2(3)2=(1−6)2−2(9)=25−18=7S3:α,β,γ are the roots of x3−7x2+16x−12=0 α+β+γ=−ba=7 αβ+βγ+γα=ca=16and αβγ=−da=12α2+β2+γ2=(α+β+γ)2−2(αβ+βγ+γα)=72−2(16)=49−32=17So, S1,S2,S3 all are true (Option A)

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