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Question

Consider the following statements:
S1: if the roots fo x2bx+c=0 are two consecutive integers, then value of b24c is equal to 1
S2: If α,β are roots of x2x+3=0 then value of α4+β4 is equal 7.
S3: If α,β,γ are roots of x37x2+16x12=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 True
S1: If the roots of x2bx+c=0 are two consecutive integers

i.e., |αβ|=b24aca=1

∣ ∣(b)24c1∣ ∣=1

b24c=1

b24c=1


S2:α,β are the roots of x2x+3=0

α+β=ba=1 and αβ=ca=3


α4+β4=(α2+β2)22α2β2

=[(α+β)22αβ]22(αβ)2

=[(1)22(3)]22(3)2

=(16)22(9)

=2518

=7

S3:α,β,γ are the roots of x37x2+16x12=0

α+β+γ=ba=7
αβ+βγ+γα=ca=16
and αβγ=da=12

α2+β2+γ2=(α+β+γ)22(αβ+βγ+γα)

=722(16)

=4932

=17

So, S1,S2,S3 all are true (Option A)

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