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Let α,β be the roots of the equation ax2+bx+c=0. Let Sn=αn+βn for n1 and Δ=∣ ∣31+S11+S21+S11+S21+S31+S21+S31+S4∣ ∣. If Δ<0, then the equation ax2+bx+c=0 has

A
positive real roots
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B
negative real roots
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C
equal roots
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D
imaginary roots
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Solution

The correct option is D imaginary roots
Δ=∣ ∣ ∣1+1+11+α+β1+α2+β21+α+β1+α2+β21+α3+β31+α2+β21+α3+β31+α4+β4∣ ∣ ∣=∣ ∣1111αβ1α2β2∣ ∣×∣ ∣1111αβ1α2β2∣ ∣ [multiplying row by row]=D2 (say)
Now,
D=∣ ∣1111αβ1α2β2∣ ∣=(1α)(αβ)(β1)=(βα)[αβαβ+1]=(βα)(ca+ba+1)=(βα)a(a+b+c)Δ=D2=(βα)2a2(a+b+c)2=1a2(a+b+c)2[b2a24ca]=1a4(a+b+c)2(b24ac)

If Δ<0, i.e., b24ac<0, then the roots of ax2+bx+c=0 are imaginary.

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