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Question

# Let α,β be the roots of the equation ax2+bx+c=0. Let Sn=αn+βn for n≥1 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 rootsΔ=∣∣ ∣ ∣∣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∣∣ ∣∣=(1−α)(α−β)(β−1)=(β−α)[αβ−α−β+1]=(β−α)(ca+ba+1)=(β−α)a(a+b+c)∴Δ=D2=(β−α)2a2(a+b+c)2=1a2(a+b+c)2[b2a2−4ca]=1a4(a+b+c)2(b2−4ac) If Δ<0, i.e., b2−4ac<0, then the roots of ax2+bx+c=0 are imaginary.

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