The nature of roots of the quadratic equation (x−sinα)(x−cosα)+√3=0 are
A
Real and equal.
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B
Real and unequal.
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C
Imaginary.
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D
Rational and equal.
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Solution
The correct option is C Imaginary. Given equation x2−(sinα+cosα)x+(sinαcosα+√3)=0 For the nature of the roots Δ=(sinα+cosα)2−4(sinαcosα+√3) =(sinα−cosα)2−4√3∵0≤(sinα−cosα)2≤2 ∴−4√3≤Δ≤2−4√3 ⇒Δ is negative.