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Question

The quadratic equation tanθ x2+2(secθ+cosθ)x+(tanθ+32cotθ) always has


A

equal roots for all θ

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B

complex roots for all θ

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C

real and distinct roots for all θ

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D

real roots or complex roots depending upon θ

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Solution

The correct option is B

complex roots for all θ


Discriminant, =b24ac
=(2(secθ+cosθ))24tanθ(tanθ+32cotθ)
=4(sec2θ+cos2θ+2)4(tan2θ+32)]
=4[(sec2θ+cos2θ+2(tan2θ+32)]
=4[(sec2θ+tan2θ+2+cos2θ+32)]
=4[1+2+cos2θ32)
=4[3+cos2θ32]
=4[cos2θ+332]

As cos2θ1
So, 4[1+332]
=4(432)
4(43×1.4)
=4(0.2)
is negative
No real roots


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