If -π

Question

If -π < θ < π , the equation (cos 3 θ + 1)x2+(2 cos ⁡ 2 θ - 1)x+(1 - 2 cos ⁡ θ )=0 has more than two roots for (A) no value of θ  (B) one value of θ  (C) two value of θ  (D) all values of θ

Tardigrade Admin · Accepted Answer

Given equation has more than two roots if it is an identity

\Rightarrow cos 3 θ + 1 = 0; 2 cos 2 θ - 1 = 0 an d 1 - 2 cos θ = 0

\Rightarrow cos 3 θ = - 1 \Rightarrow θ = \pm \frac{π}{3}

which does not satisfy

2 cos 2 θ - 1 = 0

Hence, no value possible

Q. If $- π < θ < π,$ the equation $(cos 3 θ + 1) x^{2} + (2 cos 2 θ - 1) x + (1 - 2 cos θ) = 0$ has more than two roots for

no value of $θ$

one value of $θ$

two value of $θ$

all values of $θ$

Given equation has more than two roots if it is an identity
$\Rightarrow cos 3 θ + 1 = 0; 2 cos 2 θ - 1 = 0 an d 1 - 2 cos θ = 0$
$\Rightarrow cos 3 θ = - 1 \Rightarrow θ = \pm \frac{π}{3}$ which does not satisfy $2 cos 2 θ - 1 = 0$
Hence, no value possible

Q. If −π<θ<π, the equation (cos3θ+1)x2+(2cos⁡2θ−1)x+(1−2cos⁡θ)=0 has more than two roots for

no value of θ

one value of θ

two value of θ

all values of θ