A value of $\theta \in(0, \pi / 3)$, for which $\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \cos 6 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \cos 6 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \cos 6 \theta\end{array}\right|=0$, is:

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Determinants

A value of

θ \in (0, π /3)

, for which

1 + cos^{2} θ cos^{2} θ cos^{2} θ sin^{2} θ 1 + sin^{2} θ sin^{2} θ 4 cos 6 θ 4 cos 6 θ 1 + 4 cos 6 θ = 0

, is:

Answer: (B)

Subtracting the third row from the first two collapses the determinant neatly, and it reduces to

2 + 4 cos 6 θ

. Setting

2 + 4 cos 6 θ = 0

gives

cos 6 θ = - \frac{1}{2}

, so

6 θ = \frac{2 π}{3}

and

θ = \frac{π}{9}

, which does lie in

(0, \frac{π}{3})

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option