Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations. $\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$ has a non trivial solution, then the value of $\theta$ is:

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Determinants

Let

θ \in (0, \frac{π}{2})

. If the system of linear equations.

(1 + cos^{2} θ) x + sin^{2} θ y + 4 sin 3 θ z = 0

cos^{2} θ x + (1 + sin^{2} θ) y + 4 sin 3 θ z = 0

cos^{2} θ x + sin^{2} θ y + (1 + 4 sin 3 θ) z = 0

has a non trivial solution, then the value of

θ

is:

(A) $\frac{4 π}{9}$
(B) $\frac{7 π}{18}$ ✓ Correct
(C) $\frac{π}{18}$
(D) $\frac{5 π}{18}$

Answer: (B)

Solution

A homogeneous system has a non-trivial solution only when its coefficient determinant vanishes. Subtracting the second row from the first and the third row from the second,

10 cos^{2} θ - 1 1 sin^{2} θ 0 - 1 1 + 4 sin 3 θ = (1 + 4 sin 3 θ + sin^{2} θ) + cos^{2} θ = 2 + 4 sin 3 θ .

sin 3 θ = - \frac{1}{2}

. For

θ \in (0, \frac{π}{2})

the angle

3 θ

lies in

(0, \frac{3 π}{2})

, and the only value there with sine

- \frac{1}{2}

3 θ = \frac{7 π}{6}

.
Hence

θ = \frac{7 π}{18}

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option

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JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Determinants

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