The sum of all values of θ ∈ (0, (π)/2 ) satisfying sin^2 2θ + cos^4 2θ = 3/4 is

The sum of all values of θ ∈ (0, (π)/2 ) satisfying sin^2… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 Jan · Shift I

The sum of all values of

\theta \in \left(0, \dfrac{\pi}{2}\right)

satisfying

\sin^2 2\theta + \cos^4 2\theta = \dfrac{3}{4}

A.
$\dfrac{3\pi}{8}$
B.
$\dfrac{5\pi}{4}$
C.
$\dfrac{\pi}{2}$
D.
$\pi$

Show answer and solution

Correct option: C

Solution

Let x = sin²2θ. Then cos⁴2θ = (1 - sin²2θ)² = (1-x)².

Equation: x + (1-x)² = 3/4 → x + 1 - 2x + x² = 3/4 → x² - x + 1/4 = 0 → (x - 1/2)² = 0 → x = 1/2.

So sin²2θ = 1/2 → sin2θ = 1/√2 (since θ ∈ (0, π/2), so 2θ ∈ (0, π), sin2θ > 0).

This gives 2θ = π/4 or 2θ = 3π/4, so θ = π/8 or θ = 3π/8.

Sum = π/8 + 3π/8 = π/2.

[JEE Main 2019 · 10 Jan · Shift I]

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