Number of values of θ ∈[0,2 π) such that 51- sin θ- cos θ, (1/3), 5-2+ sin θ+ cos θ (taken in that order) are in harmonic progression, is equal to

Question

Number of values of θ ∈[0,2 π) such that 51- sin θ- cos θ, (1/3), 5-2+ sin θ+ cos θ (taken in that order) are in harmonic progression, is equal to (A) 2 (B) 4 (C) 6 (D) 8

Tardigrade Admin · Accepted Answer

(1/3)=(2 ⋅ 51- sin θ- cos θ ⋅ 5-2+ sin θ+ cos θ/51- sin θ- cos θ+52+ sin θ+ cos θ) text Let λ=5 sin θ+ cos θ ∴ λ2-30 λ+125=0 ∴ λ=5,25 ∴ sin θ+ cos θ=1, sin θ+ cos θ=2 ∴ θ=0, (π/2)

Q. Number of values of θ∈[0,2π) such that 51−sinθ−cosθ,31​,5−2+sinθ+cosθ (taken in that order) are in harmonic progression, is equal to

2

4

6

8

31​=51−sinθ−cosθ+52+sinθ+cosθ2⋅51−sinθ−cosθ⋅5−2+sinθ+cosθ​ Let λ=5sinθ+cosθ ∴λ2−30λ+125=0 ∴λ=5,25 ∴sinθ+cosθ=1,sinθ+cosθ=2 ∴θ=0,2π​

Q. Number of values of $θ \in [0, 2 π)$ such that $5^{1 - s i n θ - c o s θ}, \frac{1}{3}, 5^{- 2 + s i n θ + c o s θ}$ (taken in that order) are in harmonic progression, is equal to