If θ 1 and θ 2 be respectively the smallest and the largest values of θ in 0 , 2 π -…

If θ1 and θ2 be respectively the smallest and the largest values of θ in 0,2π-π which satisfy the equation, 2cot2⁡θ-5sin⁡θ+4=0 , then ∫θ1θ2cos2⁡3θdθ is equal to:
  1. π3
  2. 2π3
  3. π3+16
  4. π9

Solution

2cot2⁡θ-5sin⁡θ+4=0
2cos2⁡θsin2⁡θ-5sin⁡θ+4=0
2cos2⁡θ-5sin⁡θ+4sin2⁡θ=0, sin⁡θ≠0
2sin2⁡θ-5sin⁡θ+2=0
2sin⁡θ-1sin⁡θ-2=0
sin⁡θ=12
θ=π6,5π6; θ∈0,2π-π
∴∫π/65π/6cos2⁡3θdθ=∫π/65π/61+cos⁡6θ2dθ
=12θ+sin⁡6θ6x/65π/6=125π6-π6+160-0=12.4π6=π3

Asked in: JEE Main 2020 (07 Jan Shift 2)

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