NTA Abhyas JEE Main2020MathematicsTrigonometric EquationsPractice
If x and y are the solutions of the equation 12 sin x + 5 cos x = 2 y 2 - 8 y + 21 , then the value of 12 c o t x y 2 is (Given, x < π )
Correct answer
5
Step-by-step solution
L H S = 12 sin ⁡ x + 5 cos ⁡ x ∈ - 1 2 2 + 5 2 , 1 2 2 + 5 2 i.e. - 13,13 i.e. maximum value of L H S is 13 R H S = 2 y 2 - 4 y + 4 + 13 = 2 y - 2 2 + 13 R H S ≥ 13 Roots of the equation exist if L H S = R H S = 13 . R H S = 13 when y = 2 L H S = 13 ⇒ 12 sin ⁡ x + 5 cos ⁡ x = 13 ⇒ 12 13 sin ⁡ x + 5 13 cos ⁡ x = 1 s i n x + α = 1 , where tan ⁡ α = 5 12 x + t a n - 1 5 12 = π 2 ⇒ x = π 2 - t a n - 1 5 12 ⇒ x y = π -