Question

If cos⁻¹(2/3x) + cos⁻¹(3/4x) = π/2 , x > 3/4
then x is equal to: (A) √(145)/12
(B) √(145)/10
(C) √(146)/12
(D) √(145)/11
[JEE Main 2019]

Answer 1

Answer:

Please see the attachment

Answer 2

answer : option (A) √145/12

given, cos-¹(2/3x) + cos-¹(3/4x) = π/2 , x > 3/4

taking cosine both sides,

cos[cos-¹(2/3x) + cos-¹(3/4x)] = cos(π/2)

⇒cos[cos-¹(2/3x) + cos-¹(3/4x)] = 0

⇒cos(cos-¹(2/3x)).cos(cos-¹(3/4x)) - sin(cos-¹(2/3x)).sin(cos-¹(3/4x)) = 0

⇒2/3x × 3/4x - sin(sin-¹{√(9x² - 4)/3x}).sin{√(16x² -9)/4x} = 0

⇒6/12x² - √(9x² - 4)/3x × √(16x² - 9)/4x = 0

⇒1/2x² = √(9x² - 4)(16x² - 9)/12x²

⇒6 = √(9x² - 4)(16x² - 9)

⇒36 = 144x⁴ - 81x² - 64x² + 36

⇒144x² = 145

⇒x = ±√145/12

but x > 3/4 so, x = √145/12

hence option (A) is correct choice.