Question

If x = acos3θsin2θ, y = asin3θcos2θ and [(x2 + y2)m/(xy)n] (m, n ∈ N, Q ∈ [0, 2π]) is independent of θ ∈ [0, 2π] then____

(a) 4m = 5n

(b) 4n = 5m

(c) m + n = 9

(d) mn = 20

Answer 1

The correct option (a) 4m = 5n

Explanation:

x = acos3θ sin2θ

y = asin3θ cos2θ

∴ x2 + y2 = a2cos6θsin4θ + a2sin6θcos4θ

= a2sin4θ cos4θ (1)

= a2 sin4θ cos4θ

also xy = a2 sin5θ cos5θ

∴ [(x2 + y2)m/(xy)n]

= [(a2m sin4mθ cos4mθ)/(a2n sin5nθ cos5nθ)]

= a(2m – n) ∙ sin(4m – 5n)θ ∙ cos(4m – 5n)θ

∵ expression is free from θ

⇒ sin(4m – 5n)θ ∙ cos(4m – 5n)θ = 1

⇒ 4m – 5n = 0

∴ 4m = 5n.