Question

The number of roots of the equation, `(81)^(sin^2x) + (81)^(cos^2x)` = 30 in the interval [0, π] is equal to ______.

Options

• 3

• 2

• 4

• 8

Answer 1

The number of roots of the equation, `(81)^(sin^2x) + (81)^(cos^2x)` = 30 in the interval [0, π] is equal to 4.

Explanation:

Given, `(81)^(sin^2x) + (81)^(cos^2x)` = 30

`(81)^(sin^2x) + (81)^(1 - sin^2x)` = 30

⇒ `(81)^(sin^2x) + (81)^1(81)^(-sin^2x)` = 30

⇒ `(81)^(sin^2x) + 81/((81)^(sin^2x)` = 30

Let `(81)^(sin^2x)` = t

⇒ `t + 81/t` = 30

⇒ t² – 30t + 81 = 0

⇒ (t – 27)(t – 3) = 0

⇒ t = 3 or t = 27

⇒ `(81)^(sin^2x)` = 3 to `(81)^(sin^2x)` = 27

⇒ `(3^4)^(sin^2x)` = 3 to `(3^4)^(sin^2x)` = 3³

⇒ `3^(4sin^2x)` = 3¹ or `3^(4sin^2x)` = 3³

⇒ 4sin²x = 1 or 4sin²x = 3

⇒ sin²x = `1/4` or sin²x = `3/4`

Now y = sinx

For y = sin²x; x∈[0, π]

From the above figure, we can say that the given equation has 4 solution.