Question

The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4  x ≤ pi/4` is ______.

Options

• 0

• 2

• 1

• 3

Answer 1

The number of distinct real roots of `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0 in the interval `pi/4  x ≤ pi/4` is 1.

Explanation:

We have, `|(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)|` = 0

Applying C₁ → C₁ + C₂ + C₃

⇒ `|(2cosx + sinx, cosx, cxosx),(2cosx + sinx, sinx, cosx),(2cosx + sinx, cosx, sinx)|`

⇒ `(2cosx + sinx) |(1, cosx, cosx),(1, sinx, cosx),(1, cosx, six)|` = 0

Applying R₂ → R₂ – R₁ and R₃ → R₃ – R₁

⇒ `(2cosx + sinx)|(1, cosx, cosx),(0, sinx - cosx, 0),(0, 0, sinx - cosx)|`

⇒ `(2 cosx + sinx)[1 * (sin x - cos x)^2]` = 0 ...(Expanding along C₁)

⇒ `(2 cosx + sinx)(sinx - cos x)^2` = 0

⇒ 2 cos x = –sin x or sin x = cos x

⇒ tan x = –2, which s not possible as for `pi/4  x ≤ pi/4` 

We get –1 tan x ≤ 1.

or tan x = 1

∴ x = `p/4`

So, only one real root exist.