We have,

Applying C₁→ C₁ + C₂ + C₃, we get

Taking (2cos X + sin X) common from the first column, we get

Applying R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

Expanding |A| along C₁, we get

⇒ (2cos X + sin X) [(1){(sin X – cos X)(sin X – cos X)}]

⇒ (2cos X + sin X)(sin X – cos X)² = 0

⇒ 2cos X = -sin X or (sin X – cos X)² = 0

⇒ tan X = -2 or tan X = 1

but tan X = -2 is not possible as for

So, tan X = 1

Hence, only one real distinct root exist.

Hence, the correct option is (c)