The number of distinct real roots of $\left|\begin{array}{ccc}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Determinants

The number of distinct real roots of

sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0

in the interval

- \frac{π}{4} \leq x \leq \frac{π}{4}

is:

Answer: (C)

Write

s = sin x

and

c = cos x

. Adding the second and third columns to the first makes every entry of that column

s + 2 c

, so

Δ = (s + 2 c) 111 c s c c c s = (s + 2 c) (s - c)^{2}

The equation

Δ = 0

therefore needs

tan x = - 2

tan x = 1

. On

- \frac{π}{4} \leq x \leq \frac{π}{4}

the tangent runs from

- 1

1

, so

tan x = - 2

has no solution there, while

tan x = 1

holds only at

x = \frac{π}{4}

. There is exactly

1

distinct real root.

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option