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 - π/4 ≤ x ≤ π/4 is :

Solution

Maths sir · 5 steps

Add all three columns first: a common factor pops out, and the rest of the determinant collapses easily.

1. Pull out a factor. Do C₁ → C₁ + C₂ + C₃. Every entry of the first column becomes sin x + 2 cos x, so take it outside, leaving 1, 1, 1 in that column.
2. Make it triangular. Now do R₂ → R₂ - R₁ and R₃ → R₃ - R₁. Row 2 becomes (0, sin x - cos x, 0) and row 3 becomes (0, 0, sin x - cos x). The determinant is the product of the diagonal.

   Δ = (sin x + 2 cos x)(sin x - cos x)² = 0

3. First factor. On [-π/4, π/4], cos x > 0, so sin x + 2 cos x = 0 means tan x = -2. But here tan x only runs from -1 to 1, so this factor gives no root.
4. Second factor. sin x = cos x means tan x = 1, so x = π/4. The interval includes its endpoint, so this root counts.
5. So. The only distinct root is x = π/4. The squared factor repeats it but does not make it a second distinct root. The answer is 1, option (b).

The idea to keep. When a determinant has one value on the diagonal and another everywhere else, add all columns to pull out (a + 2b). It equals (a + 2b)(a - b)². Then check every root against the interval.

Answer from the NTA final answer key. Solution by PrepWiser.