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JEE Main 2001 Mathematics question on Matrices and Determinants

The number of distinct real roots of ∣sin⁡xcos⁡xcos⁡xcos⁡xsin⁡xcos⁡xcos⁡xcos⁡xsin⁡x∣=0\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0 in the interval −π4≤x≤π4-\dfrac{\pi}{4} \leq x \leq \dfrac{\pi}{4} is
  1. A.
    0
  2. B.
    2
  3. C.
    1
  4. D.
    3
Show answer and solution

Correct option: C

Solution

The determinant factors as (sin x + 2cos x)(sin x - cos x)².

Setting each factor to zero: sin x - cos x = 0 gives tan x = 1, so x = π/4 (one solution in the interval). Setting sin x + 2cos x = 0 gives tan x = -2, but |tan x| ≤ 1 on [-π/4, π/4], so no solution exists here.

Thus there is exactly 1 distinct real root.

[2001]

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