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NDA 1 2024 Maths, question 20

Topic: Matrices and determinants

Q20

If ABCABC is a triangle, then what is the value of the determinant

∣cos⁡Csin⁡B0tan⁡A0sin⁡B0tan⁡(B+C)cos⁡C∣\left| \begin{matrix} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{matrix} \right| ?

  1. (a) −1-1
  2. (b) 00Correct answer
  3. (c) 11
  4. (d) 33

Solution.

Answer: (b) 00

  1. Expand along the first row: the determinant is −cos C sin B[tan(B+C)+tan A].
  2. Since B+C=π−A, tan(B+C)=−tan A wherever the entries are defined.
  3. Thus the determinant is zero.