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NDA 1 2025 Maths, question 11

Topic: Matrices and determinants

Q11

If A2+B2+C2=0A^{2} + B^{2} + C^{2} = 0, then what is the value of the following? ∣1cos⁡Ccos⁡Bcos⁡C1cos⁡Acos⁡Bcos⁡A1∣\left| \begin{matrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{matrix} \right|

  1. (a) -1
  2. (b) 0Correct answer
  3. (c) 1
  4. (d) 2

Solution.

Answer: (b) 0

  1. For real angles, squares are nonnegative.
  2. Thus A2+B2+C2=0A^{2} + B^{2} + C^{2} = 0 forces A=B=C=0A = B = C = 0.
  3. All cosine entries are 1.
  4. The determinant has identical rows and is therefore zero.