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NDA 1 2024 Maths, question 38

Topic: Trigonometry

Q38

If in a triangle ABCABC, sin⁡3A+sin⁡3B+sin⁡3C=3sin⁡Asin⁡Bsin⁡C\sin ^{3}A+\sin ^{3}B+\sin ^{3}C=3\sin A\sin B\sin C, then what is the value of the determinant

∣abcbcacab∣\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right|;

where a,b,ca,b,c are sides of the triangle ?

  1. (a) a+b+ca+b+c
  2. (b) ab+bc+caab+bc+ca
  3. (c) (a+b)(b+c)(c+a)(a+b)(b+c)(c+a)
  4. (d) 00Correct answer

Solution.

Answer: (d) 00

  1. For positive u, v, w, u³ + v³ + w³ − 3uvw = ½(u + v + w)[(u − v)² + (v − w)² + (w − u)²].
  2. Apply this to sin A, sin B and sin C.
  3. The given equality makes all three sines equal, so the triangle is equilateral.
  4. The determinant consequently has equal rows and is zero.