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NDA 2 2025 Maths, question 17

Topic: Algebra

Q17

Let 1,ω,ω21,\omega,\omega^{2} be three cube roots of unity. If x=a+bx=a+b, y=aω+bω2y=a\omega+b\omega^{2}, z=aω2+bωz=a\omega^{2}+b\omega, then what is x2+y2+z2x^{2}+y^{2}+z^{2} equal to?

  1. (a) 6ab6abCorrect answer
  2. (b) 3ab3ab
  3. (c) a2+b2a^{2}+b^{2}
  4. (d) 11

Solution.

Answer: (a) 6ab6ab

  1. Expanding the three squares, the coefficients of a² and b² contain 1+ω+ω²=0.
  2. Each cross term is 2ab because ω³=1.
  3. Thus the sum is 3×2ab=6ab.