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NDA 1 2024 Maths, question 32

Topic: Algebra

Q32

If a,ba,b and cc (a>0,c>0)(a>0,c>0) are in GP, then consider the following in respect of the equation ax2+bx+c=0ax^{2}+bx+c=0 :

  1. The equation has imaginary roots.

  2. The ratio of the roots of the equation is 1:ω1:\omega where ω\omega is a cube root of unity.

  3. The product of roots of the equation is (b2a2)\left( \frac{b^{2}}{a^{2}} \right).

Which of the statements given above are correct ?

  1. (a) 1 and 2 only
  2. (b) 2 and 3 only
  3. (c) 1 and 3 only
  4. (d) 1, 2 and 3Correct answer

Solution.

Answer: (d) 1, 2 and 3

  1. The GP condition gives b²=ac.
  2. The discriminant is b²−4ac=−3ac<0, so the roots are nonreal.
  3. Their ratio is a nonreal cube root of unity (either ω or ω², depending on root order).
  4. Their product is c/a=b²/a².
  5. All three statements are correct.