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NDA 2 2025 Maths, question 22

Topic: Algebra

Passage · Q21 to Q22

Let α\alpha and β\beta be the roots of the quadratic equation

x2+(log⁡0.5(a2))x+(log⁡0.5(a2))4=0x^{2}+(\log _{0.5}(a^{2}))x+(\log _{0.5}(a^{2}))^{4}=0

where a2≠1a^{2}\ne 1 and log⁡0.5(a2)>0\log _{0.5}(a^{2})>0. Further, β2=α(log⁡a2(0.5))\beta^{2}=\alpha(\log _{a^{2}}(0.5)).

Q22Not scored: no correct option as printed

What is the relation between α\alpha and β\beta?

  1. (a) α=2β\alpha=2\beta
  2. (b) 2α=β2\alpha=\beta
  3. (c) α=−2β\alpha=-2\beta
  4. (d) 2α=−β2\alpha=-\beta

Solution.

Not scored: no correct option as printed.

Answer: not scored. No option is correct as printed.

  1. With L as above, the real-root conditions are inconsistent.
  2. If complex roots are admitted, β=Lω or Lω² and Vieta’s sum forces L=1, giving the two nonreal cube roots of unity.
  3. Their ratio is nonreal, so none of α=±2β or 2α=±β holds.
  4. No listed relation follows from the supplied source.