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NDA 2 2025 Maths, question 21

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)).

Q21Not scored: no correct option as printed

What is β\beta equal to?

  1. (a) log⁡a2(0.5)\log _{a^{2}}(0.5)
  2. (b) log⁡0.5(a2)\log _{0.5}(a^{2})
  3. (c) 2(log⁡a2(0.5))2(\log _{a^{2}}(0.5))
  4. (d) 2log⁡0.5(a2)2\log _{0.5}(a^{2})

Solution.

Not scored: no correct option as printed.

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

  1. Let L=log⁡0.5(a2)>0L = \log_{0.5}(a^{2}) > 0. Then α+β=−L\alpha + \beta = -L and αβ=L4\alpha\beta = L^{4}.
  2. The extra condition gives α=Lβ2\alpha = L\beta^{2}, so β3=L3\beta^{3} = L^{3} and, for real roots, β=L\beta = L, α=L3\alpha = L^{3}.
  3. Then α+β=L3+L>0\alpha + \beta = L^{3} + L > 0, which contradicts α+β=−L\alpha + \beta = -L.
  4. So no value fits the given conditions, and none of the options is correct.