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UPSC NDA II 2025 Mathematics Paper

Question Numbers: 21-22

For the following two (02) items:

Let

\alpha

and

\beta

be the roots of the quadratic equation

(x^{2}+(log_{0.5}(\alpha ^{2}))x+(log_{0.5}(\alpha ^{2}))^{4}=0

where

a^{2}≠1

and

log_{0.5}(\alpha ^{2})>0

Further,

\beta ^{2}=\alpha (log_{\alpha ^{2}}(0.5))

Question : 21 of 120

Marks: +1, -0

What is ẞ equal to?

Solution:

Concept:

Use logarithmic properties and the relation between roots of a quadratic equation to express β in terms of α.

Explanation:

Let

L = \log_{0.5}(\alpha^2)

. Then

\log_{\alpha^2}(0.5) = \frac{1}{L}

The quadratic equation is

x^2 + Lx + L^4 = 0

with roots α and β.

From sum and product of roots:

\alpha + \beta = -L

and

\alpha\beta = L^4

. So

\beta = \frac{L^4}{\alpha}

Given

\beta^2 = \alpha \cdot \log_{\alpha^2}(0.5) = \frac{\alpha}{L}

Replace β:

\left(\frac{L^4}{\alpha}\right)^2 = \frac{\alpha}{L}

→

\frac{L^8}{\alpha^2} = \frac{\alpha}{L}

Multiply both sides by

\alpha^2 L

L^9 = \alpha^3

→

\alpha = L^3

Then

\beta = \frac{L^4}{L^3} = L

. Thus

\beta = \log_{0.5}(\alpha^2)

Answer:

Option B:

\log_{0.5}(a^{2})

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