If α, β, are cube roots of -8 , then what is {α² p²+β² q²+² r²}{β² p²+² q²+α² r²} equal to ?

Correct Answer is : γβ/ββγ

Correct Answer is : γβ/ββγ

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UPSC NDA I 2026 Mathematics Paper

Question : 5 of 120

Marks: +1, -0

\alpha, \beta, \gamma

\frac{\alpha^{2} p^{2}+\beta^{2} q^{2}+\gamma^{2} r^{2}}{\beta^{2} p^{2}+\gamma^{2} q^{2}+\alpha^{2} r^{2}}

equal to ?

Solution:

Concept:

For cube roots of

-8

\alpha^2=\beta\gamma

\beta^2=\gamma\alpha

\gamma^2=\alpha\beta

Explanation:

Since

\alpha^3=\beta^3=\gamma^3=-8

and

\alpha\beta\gamma=-8

\alpha^{2}=\beta\gamma,\quad \beta^{2}=\gamma\alpha,\quad \gamma^{2}=\alpha\beta.

Multiplying the denominator coefficients by

\frac{\gamma}{\beta}

\frac{\gamma}{\beta}(\beta^{2},\gamma^{2},\alpha^{2}) =(\alpha^{2},\beta^{2},\gamma^{2}).

Therefore,

\alpha^{2}p^{2}+\beta^{2}q^{2}+\gamma^{2}r^{2} =\frac{\gamma}{\beta}\left(\beta^{2}p^{2}+\gamma^{2}q^{2}+\alpha^{2}r^{2}\right).

So the required value is

\frac{\gamma}{\beta}

Answer:

Option B:

\frac{\gamma}{\beta}

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