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Q.

If the system of linear equations x+2ay+az = 0, x+3by+bz = 0, x+4cy+cz = 0 has non – trivial solution and a,b,c > 0 then the minimum value of $\frac{a^{3} + c^{3}}{b^{3}}$ is…………..

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answer is 2.

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Detailed Solution

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$Δ = 0 \Rightarrow b = \frac{2 ac}{a + c}$
$\begin{array}{ll} ∴ G \cdot M \geq H \cdot M A.M \geq G.M \\ \sqrt{a c} \geq \frac{2 ac}{a + c} \frac{a^{3} + c^{3}}{2} \geq \sqrt{a^{3} c^{3}} \\ \sqrt{a c} \geq b \frac{a^{3} + c^{3}}{2} \geq b^{3} \\ \frac{a^{3} + c^{3}}{b^{3}} \geq 2 \end{array}$
$(\begin{array}{l} ∴ AM \geq G.M \geq HM \\ \frac{a + b}{2} \geq \sqrt{a b} \geq \frac{2 a + b}{2} \end{array})$

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