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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Sequences and Series

NDA 2016 · Mathematics · Sequences and Series

If m is the geometric mean of ( y z )^ (yz) ,( z x )^ (zx) and ( x...

If misthegeometricmeanof(yz)log⁡(yz),(zx)log⁡(zx) and (xy)log⁡(xy)m is the geometric mean of \left(\frac{\mathrm{y}}{\mathrm{z}}\right)^{\log (\mathrm{yz})},\left(\frac{\mathrm{z}}{\mathrm{x}}\right)^{\log (\mathrm{zx})} \text{ and }\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{\log (\mathrm{xy})} then what is the value of mm ?

Options

  1. A1
  2. B3
  3. C6
  4. D9

Correct answer

A. 1

Step-by-step solution

Three terms are G1=(yz)log⁡yzG2=(zx)log⁡zxG3=(xy)log⁡xyG_1=\left(\frac{y}{z}\right)^{\log y z} G_2=\left(\frac{z}{x}\right)^{\log z x} G_3=\left(\frac{x}{y}\right)^{\log x y} Geometric mean of three terms is \begin{aligned} & m=\sqrt[3]{G_1 G_2 G_3}...(1) \\ & \begin{aligned} \therefore G_1 G_2 G_3 & =\left(\frac{y}{z}\right)^{\log y z} \cdot\left(\frac{z}{x}\right)^{\log z x} \cdot\left(\frac{x}{y}\right)^{\log x y} \\ & =\frac{y^{\log y} \cdot y^{\log z}}{z^{\log y} \cdot z^{\log z}} \times \frac{z^{\log z} \cdot z^{\log x}}{x^{\log z} \cdot x^{\log x}} \times \frac{x^{\log x} \cdot x^{\log y}}{y^{\log x} \cdot y^{\log y}} \\ & =\left(\frac{y}{x}\right)^{\log z} \cdot\left(\frac{z}{y}\right)^{\log x} \cdot\left(\frac{x}{z}\right)^{\log y} \end{aligned} \end{aligned}TakinglogbothsidesTaking log both sideslog⁡G1G2G3=log⁡[(yx)log⁡z]+log⁡[(zy)log⁡x]+log⁡[(xz)log⁡y]=log⁡zlog⁡y−log⁡zlog⁡x+log⁡xlog⁡z−log⁡xlog⁡y+log⁡ylog⁡x−log⁡ylog⁡z\begin{aligned} \log G_1 G_2 G_3= & \log \left[\left(\frac{y}{x}\right)^{\log z}\right]+\log \left[\left(\frac{z}{y}\right)^{\log x}\right]+\log \left[\left(\frac{x}{z}\right)^{\log y}\right] \\ = & \log z \log y-\log z \log x+\log x \log z -\log x \log y+\log y \log x-\log y \log z \end{aligned}$\begin{aligned} & \log G_1 G_2 G_3=0 \\ & G_1 G_2 G_3=\mathrm{e}^0=1 \\ & \text{Hence } m=\sqrt[3]{G_1 G_2 G_3}=(1)^{\frac{1}{3}} \\ & m=1\end{aligned}$
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