If x- 0 and - 1- y- 0 and - 1- z- 0 and - 1 then the value of - 1 logxy logxz logyx 1 logyz logzx logzy 1 - is

The correct option is B 0 | 1 amp; log_ x y amp; log_ x z log_ y x amp; 1 amp; log_ y z log_ z x amp; log_ z y amp; 1 ...

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Byju's Answer

Standard VI

Mathematics

Idea of a Set

If x> 0 and...

Question

If $x > 0$ and $\neq 1, y > 0$ and $\neq 1, z > 0$ and $\neq 1$ then the value of
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

0

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1

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- 1

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none of these

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Solution

The correct option is B $0$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & \frac{{log}_{e} y}{{log}_{e} x} & \frac{{log}_{e} z}{{log}_{e} x} \frac{{log}_{e} x}{{log}_{e} y} & 1 & \frac{{log}_{e} z}{{log}_{e} y} \frac{{log}_{e} x}{{log}_{e} z} & \frac{{log}_{e} y}{{log}_{e} z} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{{log}_{e} x {log}_{e} y {log}_{e} z} ∣ ∣ ∣ ∣ ∣ \begin{matrix} {log}_{e} x & {log}_{e} y & {log}_{e} z {log}_{e} x & {log}_{e} y & {log}_{e} z {log}_{e} x & {log}_{e} y & {log}_{e} z \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 & 1 & 1 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣$
$= 0$

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If x>0 and ≠1,y>0 and ≠1,z>0 and ≠1 then the value of∣∣ ∣ ∣∣1logxylogxzlogyx1logyzlogzxlogzy1∣∣ ∣ ∣∣ is

If $x > 0$ and $\neq 1, y > 0$ and $\neq 1, z > 0$ and $\neq 1$ then the value of
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ is