Question

If $p^{x–1}=qr,q^{y–1}=rp,r^z–1=pq,$ then the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ equals

• A. 2
• B. 1
• C. 3
• D. 0

Answer 1

The correct option is B 1

Given: $p^{x–1}=qr,q^{y–1}=rp,r^z–1=pq,$
$p^{x-1}=qr$
$\Rightarrow p^{x-1}\times p=pqr$
$\Rightarrow p^x=pqr$
$\Rightarrow p=(pqr{)}^{\frac{1}{x}}$ ....(i)
Similiarly, $q^{y-1}=rp$
$\Rightarrow q=(pqr{)}^{\frac{1}{y}}$ ....(ii)
Also, $r^{z-1}=pq$ ....(iii)
$\Rightarrow r=(pqr{)}^{\frac{1}{z}}$
Multiplying (i), (ii) and (iii):
$p\times q\times r={\left(pqr\right)}^{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}$ $(\because a^m\times a^n=a^{m-n})$
$\Rightarrow {\left(pqr\right)}^1={\left(pqr\right)}^{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}}$
$\Rightarrow 1=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
Hence, the correct answer is option (b).