GATE CS 2018 Q.7 (GA Q.7): Quantitative Aptitude

If $pqr \neq 0$ and $p^{-x} = \frac{1}{q}$ , $q^{-y} = \frac{1}{r}$ , $r^{-z} = \frac{1}{p}$ , what is the value of the product $xyz$ ?

$-1$

$\frac{1}{pqr}$

$1$

$pqr$

Official answer and solution

Solution

Express each variable in terms of the next using the given exponential relations.

From $p^{-x} = \frac{1}{q}$ , we take reciprocals on both sides: $p^x = q$ .

From $q^{-y} = \frac{1}{r}$ , we take reciprocals: $q^y = r$ .
Substitute $q = p^x$ into this equation:

r = (p^x)^y = p^{xy}

From $r^{-z} = \frac{1}{p}$ , we take reciprocals: $r^z = p$ .
Substitute $r = p^{xy}$ into this equation:

(p^{xy})^z = p \implies p^{xyz} = p^1

Equating the exponents gives $xyz = 1$ .

Final answer

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