CUET PG 2022 — Mathematics PYQ

1. Given Relationship from Options:

Since $yz=z^{-2}$, we can write:

2. Convert all terms to base $z$:

• First term ($a$): ${\log }_{x}y=\frac{{\log }_{z}y}{{\log }_{z}x}=\frac{-3}{{\log }_{z}x}$

• Second term ($b$): ${\log }_{z}y=-3$

• Third term ($c$): $-15{\log }_{x}z=\frac{-15}{{\log }_{z}x}$

3. Apply AP Condition ($2b=a+c$):

Substitute the terms into the formula:

4. Solve for ${\log }_{z}x$:

This means $x=z^3$.

Verification

If $x=z^3$ and $y=z^{-3}$, let's check the original terms:

• $a={\log }_{x}y={\log }_{z^3}{z}^{-3}=-1$

• $b={\log }_{z}y={\log }_z{z}^{-3}=-3$

• $c=-15{\log }_{x}z=-15{\log }_{z^3}z=-15\left(\frac{1}{3}\right)=-5$

The terms are $-1,-3,-5$, which are clearly in AP with a common difference of $-2$.

Final Answer:

Based on the AP condition being satisfied: