CUET PG 2022 — Mathematics PYQ

1. Simplify the Terms

Let $u={\log }_{x}y$ and $v={\log }_{x}z$.

• $a_1={\log }_{x}y=u$

• $a_2={\log }_{z}y=\frac{{\log }_{x}y}{{\log }_{x}z}=\frac{u}{v}$

• $a_3=-15{\log }_{x}z=-15v$

2. Set up the AP Equation

Since $a_1,a_2,a_3$ are in AP:

3. Solve for $u$

Multiply by $v$:

4. Find the constant value

In this specific problem, for the terms to maintain a consistent AP regardless of the values of the variables, we test for the condition where $u$ results in a characteristic value.

Setting $u=-1$ (which leads to $xy=1$):

5. Solve the Quadratic for $v$

Using the factorization method:

Since these roots provide a valid $v$, it confirms that: