CUET PG 2023 — Mathematics PYQ

Step 1: Relationships from Sum and Product of Roots

Using the properties of quadratic equations:

For equation (1):

• Sum of roots: $\alpha +\beta =-p$

• Product of roots: $\alpha \beta =qr$

For equation (2):

• Sum of roots: $\beta +\gamma =-q$

• Product of roots: $\beta \gamma =rp$

For equation (3):

• Sum of roots: $\gamma +\alpha =-r$

• Product of roots: $\gamma \alpha =pq$

Step 2: Finding the Product of the Common Roots

We need to find the product of the three common roots, which is $\alpha \beta \gamma$.

Multiply the three product equations together:

Taking the square root on both sides:

Step 3: Determining the Sign

Let's check the sum of the roots. Adding the sum equations:

Also, we can find $\alpha ,\beta ,\gamma$ in terms of $p,q,r$ by subtracting equations. For example:

From $\alpha \beta =qr$ and $\beta \gamma =rp$, dividing them gives $\alpha /\gamma =q/p⟹p\alpha =q\gamma$.

By substituting these back into the original equations, it can be shown that $\alpha \beta \gamma =pqr$ is the consistent solution for these types of symmetric cyclic equations.

Answer: The product of the three common roots is $pqr$.