CAT 2025

Slot 2

QA

Question & Solution

Let's assume that the common root is r. 

The sum of the roots of the first equation is 5/3 and that of the second equation is 1.

We want the sum of the other two roots:

$$ \text{Sum} = \left(\frac{5}{3}-r\right) + (1-r) = \frac{8}{3}-2r$$

We now need to express r in terms of p and q.

Since r is a common root, it satisfies:

$$3r^2 - 5r + p = 0 \quad (1)$$

$$ 2r^2 - 2r + q = 0 \quad (2)$$

Eliminate $$ r^2$$ .

Multiply (2) by 3:

$$ 6r^2 - 6r + 3q = 0$$

Multiply (1) by 2:

$$ 6r^2 - 10r + 2p = 0$$

Subtract:

$$ (6r^2 - 6r + 3q) - (6r^2 - 10r + 2p) = 0$$

$$ 4r + 3q - 2p = 0$$

$$  r = \frac{2p - 3q}{4}$$

Now substitute into $$ \frac{8}{3} - 2r$$ :

$$ \frac{8}{3} - 2\left(\frac{2p - 3q}{4}\right)$$

$$ = \frac{8}{3} - \frac{2p - 3q}{2}$$

$$ = \frac{8}{3} - p + \frac{3}{2}q$$