CAT 2024 Slot 1 · Quant · Question 20

Let p be the common root, so it satisfies all three equations.

From the first two, {p}^{2}+mp+9=0 and {p}^{2}+np+17=0.

Adding them, 2{p}^{2}+\left( m+n \right)p+26=0.

The third equation says {p}^{2}+\left( m+n \right)p+35=0.

Subtracting the third from the sum removes the awkward \left( m+n \right)p term entirely: {p}^{2}-9=0.

So p=\pm 3, and the root is negative, giving p=-3.

Substituting into the first equation, 9-3m+9=0, so m=6.

Substituting into the second, 9-3n+17=0, so n=\frac{26}{3}.

Hence 2m+3n=12+26=38.

Shortcut. Adding the first two equations produces exactly the \left( m+n \right)p term that the third one carries, so one subtraction eliminates both unknowns at once and leaves {p}^{2}=9. There is no need to solve for m and n before finding p.