A quadratic equation with integral coefficients has two different prime numbers as its roots. If the sum of the coefficients of the equation is prime, then the sum of the roots is

Let equation is $\mathrm{a}(\mathrm{x}-\mathrm{\alpha })(\mathrm{x}-\mathrm{\beta })=0,\text{ where }\mathrm{\alpha },\mathrm{\beta }$ are prime numbers 

Sum of coefficients $=\mathrm{a}-\mathrm{a}(\mathrm{\alpha }+\mathrm{\beta })+\mathrm{a\alpha \beta }$

$=\mathrm{a}(\mathrm{\alpha }-1)(\mathrm{\beta }-1)$, which is prime number 

$\Rightarrow a=1\text{ and one of }(\alpha -1)\text{ and }(\beta -1)\text{ will be }1$

Let $\mathrm{\alpha }-1=1\text{ or }\mathrm{\alpha }=2$

then $\beta -1$ will be 2

So $\beta =3$

Hence, sum of roots = 5