IIT-JEE 2008 · Paper 2, Mathematics question 11

Statement 2. If both $b=pa$ and $c=qa$, the two equations would be the same, so $\{\alpha,\beta\}=\{\alpha,\frac1\beta\}$, forcing $\beta^2=1$: excluded. So $b\ne pa$ or $c\ne qa$. True.

Statement 1. If α were non-real, then $\beta=\bar\alpha$ (real coefficients), and the second equation's roots $\alpha$ and $\frac1{\bar\alpha}$ would also have to be conjugates, so $\frac1{\bar\alpha}=\bar\alpha$, i.e. $\beta^2=1$: excluded. So all roots are real, both discriminants $4(p^2-q)$ and $4(b^2-ac)$ are $\ge0$, and their product is $\ge0$. True.

Explanation? Statement 2 is not the reason for Statement 1.

(B)