QJEEAdv08Ma1895: JEE Advanced 2008 Mathematics - Class 11 JEEAdv08Ma1895 Solution: Let , be real numbers. Suppose are the roots of the equation…

As a, b, c, p, q $\in$ R and the two given equations have exactly one common root.

$\Rightarrow$ Either both equations have real roots.

Or both equations have imaginary roots.

$\Rightarrow$ Either $\Delta_1\ge0$ and \Delta_2 > 0 or

\Delta_1 < 0 and \Delta_2 < 0

$\Rightarrow~p^2-q\ge0$ and $b^2-ac\ge0$

or $p^2-q\le0$ and $b^2-ac\le0$

$\Rightarrow~(p^2-q)(b^2-ac)\ge0$

$\therefore$ Statement 1 is true.

Also, we have $\alpha\beta=q$ and ${\alpha \over \beta } = {c \over a}$

$\therefore$ ${{\alpha \beta } \over {{\alpha \over \beta }}} = {q \over c} \times a \Rightarrow {\beta ^2} = {{qa} \over c}$

As $\beta \ne 1$ or $-1$

$\Rightarrow {\beta ^2} \ne 1$

$\Rightarrow {{qa} \over c} \ne 1$ or $c \ne qa$

Again as exactly one root $\alpha$ is common and $\beta\ne1$

$\therefore$ $\alpha + \beta \ne \alpha + {1 \over \beta } \Rightarrow {{ - 2b} \over a} \ne - 2P$

$\Rightarrow b \ne ap$

$\therefore$ Statement 2 is correct.

But Statement 2 is not correct explanation of Statement 1.