Question

For three irrational numbers $p,q$ and $r$ then $p.(q+r)$ can be

• A. A rational number
• B. An irrational number
• C. An integer
• D. All of the above

Answer 1

The correct option is D All of the above

$p,q$ and $r$ are all irrational
Let $p=q=r=\sqrt{2}$
$p(q+r)=p.q+p.r$
$\Rightarrow \sqrt{2}(\sqrt{2}+\sqrt{2})=\sqrt{2}.\sqrt{2}+\sqrt{2}.\sqrt{2}=2+2=4$
which is rational as well as integer
Let us take another case in which $p=\sqrt{2}$ and $q=r=\sqrt{3}$
$\Rightarrow \sqrt{2}(\sqrt{3}+\sqrt{3})=\sqrt{2}.\sqrt{3}+\sqrt{2}.\sqrt{3}=\sqrt{6}+\sqrt{6}=2\sqrt{6}$
which is an irrational number.
So on applying distributive property on three irrational numbers we can get an integer,a rational as well as an irrational number .
So option $D$ is correct.