Question

If $p$ and $q$ are two distinct irrational numbers, then which of the following is always is an irrational number

• A. $\frac{p}{q}$
• B. $pq$
• C. $(p+q{)}^2$
• D. $\frac{p^{2}q+qp}{pq}$

Answer 1

Answer: (D)

$\frac{p^{2}q+qp}{pq}$

As, given $p$ and $q$ are two distinct irrational numbers.

Let $p=2+\sqrt{3}$ and $q=2-\sqrt{3}$

Then,

Option $A$

$\frac{p}{q}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$

$\frac{p}{q}=\frac{4+3+4\sqrt{3}}{4-3}=7+4\sqrt{3}$

Option $B$

$pq=(2+\sqrt{3})(2-\sqrt{3})=4-3=1$

Option $C$

$(p+q{)}^2=(2+\sqrt{3}+2-\sqrt{3}{)}^2=4^2=16$

Option $D$

$\frac{p^{2}q+pq}{pq}=p+1$ is always an irrational number, because sum of rational and irrational is always irrational.

Hence, this is irrational.

Hence, this is the answer.