Question

Which of the following statements are true always?

• A. Product of 2 irrational numbers is an irrational number.
• B. The sum of two irrational numbers is irrational.
• C. The sum or difference of a rational and irrational number is rational.
• D. The product and quotient of a non-zero rational and irrational number is irrational.

Answer 1

The correct option is D The product and quotient of a non-zero rational and irrational number is irrational.

Option A does not always hold true as $\sqrt{2}\times \sqrt{2}=2$, which is rational.
Sum of two irrational number can also be rational. Here is one example among many possibilities: $(5-\sqrt{2})+(4-\sqrt{2})+2\sqrt{2}=9$. Thus option B does not always hold true.
For different rational and irrational values, statements in options A,B,C fail to hold. But option D is true always.