Which of the following statements is true?

MCQ

A
Product of two irrational numbers is always irrational.
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Product of a rational and an irrational number is always irrational.
C
Sum of two irrational numbers can never be irrational.
D
Sum of an integer and a rational number can never be an integer.

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Answer

Correct option: B.

Product of a rational and an irrational number is always irrational.

$a$.Is incorrect, Product of two irrational numbers is not always irrational, it can be also rational sometimes.
when an irrational number is multiplied to itself, or multiplied by another irrational, that product becomes a perfect square.
Example:
$\sqrt{2}\times\sqrt{2}=2 ($Rational$)$
$\sqrt{2}\times\sqrt{8}=\sqrt{16}=\pm4 ($Rational$)$
$b$.Is correct, because when a rational number is multiplied to an irrational number, it can not make an irrational number terminating or Non$-$terminating Repeating.
Product again becomes a Non$-$terminating Non$-$Repeating number.
as: $2\times\sqrt{3}=2\sqrt{3}$
$\frac{2}{3}\times\sqrt{3}=\frac{2}{\sqrt{3}}$
So, product of a rational number and an irrational number is always an irrational, because irrational number is just changed in magnitude not in properties.
$c$.Is incorrect, Sum of two irrational numbers can be an irrational number.
i.e. if we add $\sqrt{2}$ and $\sqrt{3},$ we will get $\sqrt{2}+\sqrt{3}$ which is also an irrational.
$d$.Is incorrect, Sum of an integer and a rational number can be a integer.
Because all integers are rational numbers and also we can say some rational numbers are integers.
So their sum with integer would be a integer
i.e. $2 + 3 = 5$
Hence, correct option is $(b).$