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The product of a non-zero rational and an irrational number is

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always irrational

always rational

rational or irrational

none

answer is A.

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Detailed Solution

We need to find out the product of a non-zero rational and an irrational number.
Let

x

be a rational number and

y

be an irrational number.
Let

xy = a

.
Let us assume that

a

is rational.
Since,

a

is a rational number, it can be expressed as

\frac{p}{q}

, where

p

and

q

are integers.
Let

x = \frac{m}{n}

, where

m

and

n

are integers.
Now,

xy = a

\frac{my}{n} = \frac{p}{q}

\Rightarrow y = \frac{pn}{qm}

Now,

pn

and

qm

are integers. Therefore,

\frac{pn}{qm}

is a rational number.
However,

y

is irrational. So, our assumption is incorrect.
Therefore, the product of a non-zero rational and an irrational number is always an irrational number.
Hence, option 1 is correct.

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