Question

Assertion (A): `sqrt2(5 - sqrt2)` is an irrational number.

Reason (R): Product of two irrational number is always irrational.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

• Assertion (A) is true but Reason (R) is false.

• Assertion (A) is false but Reason (R) is true.

Answer 1

Assertion (A) is true but Reason (R) is false.

Explanation:

Assertion (A): Simplifying the expression 

`sqrt2(5 - sqrt2)`

= `5sqrt2 - (sqrt2)^2`

= `5sqrt2 - 2`

It is known that `sqrt2` is an irrational number and multiplying an irrational number by a rational number. (here, 5) results in an irrational number. Subtracting a rational number (2) from an irrational number still gives us an irrational number.

Thus, `sqrt2(5 - sqrt2)` is an irrational number.

Hence, Assertion (A) is true.

Reason (R): It is known that, `sqrt2` is an irrational number.

Product of `sqrt2` and `sqrt2` is equal to 2, which is a rational number.

Therefore, product of two irrational numbers is always an irrational number is false.

Hence, Reason (R) is false.