Question

Assertion (A): `(sqrt(2) + sqrt(3))` is an irrational number.

Reason (R): The sum of two irrational numbers is always an irrational number.

पर्याय

• 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:

Evaluate the Assertion (A):

The statement `(sqrt(2) + sqrt(3))` is an irrational number is true.

To prove this, we can assume it is rational (equal to x) and square both sides:

`(sqrt(2) + sqrt(3))^2 = x^2`

`2 + 3 + 2sqrt(6) = x^2`

`2sqrt(6) = x^2 - 5`

`sqrt(6) = (x^2 - 5)/2`

Since x  is rational, the right side is rational, but `sqrt(6)` is irrational. This contradiction proves `(sqrt(2) + sqrt(3))` must be irrational.

Evaluate the Reason (R):

The statement “The sum of two irrational numbers is always an irrational number” is false. While it is often true, it is not always true. We can disprove this with a simple counterexample:

Let the first irrational number be `sqrt(2)`.

Let the second irrational number be `-sqrt(2)`.

Their sum is `sqrt(2) + (-sqrt(2)) = 0`. 

Since 0 is a rational number, the “always” part of the statement fails.