Question

Assertion (A): `(sqrt(3) + sqrt(5))` is an irrational number.

Reason (R): Sum of the any two irrational numbers is always irrational.

Options

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

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

• 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): `(sqrt(3) + sqrt(5))` is an irrational number.

Let us assume `(sqrt(3) + sqrt(5))` is a rational number x.

`x = sqrt(3) + sqrt(5)`

Squaring both sides:

`x^2 = (sqrt(3) + sqrt(5))^2`

`x^2 = 3 + 5 + 2sqrt(15)`

`x^2 = 8 + 2sqrt(15)`

`x^2 - 8 = 2sqrt(15)`

`(x^2 - 8)/2 = sqrt(15)`

Since x is rational, `(x^2 - 8)/2` is also rational.

But `sqrt(15)` is irrational.

This is a contradiction.

So, `(sqrt(3) + sqrt(5))` is irrational.

Assertion (A) is true.

Reason (R): Sum of any two irrational numbers is always irrational.

Let us consider two irrational numbers: `sqrt(2)` and `-sqrt(2)`.

Sum = `sqrt(2) + (-sqrt(2)) = 0`

Since 0 is a rational number, the sum of two irrational numbers is not always irrational.

Reason (R) is false.