Question

Assertion: All integers are rational.

Reason: Square root of all positive integers are irrational.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

A is true but R is false.

Explanation:

Assertion: All integers are rational.

• A rational number is any number that can be expressed in the form `p/q`, where p, q ∈ ℤ, q ≠ 0.
• Example: `5 = 5/1, -3 = (-3)/1, 0 = 0/1`.
  Hence, all integers are rational.

Reason: Square root of all positive integers are irrational.

• This is not always true.
• If the integer is a perfect square (like 1, 4, 9, 16, ...), its square root is an integer, hence rational.
• Example: `sqrt(9) = 3` (rational), but `sqrt(2), sqrt(3), sqrt(5), ...` are irrational.
  Therefore, the reason is false.