Advertisements
Advertisements
Question
(i) `sqrt3` is a rational number.
(ii) The rationalising factor of `sqrt3 + 1` is `sqrt3 - 1`.
Options
Only (i)
Only (ii)
Both (i) and (ii)
Neither (i) nor (ii)
Advertisements
Solution
Only (ii)
Explanation:
(i) `sqrt(3)` is not a rational number; it is an irrational number.
This is because `sqrt(3)` cannot be expressed as a fraction of two integers.
A proof by contradiction shows that if it were rational, it would imply a common factor contradicting the simplest form assumption.
Therefore, statement (i) is false.
(ii) The rationalising factor of `sqrt(3) + 1` is `sqrt(3) - 1`.
Multiplying by this conjugate removes the irrational part in the denominator or numerator since `(sqrt(3) + 1)(sqrt(3) - 1) = 3 - 1 = 2`, a rational number.
Therefore, statement (ii) is true.
Hence, only statement (ii) is valid.
