Question

Assertion: 2 is a rational number.
Reason: The square roots of all positive integers are irrationals.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.

Answer 1

Hint: Integers are the complete numbers, i.e. they are free of any decimal parts.
e.g. $...,-2,-1,0,1,2,...$
Rational numbers are the numbers which can be expressed as a ratio of two integers. i.e. they can be written in the form of ${\displaystyle \frac{p}{q}},(p,q)\in \mathbb{Z},q\ne 0$.
Irrational numbers are the numbers which cannot be expressed as a ratio of two integers.
e.g. $\sqrt{2}=1.4142135...,\pi =3.141592...,e=2.71828...,\log 5=0.69897...$

Complete step by step answer:
Let us examine both the statements one by one:
Assertion: 2 is a rational number.: CORRECT, because we can write 2 as ${\displaystyle \frac{2}{1}}$, where both 2 and 1 are integers.
Reason: The square roots of all positive integers are irrationals.: INCORRECT, because square roots of perfect square positive integers are themselves integers, not irrational. e.g. $\sqrt{4}=2$.
Since the reason is incorrect, the question of it being a correct explanation does not arise at all.

The correct answer is C. Assertion is correct but Reason is incorrect.

Note: Rational numbers are either terminating or non-terminating and repeating in decimal form.
e.g. 3.4, 5.7777... = $5.\overline{7}$, 8.13 etc.
Irrational numbers are neither terminating nor repeating in decimal form.
e.g. $\sqrt{2}=1.4142135...,\pi =3.141592...,e=2.71828...,\log 5=0.69897...$
An even root of a negative number is a complex number. The complex unit is defined as $i=\sqrt{-1}$.