Question

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

Answer 1

Hint: To find the relation between assertion and reason, we have to analyze each statement. We can see that the assertion states that 2 is a rational number. A rational number is a real number that can be represented in the form of $\dfrac{P}{Q}$ where P and Q are integers and $Q \ne 0$. From this, we can analyze whether the assertion is correct or not. Irrational numbers are real numbers that can be written in decimals but cannot be written as the ratio of two integers. They have endless non-repeating digits after the decimal point. We can write down the square roots of a few numbers like the first 10 real numbers to check whether the reason is true or not.

Complete step-by-step solution
We need to find the relation between assertion and reason. In the assertion section, it is given that 2 is a rational number. Let us find what is a rational number.
A rational number is a real number that can be represented in the form of $\dfrac{P}{Q}$ where P and Q are integers and $Q \ne 0$. In other words, a rational number is a number that can be represented as the ratio of two integers.
Now, let us examine 2.
2 is a real number and can be written as \[\dfrac{2}{1}\], that is, the ratio of two integers. Hence 2 is a rational number.
We can see that the assertion statement is true.
Now, let us examine the reason statement.
It is given that the square roots of all positive integers are irrational.
Irrational numbers are real numbers that can be written in decimals but cannot be written as the ratio of two integers. They have endless non-repeating digits after the decimal point. For example, let us consider $\sqrt{2}=1.414....$ . We can see that they have long digits after the decimals.
Now, let us come back to the statement. Let us consider the square root of the first 10 positive integers as follows.
$\begin{align}
  & \sqrt{1}=1 \\
 & \sqrt{2}=1.414... \\
 & \sqrt{3}=1.732... \\
 & \sqrt{4}=2 \\
 & \sqrt{5}=2.236... \\
 & \sqrt{6}=2.449... \\
 & \sqrt{7}=2.645... \\
 & \sqrt{8}=2.828... \\
 & \sqrt{9}=3 \\
 & \sqrt{10}=3.162... \\
\end{align}$
We can see that all the square roots of positive integers are not irrational. For, example, let us observe 1, 4,9,16, etc. They form the square root of these perfect squares. They are rational numbers. We can also note that all perfect squares are rational numbers.
Hence, we can say that the square roots of all positive integers are not irrational. Hence, the correct option is C.

Note: Do not get confused with rational and irrational numbers. You may interchange the definitions of these like rational numbers that can be written in decimals but cannot be written as the ratio of two integers while irrational numbers can be expressed as a ratio of two integers. We can also observe that the numbers that form perfect squares are always rational.