Question

Determine whether the argument used to check the validity of the following statement is correct.
P: If $ {{x}^{2}} $ is irrational, then x is rational’
The statement is true because the number $ {{x}^{2}}={{\pi }^{2}} $ is irrational, therefore $ x=\pi $ irrational.
A. True
B. False

Answer 1

Hint: We had to check whether the given argument or reason given for the statement is true or false. For which we will take some general example which will satisfy or say that statement is right then it is true otherwise it is false.

Complete step by step answer:
Moving ahead with the question in the step wise manner;
The statement says that if $ {{x}^{2}} $ is irrational, then x is rational. And according to argument it is true because the number $ {{x}^{2}}={{\pi }^{2}} $ is irrational, therefore $ x=\pi $ irrational, means they had taken the example of $ \pi $ .
Now to check whether the given statement is true or false let us check for it by taking a general example. Let $ x=\sqrt{k} $ , in which k is a rational number and $ \sqrt{k} $ is an irrational number, which means ‘x’ is also an irrational number.
Now according to the statement if we get the $ {{x}^{2}} $ , it will also become irrational. So let us find out $ {{x}^{2}} $ by squaring both sides. For example we took i.e. $ x=\sqrt{k} $ . So we will get;
 $ \begin{align}
  & {{\left( x \right)}^{2}}={{\left( \sqrt{k} \right)}^{2}} \\
 & {{x}^{2}}=k \\
\end{align} $
So we got $ {{x}^{2}}=k $ and as we had taken ‘k’ as a rational number. Which means $ {{x}^{2}} $ is also rational, which contradicts the statement which says that $ {{x}^{2}} $ should be irrational.
So it is a false statement.

So, the correct answer is “Option B”.

Note: In order to get the answer of such assertion and reasoning type questions we had to go in the similar process, by taking general or particular examples and finding out whether it will satisfy or not. If it satisfies then the statement is true otherwise it is false.