Question

Which one of the following is not true?
A. $$\sqrt{2}$$ is an irrational number
B. If a is a rational number and $$\sqrt{b}$$ is an irrational number then $$a\sqrt{b}$$ is irrational number
C. Every surd is an irrational number
D. The square root of every positive integer is always irrational.

Answer 1

Hint:
This is a statement based; hence we will look into each statement and select or eliminate the ones which are true or false. In this question, the number system is used, we know that an irrational number is the one that cannot be simplified into $${\displaystyle \frac{p}{q}}$$ form. We will find the statement which is NOT correct to get to the final answer.

Complete step by step solution:
Statement-1:
  $$\sqrt{2}$$ is an irrational number

We know that $$\sqrt{2}$$ cannot be further simplified to eliminate its root, Since the root is not eliminated, the number cannot be simplified into $${\displaystyle \frac{p}{q}}$$ form. Hence, $$\sqrt{2}isanirrationalnumber$$ .
Therefore, Statement-1 is true
Since statement-1 is true, Statement-1 is eliminated

Statement-2:
If a is a rational number and $$\sqrt{b}$$ is an irrational number then $$a\sqrt{b}$$ is irrational number

We know that even after multiplying a rational number $$a$$to an irrational number $$\sqrt{b}$$ , $$\sqrt{b}$$ is still present in the product and cannot be further simplified to eliminate its root, Since the root is not eliminated, the number cannot be simplified into $${\displaystyle \frac{p}{q}}$$ form.
Hence, If a is a rational number and $$\sqrt{b}$$ is an irrational number then $$a\sqrt{b}$$ is an irrational number.

Therefore, Statement-2 is true .
Since statement-2 is true, Statement-2 is eliminated.


Statement-3: Every surd is an irrational number
According to the definition of a surd, when we can't simplify a number to remove a square root (or cube root etc) then it is a surd. Hence, every surd is an irrational number.

Therefore, Statement-3 is true.
Since statement-3 is true, Statement-3 is eliminated.

Statement-4:
The square root of every positive integer is always irrational.

Let us consider a positive integer as 4. Hence, taking the square root of 4, we get $$\sqrt{4}$$. We know that $$\sqrt{4}$$ can be further simplified as 2 and hence, it can be simplified into $${\displaystyle \frac{p}{q}}$$ form.
Therefore, Statement-4 is false
Since statement-4 is false, Statement-4 is selected.

Hence, our final answer is D.

Note:
In questions like these, we need to read the questions very carefully because students in hurry may miss out on the NOT present in the question, and then they will get the first statement as true assuming that to be the final answer and skipping to check all the other options. Hence, may end up predicting a WRONG answer for such an easy question.