Consists of two statements, namely, Assertion (A) and Reason (R).… - Vidyadip

MCQ

Consists of two statements, namely, Assertion $(A)$ and Reason $(R).$ For selecting the correct answer, use the following code:

Assertion (A)	Reason (R)
$\sqrt{3}$ is an irrational number.	The sum of rational number and an irrational number is an irrational number.

The correct answer is: $(a), (b), (c), (d).$

A
Both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is a correct explanation of Assertion $(A).$
✓
Both Assertion $(A)$ and Reason $(R)$ are true but Reason is not a correct explanation of Assertion $(A).$
C
Assertion $(A)$ is true and Reason $(R)$ is false.
D
Assertion $(A)$ is false and Reason $(R)$ is true.

✓

Answer

Correct option: B.

Both Assertion $(A)$ and Reason $(R)$ are true but Reason is not a correct explanation of Assertion $(A).$

If possible, let $\sqrt{3}$ be a rational number and its simplest form is $\frac{\text{a}}{\text{b}}.$
Then, $\sqrt{3}=\frac{\text{a}}{\text{b}}\Rightarrow\frac{\text{a}^2}{\text{b}^2}=3\Rightarrow\frac{\text{a}^2}{\text{b}}=3\text{b}$
Clearly, $3b$ is an integer and $\frac{\text{a}^2}{\text{b}}$ is not an integer since $(a, b) = 1$
Thus, we arrive at a contradiction
So, our supposition is wrong
Hence, $\sqrt{3}$ is an irrational number
So, the Assertion $(A)$ is true.
If possible, let the sum of a rational number a and an irrational number $\sqrt{\text{b}}$ be a rational number
Then, $\text{a}+\sqrt{\text{b}}=\text{c}\Rightarrow\sqrt{\text{b}}=\text{c}-\text{a}$
But, the difference of two irrational is a rational
So, $(c - a)$ is rational and thus, $\sqrt{\text{b}}$ is rational
Thus, we arrive at a contradiction
So, our supposition is wrong
Hence, the sum of a rational and an irrational is irrational
So, the reason $(R)$ is true.
Hence, the Assertion $(A)$ and Reason $(R)$ are true but Reason is not a correct explanation of Assertion $(A).$

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