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Assertion (A) : $(a+ \sqrt{b}) . (a - \sqrt{b})$ is a rational number, where a and b are positive integers. Reason (R) : Product of two irrationals is always rational
Directions : In question below a statement of Assertion (A)
is followed by a statement of Reason (R). Choose the correct option :
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct
explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not
correct explanation for Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
Assertion (A) : $(a+ \sqrt{b}) . (a - \sqrt{b})$ is a rational number, where a and b are
positive integers.
Reason (R) : Product of two irrationals is always rational
This Question has 1 answers.
### Assertion (A):
$ (a + \sqrt{b}) \cdot (a - \sqrt{b}) $ is a rational number, where $a$ and $b$ are positive integers.
### Proof:
Using the identity for the difference of squares:
$ (a + \sqrt{b}) \cdot (a - \sqrt{b}) = a^2 - (\sqrt{b})^2 $
$ = a^2 - b $
Since $a$ and $b$ are positive integers, $a^2 - b$ is also an integer, which is a rational number.
Thus, Assertion (A) is **true**.
---
### Reason (R):
"Product of two irrationals is always rational."
This statement is **false**. The product of two irrational numbers is **not always** rational. For example:
$ \sqrt{2} \times \sqrt{3} = \sqrt{6} $
which is irrational.
Thus, Reason (R) is **false**.
---
### Conclusion:
- Assertion (A) is **true**.
- Reason (R) is **false**.
- Since Reason (R) is incorrect, it does not explain Assertion (A).
**Final Answer:** Assertion is true, but Reason is false.
$ (a + \sqrt{b}) \cdot (a - \sqrt{b}) $ is a rational number, where $a$ and $b$ are positive integers.
### Proof:
Using the identity for the difference of squares:
$ (a + \sqrt{b}) \cdot (a - \sqrt{b}) = a^2 - (\sqrt{b})^2 $
$ = a^2 - b $
Since $a$ and $b$ are positive integers, $a^2 - b$ is also an integer, which is a rational number.
Thus, Assertion (A) is **true**.
---
### Reason (R):
"Product of two irrationals is always rational."
This statement is **false**. The product of two irrational numbers is **not always** rational. For example:
$ \sqrt{2} \times \sqrt{3} = \sqrt{6} $
which is irrational.
Thus, Reason (R) is **false**.
---
### Conclusion:
- Assertion (A) is **true**.
- Reason (R) is **false**.
- Since Reason (R) is incorrect, it does not explain Assertion (A).
**Final Answer:** Assertion is true, but Reason is false.
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