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Mathematical Reasoning — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsMathematical Reasoning3 Marks
Question
Check the validity of the statements given below by the method given against it.
p: The sum of an irrational number and a rational number is irrational (by contradiction method).
✓

Answer

The given statement is as follows.
p: the sum of an irrational number and a rational number is irrational.
Let us assume that the given statement, p, is false. That is, we assume that the sum of an irrational number and a rational number is rational.
Therefore, $\sqrt{\text{a}}+\frac{\text{b}}{\text{c}}=\frac{\text{d}}{\text{e}},$ where $\sqrt{\text{a}}$ is irrational and and b, c, d, e are integers.
⇒ de − bc = a
But here, $\frac{\text{d}}{\text{e}}-\frac{\text{b}}{\text{c}}$ is a rational number and $\sqrt{\text{a}}$ is an irrational number.
This is a contradiction. Therefore, our assumption is wrong.
Therefore, the sum of an irrational number and a rational number is rational.
Thus, the given statement is true.

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