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प्रश्न
Prove that of the numbers `2 -3 sqrt(5)` is irrational:
उत्तर
Let `2 -3 sqrt(5)` be rational.
Hence 2 and `2 -3 sqrt(5)` are rational.
∴ 2 – ( `2 -3 sqrt(5) )` = 2 – `2 + 3 sqrt(5)` = ` 3 sqrt(5)` = rational [∵Difference of two rational is rational]
∴ `1/3 × 3sqrt(5) = sqrt(5)` = rational [∵Product of two rational is rational]
This contradicts the fact that `sqrt(5)` is irrational.
The contradiction arises by assuming `2 -3 sqrt(5)` is rational.
Hence, `2 -3 sqrt(5)` is irrational.
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