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प्रश्न
Prove that `(2+sqrt3)/5` is an irrational number, given that `sqrt 3` is an irrational number.
उत्तर
To prove `(2+sqrt3)/5` is irrational, let us assume that `(2+sqrt3)/5` is rational.
`(2+sqrt3)/5 = "a"/"b"; "b" ≠ 0` and a and b are integers.
`=> 2"b" + sqrt3 "b" = 5"a"`
`=> sqrt 3 "b" = 5"a" - 2 "b"`
`=> sqrt 3 = (5"a" - 2"b")/"b"`
Since a and b are integers so, 5a - 2b will also be an integer.
So, `(5"a" - 2"b")/"b"` will be rational which means `sqrt 3` is also rational.
But we know `sqrt 3` is irrational (given).
Thus, a contradiction has risen because of incorrect assumptions.
Thus, `(2+sqrt3)/5` is irrational
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