State True or False.

Given that √5 is irrational, 2√5 − 3 is also an irrational number.

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True

False

answer is A.

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Detailed Solution

To prove the Irrationality of numbers, we need to apply the method of contradiction.
Let us assume that the given number is rational and show that our assumption is wrong.
So, assume 2√5 - 3 is rational.
Then, it can be represented in the form a/b, where a, b are co-primes integers and b ≠ 0.

25 − 3 = a b

By rearranging the terms, we get,

25 = a b + 3 25 = a + 3 b b 5 = 12 a + 3 b b 5 = a + 3 b 2 b

Since a and b are integers, a + 3b and 2b are also integers.
So,

a + 3 b 2 b

is a rational number.
∴ √5 is rational.
But it is given that, √5 is irrational.
This contradicts the fact that √5 is an irrational number.
Therefore, the statement is true.
So, the correct option is (1).

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