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RD Sharma Solutions Class 10 Mathematics Solutions for Real Numbers Exercise 1.5 in Chapter 1 - Real Numbers

Question 7 Real Numbers Exercise 1.5

Prove that 2 − 3√5 is an irrational number

Answer:

Let’s assume on the contrary that 2 - 3√5 is a rational number. Then, there exist co-prime

positive integers a and b such that 2 - 3√5 = a/b

⇒ 3√5 = 2 - a/b

⇒ √5 = (2b – a)/(3b)

⇒ √5 is rational [∵ 3, a and b are integers ∴ (2b - a)/b is a rational number] This contradicts the fact that √5 is irrational. So, our assumption is incorrect.

Hence, 2 - 3√5 is an irrational number

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