# Extra Questions For Class 10 Maths Chapter 1 Real Numbers Based On Irrationality of Numbers

Extra Questions On Irrationality of Numbers

Question: Prove that √5 is an irrational number.

Solution: Let √5 is a rational number then we have

√5=p/q, where p and q are co-primes.

⇒ p =√5q

Squaring both sides, we get

p2=5q2

⇒ p2 is divisible by 5

⇒ p is also divisible by 5

So, assume p = 5m where m is any integer.

Squaring both sides, we get p2 = 25m2

But p2 = 5q2

Therefore, 5q2 = 25m2

⇒ q2 = 5m2

⇒ q2 is divisible by 5

⇒ q is also divisible by 5

From above we conclude that p and q has one common factor i.e. 5 which contradicts that p and q are co-primes.

Therefore our assumption is wrong.

Hence, √5 is an irrational number.

Q.1. Prove that √2 is an irrational number.

Q.2. Prove that √3 is an irrational number.

Q.3. Prove that 2 + 5√3 is an irrational number.

Q.4. Prove that 3- 2√5 is an irrational number.

Q.5. Prove that √2 + √3 is an irrational number.

Q.6. Prove that √3 + √5 is an irrational number.

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