CBSE Class 10-Mathematics: Chapter – 1 Real Numbers Part 10 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 9:

Use Euclid՚s Division Lemma to show that the square of any positive integer of the form or for some integer .

Let

Question 10:

Prove that in is not a rational number, if n is not perfect square.

Let

From (i) and (ii) is a common factor of both .

This contradicts the Assumption that are co-prime.

So, our supposition is wrong.

Question 11:

Prove that the difference and quotient of are irrational.

Difference of

which is irrational and quotient is

Question 12:

Show that is divisible by , if n is an odd positive integer.

Let (an odd integer)

which is divisible by

Question 13:

Use Euclid Division Lemma to show that cube of any positive integer is either of the form .