NCERT Class 10 Chapter 1 Real Numbers Official CBSE Board Sample Problems Short Answer

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Question

Find the HCF of 392 and 732 using Euclid’s Division lemma.

Solution

By Euclid’s Division Algorithm we get

Question

Can the HCF and LCM of two numbers be 27 and 288?

Solution

HCF is always a factor of the LCM but we observe that 27 is not a factor of 288. So these are not the I-KF and LCM of two numbers

Question

Show that is an irrational number.

Solution

We will do a proof by contradiction Assume is rational then it can be expressed as a ratio, a fraction

Since p and q are positive integers, p-3q is also a positive integer. Let’s call it r for simplicity.

LHS is irrational, but RHS is rational so LHS is not equal to RHS.

But it leads to a contradiction . Therefore is an irrational number.

Question

Prove that

is irrational.

Solution

Let be rational,

where a and b are integers which are coprime and .

and 2b are integers which implies that is a rational number.

Which implies that is also rational.

This contradicts the fact that is an irrational number, hence our assumption is wrong.

Thus is an irrational.

Question

Show that every positive even integer is of the form 2m and that every positive odd integer is of the form, where m is some integer.

Solution

Let ‘a’ be any positive integer and

By Euclid’s division algorithm

(i.e.)

(where )

(where )

If a is the form of 2m then ‘a’ is an even integer and positive odd integer is of the form

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