CBSE Class 10-Mathematics: Chapter – 1 Real Numbers Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 10:

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form , what can you say about the prime factors of ?

(i)

(ii)

(iii)

Answer:

(iii)

It is rational because decimal expansion is terminating. Therefore, it can be expressed is form where factors of are of the form where and are non-negative integers.

(ii)

It is irrational because decimal expansion is neither terminating nor non-terminating repeating.

(iii)

It is rational because decimal expansion is non-terminating repeating. Therefore, it can be expressed in form where factors of q are not of the form where and are non-negative integers.

Question 11:

Show that every positive even integer is of the from and that every positive odd integer is of the form for some integer .

Answer:

Let

If (which is even)

If (which is odd)

So, every positive even integer is of the form and odd integer is of the form .

Question 12:

Show that any number of the form can never end with the digit 0.

Answer:

It does not contain . So can never end with the digit 0

Question 13:

Use Euclid՚s Division Algorithm to find the HCF of 4052 and 12576.

Answer:

HCF of 12576 and 4052 is ‘4’ .

Questions 14:

Given that HCF of two numbers is and their LCM is . If one of the numbers is , find the other.

Answer:

Other number is

Question 15:

Show that every positive odd integer is of the form for some integer .

Answer:

Let

Question 16:

Show that any number of the form can never end with the digit .

Answer: