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: