CBSE Class 10-Mathematics: Chapter –1 Real Numbers Part 3
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Question 15:
is
(a) An integer
(b) An irrational number
(c) A rational number
(d) None of these
Answer:
(c) A rational number
Question 16:
is
(a) A rational number
(b) An irrational number
(c) Both (a) & (b)
(d) Neither rational nor irrational
Answer:
(b) An irrational number
Question 17:
is
(a) A rational number
(b) An irrational number
(c) An integer
(d) Not real number
Answer:
(b) An irrational number
2 Mark Questions
Question 1:
Show that any positive odd integer is of the form , or , where is some integer.
Answer:
Let be any positive integer and . Then, by Euclid’s algorithm,
for some integer , and because .
Therefore, or or
Also, , where is a positive integer
, where is an integer
, where is an integer
Clearly, are of the form , where is an integer.
Therefore, are not exactly divisible by . Hence, these expressions of numbers are odd numbers.
And therefore, any odd integer can be expressed in the form , or ,
Or
Question 2:
An army contingent of members is to march behind an army band of members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Answer:
We have to find the to find the maximum number of columns in which they can march.
To find the , we can use Euclid’s algorithm.
The is .
Therefore, they can march in 8 columns each.
Question 3:
Use Euclid’s division lemma to show that the cube of any positive integer is of the form .
Answer:
Let a be any positive integer and
, where and
Therefore, every number can be represented as these three forms.
We have three cases.
Case 1: When
Where m is an integer such that
Case 2: When ,
Where is an integer such that
Case 3: When ,
Where m is an integer such that
Therefore, the cube of any positive integer is of the form , or .