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

(c) A rational number

Question 16:

is

(a) A rational number

(b) An irrational number

(c) Both (a) & (b)

(d) Neither rational nor irrational

(b) An irrational number

Question 17:

is

(a) A rational number

(b) An irrational number

(c) An integer

(d) Not real number

(b) An irrational number

2 Mark Questions

Question 1:

Show that any positive odd integer is of the form , or , where is some integer.

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?

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 .

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 .