NCERT Mathematics Class 10 Exemplar Ch 1 Real Numbers Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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EXERCISE 1.2

1. Write whether every positive integer can be of the form , where q is an integer. Justify your answer.

Answer: No, because an integer can be written in the form , , , .

2. “The product of two consecutive positive integers is divisible by 2” . Is this statement true or false? Give reasons.

Answer: True, because will always be even, as one out of or must be even.

3. ″ The product of three consecutive positive integers is divisible by 6 ″ . Is this statement true or false ″ ? Justify your answer.

Answer: True, because will always be divisible by 6, as at least one of the factors will be divisible by 2 and at least one of the factors will be divisible by 3.

4. Write whether the square of any positive integer can be of the form , where m is a natural number. Justify your answer.

Answer: No. Since any positive integer can be written as , , , therefore, square will be , , .

5. A positive integer is of the form , q being a natural number. Can you write its square in any form other than , i.e.. , 3m or for some integer m? Justify your answer.

Answer: No. .

6. The numbers 525 and 3000 are both divisible only by . What is ? Justify your answer.

Answer: HCF = 75, as HCF is the highest common factor.

7. Explain why is a composite number.

Answer: which has more than two factors?

8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

Answer: No, because HCF (18) does not divide LCM (380) .

9. Without actually performing the long division, find if 987 10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

Answer: Terminating decimal expansion, because

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10. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form ? Give reasons.

Answer: Since 327.7081 is a terminating decimal number, so q must be of the form ; m, n are natural numbers.