NCERT Mathematics Class 10 Exemplar Ch 1 Real Numbers Part 2

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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.

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