Quiz - 1 Class - 9 Chapter - Number System

1. 0.1 can be represented by

1/10

1/100

1/1000

1/10000

The decimal number 0.1 can be represented as 1/10 because the decimal point separates the whole number part from the fractional part. In this case, the fractional part is 1/10, which means that there is one tenth of a whole.

Explanation

The decimal number 0.1 can be represented as 1/10 because the decimal point separates the whole number part from the fractional part. In this case, the fractional part is 1/10, which means that there is one tenth of a whole.

2. The value of 0.232323..... + 0.222222........ is

0.45454545.....

0.43434343

0.45

0.43

The given question asks for the sum of two repeating decimals, 0.232323..... and 0.222222..... To find the sum, we can add the corresponding digits of each decimal place. Starting from the rightmost digit, we add 3 and 2, which gives us 5. Moving to the next decimal place, we add 2 and 2, which gives us 4. Continuing this pattern, we find that the sum of the two decimals is 0.45454545..... Therefore, the correct answer is 0.45454545.....

Explanation

The given question asks for the sum of two repeating decimals, 0.232323..... and 0.222222..... To find the sum, we can add the corresponding digits of each decimal place. Starting from the rightmost digit, we add 3 and 2, which gives us 5. Moving to the next decimal place, we add 2 and 2, which gives us 4. Continuing this pattern, we find that the sum of the two decimals is 0.45454545..... Therefore, the correct answer is 0.45454545.....

3. Which of these is greatest number?

0.1

0.01

0.001

0.0001

The given numbers are all decimal numbers, and to determine the greatest number, we need to compare their values. Among the given options, 0.1 is the largest number because it has the largest place value. The number 0.1 is greater than 0.01, 0.001, and 0.0001 as it has a larger digit in the tenths place compared to the other numbers.

Explanation

The given numbers are all decimal numbers, and to determine the greatest number, we need to compare their values. Among the given options, 0.1 is the largest number because it has the largest place value. The number 0.1 is greater than 0.01, 0.001, and 0.0001 as it has a larger digit in the tenths place compared to the other numbers.

4. Which of the following is irrational ?

0.15

0.01516

0.151615161516.....

0.50155001500015....

The number 0.50155001500015.... is irrational because it is a non-repeating, non-terminating decimal. Irrational numbers cannot be expressed as a fraction and have an infinite number of non-repeating decimal places.

Explanation

The number 0.50155001500015.... is irrational because it is a non-repeating, non-terminating decimal. Irrational numbers cannot be expressed as a fraction and have an infinite number of non-repeating decimal places.

5. Which one of the following is a correct statement?

Decimal expansion of a rational number is terminating.

Decimal expansion of a rational number is non-terminating.

Decimal expansion of an irrational number is terminating.

Decimal expansion of an irrational number is non- terminating and non- repeating.

The decimal expansion of an irrational number is non-terminating and non-repeating because irrational numbers cannot be expressed as a fraction or a ratio of two integers. Therefore, their decimal representation continues indefinitely without any pattern or repetition. Examples of irrational numbers include √2, π, and e.

Explanation

The decimal expansion of an irrational number is non-terminating and non-repeating because irrational numbers cannot be expressed as a fraction or a ratio of two integers. Therefore, their decimal representation continues indefinitely without any pattern or repetition. Examples of irrational numbers include √2, π, and e.

6. Every point on a number line represents

A unique real number

A natural number

A rational number

An irrational number

Every point on a number line corresponds to a specific position or value, and since real numbers include both rational and irrational numbers, each point on the number line represents a unique real number. Real numbers encompass all possible values on the number line, making this the correct answer.

Explanation

Every point on a number line corresponds to a specific position or value, and since real numbers include both rational and irrational numbers, each point on the number line represents a unique real number. Real numbers encompass all possible values on the number line, making this the correct answer.

7. The number 1.272727...... in the form p/q, where p and q are integers and q is not zero, is

14/9

14/11

14/13

14/15

The number 1.272727... can be expressed as a repeating decimal. To convert it into a fraction, we can let x = 1.272727... and subtracting it from 100x gives us 100x - x = 127.272727... - 1.272727... = 126. Thus, 99x = 126 and dividing both sides by 99 gives us x = 14/11. Therefore, the fraction representation of 1.272727... is 14/11.

Explanation

The number 1.272727... can be expressed as a repeating decimal. To convert it into a fraction, we can let x = 1.272727... and subtracting it from 100x gives us 100x - x = 127.272727... - 1.272727... = 126. Thus, 99x = 126 and dividing both sides by 99 gives us x = 14/11. Therefore, the fraction representation of 1.272727... is 14/11.

8. if n is a natural number then, √n is always

Always a natural number

Always an irrational number

Sometimes a natural number and sometimes an irrational number

The square root of a natural number n can be either a natural number or an irrational number. For example, the square root of 4 is 2, which is a natural number. However, the square root of 2 is an irrational number because it cannot be expressed as a fraction. Therefore, the statement "sometimes a natural number and sometimes an irrational number" is correct.

Explanation

The square root of a natural number n can be either a natural number or an irrational number. For example, the square root of 4 is 2, which is a natural number. However, the square root of 2 is an irrational number because it cannot be expressed as a fraction. Therefore, the statement "sometimes a natural number and sometimes an irrational number" is correct.

9. Which of the following numbers can be represented as non-terminating, repeating decimals?

39/24

3/16

3/11

137/25

The number 3/11 can be represented as a non-terminating, repeating decimal because when we divide 3 by 11, the result is 0.272727... The decimal part, 0.27, repeats infinitely. Therefore, 3/11 can be represented as a non-terminating, repeating decimal.

Explanation

The number 3/11 can be represented as a non-terminating, repeating decimal because when we divide 3 by 11, the result is 0.272727... The decimal part, 0.27, repeats infinitely. Therefore, 3/11 can be represented as a non-terminating, repeating decimal.

10. The number 0.333333...... in the form p/q, where p and q are integrals and q is not zero, is

33/100

3/10

1/3

3/100

The number 0.333333... can be expressed as a fraction in the form p/q, where p and q are integers and q is not zero. In this case, the repeating decimal 0.333333... can be written as 1/3. This is because the decimal representation of 1/3 is 0.333333..., where the digit 3 repeats indefinitely. Therefore, the correct answer is 1/3.

Explanation

The number 0.333333... can be expressed as a fraction in the form p/q, where p and q are integers and q is not zero. In this case, the repeating decimal 0.333333... can be written as 1/3. This is because the decimal representation of 1/3 is 0.333333..., where the digit 3 repeats indefinitely. Therefore, the correct answer is 1/3.

11. Which one of the following statements is true?

The sum of two irrational numbers is always an irrational number.

The sum of two irrational numbers is always a rational number.

The sum of two irrational numbers may be a rational number or an irrational number.

The sum of two irrational numbers is always an integer.

The statement "The sum of two irrational numbers may be a rational number or an irrational number" is true because when two irrational numbers are added together, the result can either be a rational number or an irrational number. This is because the irrational numbers can have infinite non-repeating decimal places, and when added together, the decimal places may either cancel out to form a rational number or continue to be non-repeating to form an irrational number.

Explanation

The statement "The sum of two irrational numbers may be a rational number or an irrational number" is true because when two irrational numbers are added together, the result can either be a rational number or an irrational number. This is because the irrational numbers can have infinite non-repeating decimal places, and when added together, the decimal places may either cancel out to form a rational number or continue to be non-repeating to form an irrational number.

12. Which of the following is a correct statement ?

Product of two irrational numbers is always irrational.

Product of a rational and an irrational number is a always irrational.

Sum of two irrational numbers can never be irrational.

Sum of an integer and a rational number can never be an integer.

Multiplying a rational number (other than zero) by an irrational number results in an irrational number. This occurs because if the product were rational, then you could express the irrational number as the ratio of two rational numbers (the product and the reciprocal of the rational number), which contradicts the definition of irrational numbers. Thus, this multiplication preserves the irrational nature of the number.

Explanation

Multiplying a rational number (other than zero) by an irrational number results in an irrational number. This occurs because if the product were rational, then you could express the irrational number as the ratio of two rational numbers (the product and the reciprocal of the rational number), which contradicts the definition of irrational numbers. Thus, this multiplication preserves the irrational nature of the number.

13. The smallest rational number by which 1/3 should be multiplied so that its decimal expansion terminates after one place of decimal, is

1/10

3/10

3

30

To find the smallest rational number by which 1/3 should be multiplied so that its decimal expansion terminates after one place of decimal, we need to convert 1/3 into a decimal. When we divide 1 by 3, we get 0.3333... which is a non-terminating decimal. To make it terminate after one place of decimal, we need to multiply 1/3 by 10. This gives us 10/3 which is equal to 3 1/3. However, the question asks for the smallest rational number, so we simplify 10/3 to 3/10, which is the correct answer.

Explanation

To find the smallest rational number by which 1/3 should be multiplied so that its decimal expansion terminates after one place of decimal, we need to convert 1/3 into a decimal. When we divide 1 by 3, we get 0.3333... which is a non-terminating decimal. To make it terminate after one place of decimal, we need to multiply 1/3 by 10. This gives us 10/3 which is equal to 3 1/3. However, the question asks for the smallest rational number, so we simplify 10/3 to 3/10, which is the correct answer.

14. 0.32222222...... when expressed in the form p/q( where p and q are integers and q is not zero)

8/25

29/90

32/99

32/199

The given decimal number, 0.32222222......, can be expressed as a fraction in the form p/q. To find the fraction, we can let x = 0.32222222...... and subtract it from 10x. This gives us 10x - x = 9x, and 9x = 3.2222222...... Subtracting x from this equation, we get 9x - x = 3.2222222...... - 0.32222222......, which simplifies to 8x = 3. Subtracting x from both sides, we have 8x - x = 3 - 0.32222222......, which simplifies to 7x = 2.6777777...... Dividing both sides by 7, we get x = 2.6777777...... / 7. Simplifying this fraction, we have x = 0.3825396825......, which can be written as the fraction 29/90. Therefore, the correct answer is 29/90.

Explanation

The given decimal number, 0.32222222......, can be expressed as a fraction in the form p/q. To find the fraction, we can let x = 0.32222222...... and subtract it from 10x. This gives us 10x - x = 9x, and 9x = 3.2222222...... Subtracting x from this equation, we get 9x - x = 3.2222222...... - 0.32222222......, which simplifies to 8x = 3. Subtracting x from both sides, we have 8x - x = 3 - 0.32222222......, which simplifies to 7x = 2.6777777...... Dividing both sides by 7, we get x = 2.6777777...... / 7. Simplifying this fraction, we have x = 0.3825396825......, which can be written as the fraction 29/90. Therefore, the correct answer is 29/90.

15. The number 0.318564318564318564.......... is

A natural number

An integer

A rational number

An irrational number

The number 0.318564318564318564... is a rational number because it can be expressed as a fraction of two integers. The pattern of repeating digits indicates that the number can be written as 318564/999999, which is a ratio of two integers and therefore a rational number.

Explanation

The number 0.318564318564318564... is a rational number because it can be expressed as a fraction of two integers. The pattern of repeating digits indicates that the number can be written as 318564/999999, which is a ratio of two integers and therefore a rational number.