WBBSE Class 9 Maths Arithmetic Chapter 1 Real Numbers Multiple Choice Questions

WBBSE Class 9 Maths Arithmetic Chapter 1 Real Numbers Multiple Choice Questions

Example 1. The decimal expansion of √5 is

1. A terminating decimal
2. A terminating or recurring decimal
3. A non-terminating and non-recurring decimal
4. None of them

Solution:

√5 = 2.236067……….

⇒ Therefore the decimal expansion of √5 is a non-terminating and non-recurring decimal.

∴ So the correct answer is 3. A non-terminating and non-recurring decimal

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The decimal expansion of √5 is a non-terminating and non-recurring decimal.

Example 2. The product of two irrational numbers is

1. Always irrational number
2. Always rational number
3. Always an integer
4. Rational or irrational number

Solution: The product of two irrational number √18 and √2 is √18 x √2 = √36 = 6 [rational number]

⇒ Again √7 x √2 = √14 [irrational number]

⇒ Therefore product of two irrational number is rational or irrational number.

∴ So the correct answer is 4. None of them

The product of two irrational numbers is rational or irrational number

Example 3. π and \(\frac{22}{7}\)

1. Both are rational number
2. Both are always irrational number
3. π is rational and \(\frac{22}{7}\) is irrational
4. π is irrational and \(\frac{22}{7}\) is irrational

Solution: The ratio of perimeter and diameter of each circle is a fixed number and is denoted by π where π= (approx) or 3.14 (approx).

⇒ Therefore the value of π can not be expressed as the ratio of two integers so π is a irrational number.

⇒ As \(\frac{22}{7}\) is a ratio of two integers 22 and 7, therefore \(\frac{22}{7}\) is a rational number.

∴ So the correct answer is 4. π is irrational and \(\frac{22}{7}\) is irrational

π and \(\frac{22}{7}\) is irrational and \(\frac{22}{7}\) is irrational

Example 4. Between two rational numbers, there exist.

1. No rational number
2. Only one rational number
3. Infinite numbers of rational numbers.
4. No irrational number

Solution: If x and y are two rational numbers and x < y, then the rational numbers between x and y are (x + d), (x+2d),……, (x + nd).

⇒ where \(d=\frac{y-x}{x+1}\)

⇒ It is possible to take the value of n as large as we like, the number of rational numbers lying between x and y is infinite.

∴ So the correct answer is 3. Infinite numbers of rational numbers.

Between two rational numbers, infinite numbers of rational numbers exist.

Example 5. Between two irrational numbers, there exists

1. No rational number
2. Only one irrational number
3. Infinite numbers of irrational
4. No irrational number

Solution: There are infinite numbers of irrational numbers between two irrational numbers.

⇒ √2 = 1.4142103………

⇒ √3 = 1.732050807……..

⇒ The irrational numbers between √2 and √3 are 1.41421030030003……, 1.4142126122612226…..

∴ So the correct answer is 3. Infinite numbers of irrational

Between two irrational numbers, there exists Infinite numbers of irrational

Example 6. The number 0 is

1. Whole number but not an integer
2. Integer but not rational
3. Rational but not a real number
4. Whole numbers, integers, rational, and real numbers but not irrational

Solution: 0 = \(\frac{0}{1}\) = \(\frac{0}{2}\) = \(\frac{0}{17}\)

⇒ So 0 is a rational number,

⇒ Again 0 is a whole number and 0 is an integer that is neither positive nor negative.

∴ So the correct answer is 4. Whole numbers, integers, rational, and real numbers but not irrational

The number 0 is Whole numbers, integers, rational, and real numbers but not irrational

Example 7. The sum of two rational numbers is

1. Always rational
2. Always irrational
3. Always integer
4. None of them

Solution: The sum of two rational numbers will always be rational.

⇒ e.g.\(\frac{2}{3}\) + \(\frac{5}{6}\) = \(\frac{9}{6}\), √25 + √16 = \(\frac{9}{1}\)

∴ So the correct answer is 1. Always rational

The sum of two rational numbers is Always rational

Example 8. Which is not an irrational number from the following number?

1. 25
2. √7 – √3
3. 4 + 2√25
4. 9

Solution: 4 + √25 = 4 + 5 = \(\frac{9}{1}\) (rational number)

⇒ √3 x √2 = √6 (irrational number)

⇒ 2√5 (irrational number)

⇒(√7-√3) is a irrational number

∴ So the correct answer is 3. 4 + 2√25

Example 9. Which is not a rational number from the following number?

1. √0.4
2. 3.06
3. \(\sqrt{1 \frac{9}{16}}\)
4. √8

Solution: √0.4 = \(\sqrt{\frac{4}{9}}\) = \(\frac{2}{3}\) (rational number)

⇒ 3.06 = 3\(\frac{6}{90}\) (rational number) = 3\(\frac{1}{5}\) = \(\frac{46}{15}\) (rational number)

⇒ \(\sqrt{1 \frac{9}{16}}=\sqrt{\frac{25}{16}}=\frac{5}{4}\) (rational number)

⇒ √8 = \(\sqrt{4 \times 2}\) = 2√2 (irrational number)

∴ So the correct answer is 4. √8

Example 10. Which of the following number is a recurring decimal?

1. \(\frac{5}{8}\)
2. \(\frac{11}{25}\)
3. \(\frac{13}{80}\)
4. \(\frac{19}{24}\)

Solution: If the rational numbers of the form \(\frac{p}{q}\) where q has the prime factors 2 and 5 only be expressed into decimals, it will be terminating decimal numbers.

⇒ But if the rational numbers of the form be expressed into decimals, it will be recurring decimal numbers, where q has prime factors other than 2 and 5.

1. The denominator of \(\frac{5}{8}\) is 8 and 8 = 2³; 8 has no prime factor except 2.

∴ a terminating decimal number will be found if \(\frac{5}{8}\) be expressed into decimal.

2. The denominator of \(\frac{11}{25}\) is 25 and 25 = 5²; 25 has no prime factor except 5.

∴ a terminating decimal number will be found if \(\frac{11}{25}\) is expressed into decimal.

3. The denominator of \(\frac{13}{80}\) is 80 and 80 = 2⁴ x 5;

⇒ 80 has no prime factor except 2 and 5.

∴ a terminating decimal number will be found if \(\frac{13}{80}\) is expressed into decimal.

4. The denominator of \(\frac{19}{24}\) is 24 and 24 = 2³ x 3;

⇒ 24 has a prime factor 3 other than 2

∴ The decimal form of \(\frac{19}{24}\) will not be terminating. It will be recurring.

∴ So the correct answer is 4. \(\frac{19}{24}\)