Show that 1 3 - 5 is an irrational number.

Given, 1 3 - 5 Rationalise the denominator and apply the identity a - b a + b = a 2 - b 2 1 3 - 5 = 1 3 - 5 × 3 + 5 3 + 5 = 3 + 5 9 - 5 = 3 + 5 4 Let us assume that 1 3 - 5 is a rational number, then 3 + 5 = r (say), where r is a rational number. ⇒ 3 + 5 = r ⇒ 5 = r - 3 As r is a rational number and r ≠ 0 , so r - 3 is also rational number. ⇒ 5 is a rational number. But this contradicts that 5 is an irrational number. Hence, our assumption is wrong. Therefore, 1 3 - 5 is an irrational number.

In each case state whether the following are rational or irrational. If they are rational and expressed in the form of p q where p , q are coprime integers then what can you say about the prime factors of q ? 279 . 034

Given number is 279 . 034 . This can be represented in p q form as 279 . 034 = 279034 1000 = 139517 500 now the factors of q i.e, 500 = 2 × 2 × 5 × 5 × 5 = 2 2 × 5 3 which is in the form of 2 m × 5 n , where m , n are positive integers. Therefore the given number is rational and the denominator has only 2 and 5 has prime factors.

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of p q where p , q are coprime integers then what can you say about the prime factors of q ? 76 . 17893 ¯

The given number can be written as 76 . 17893 ¯ = 76 . 178931789317893 . . . . . . as it has repeating or recurring decimals so it is rational number let , x = 76 . 1789317893 . . . ( 1 ) 100000 x = 7617893 . 1789317893 . . . . . ( 2 ) On subtracting ( 1 ) from ( 2 ) we get , 99999 x = 7617817 Hence, x = 7617817 99999 = p q and factors of q i.e , 99999 = 3 × 3 × 11111 = 3 2 × 11111 Therefore the given number is rational and the denominator has factors other than 2 and 5 .

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of p q where p, q are co prime integers then what can you say about the factorization of q? 3 . 010010001 . . . .

Numbers which cannot be expressed in the form of p q , where p , q are integers and q ≠ 0 are called irrational numbers. Therefore the given number has neither terminating decimals nor repeating . Hence it is an irrational number.

In each case state whether the following are rational or irrational. If they are rational , expressed in the form of p q where p, q are coprime integers then what can you say about the factorization of q? 39 . 546782

Given number is 39 . 546782 39 . 546782 = 39546782 1000000 = 19973391 500000 where p , q are coprime .. now the factors of q are i.e , 500000 = 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 × 5 × 5 = 2 5 × 5 6 which is in the form 2 m × 5 n , where m , n are positive integers. Therefore the given number is rational and the denominator has only 2 and 5 has prime factors.

In each case state whether the following are rational or irrational. If they are rational , expressed in the form of p q where p , q are co prime integers then what can you say about the factorization of q ? 2 . 3476817681 . . . . .

Numbers which cannot be expressed in the form of p q , where p , q are integers and q ≠ 0 are called irrational numbers. The given number has neither terminating decimals nor repeated. Therefore, it is an irrational number.

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of p q where p , q are co prime integers then what can you say about the factorization of q ? 59 . 120120012000 . . . .

Numbers which cannot be expressed in the form of p q , where p , q are integers and q ≠ 0 are called irrational numbers. The given number has neither terminating decimals nor repeated. Therefore, it is irrational number.

M L Aggarwal Solutions for Exercise 7: Chapter Test

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 7: Chapter Test

Attempt the practice questions from Exercise 7: Chapter Test with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 9 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Exercise 7: Chapter Test with Hints & Solutions

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 8

Show that $\frac{1}{3 - \sqrt{5}}$ is an irrational number.

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 18

If $34.0356$ expressed in the form of $\frac{p}{q},$ where $p$ and $q$ are coprime integers then what can you say about the factorization of $q$ ?

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational and expressed in the form of $\frac{p}{q}$ where $p, q$ are coprime integers then what can you say about the prime factors of $q ?$

$279.034$

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of $\frac{p}{q}$ where $p, q$ are coprime integers then what can you say about the prime factors of $q ?$

$76 . \bar{17893}$

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of $\frac{p}{q}$ where p, q are co prime integers then what can you say about the factorization of q?

$3.010010001 . . . .$

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational , expressed in the form of $\frac{p}{q}$ where p, q are coprime integers then what can you say about the factorization of q?

$39.546782$

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational , expressed in the form of $\frac{p}{q}$ where $p, q$ are co prime integers then what can you say about the factorization of $q ?$

$2.3476817681 . . . . .$

EASY

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 1 - Rational and Irrational Numbers>Chapter Test>Q 19

In each case state whether the following are rational or irrational. If they are rational ,expressed in the form of $\frac{p}{q}$ where $p, q$ are co prime integers then what can you say about the factorization of $q ?$

$59.120120012000 . . . .$

M L Aggarwal Solutions for Exercise 7: Chapter Test

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 7: Chapter Test

Questions from M L Aggarwal Solutions for Exercise 7: Chapter Test with Hints & Solutions

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