Show that ( 3 - 2 ) ( 5 + 2 6 ) 3 + 2 is rational

We have to show that ( 3 - 2 ) ( 5 + 2 6 ) 3 + 2 is rational. We have ( 3 - 2 ) ( 5 + 2 6 ) 3 + 2 = ( 3 - 2 ) ( 3 + 2 × 3 × 2 + 2 ) 3 + 2 = ( 3 - 2 ) 3 2 + 2 × 3 × 2 + 2 2 3 + 2 = ( 3 - 2 ) 3 + 2 2 3 + 2 [Use ( a + b ) 2 = a 2 + 2 a b + b 2 ] = ( 3 - 2 ) ( 3 + 2 ) = 3 2 - 2 2 [Use a + b a - b = a 2 - b 2 ] = 3 - 2 = 1 . We know that a number of the form p q , q ≠ 0 , where p and q are co-prime numbers is called a rational number. Here, we see that the number 1 can be expressed as 1 1 . Hence, 1 is a rational number. Therefore, the given expression ( 3 - 2 ) ( 5 + 2 6 ) 3 + 2 is rational. [proved]

Show that 5 - 5 3 + 5 + 2 5 is rational

We have to show that 5 - 5 3 + 5 + 2 5 is rational. We have 5 - 5 3 + 5 + 2 5 = 5 - 5 ( 3 - 5 ) 3 + 5 ( 3 - 5 ) + 2 5 [Multiply both the numerator and the denominator by ( 3 - 5 ) = 15 - 5 5 - 3 5 + 5 3 2 - 5 2 + 2 5 [Use a + b a - b = a 2 - b 2 ] = 20 - 8 5 9 - 5 + 2 5 = 4 5 - 2 5 4 + 2 5 = 5 - 2 5 + 2 5 = 5 We know that a number of the form p q , q ≠ 0 , where p and q are co-prime numbers is called a rational number. Here, we see that the number 5 can be expressed as 5 1 . Hence, 5 is a rational number. Therefore, the given expression 5 - 5 3 + 5 + 2 5 is rational. [proved]

Show that 1 1 + 2 + 1 2 + 3 + 1 3 + 4 is rational

We have to show that 1 1 + 2 + 1 2 + 3 + 1 3 + 4 is rational. We have 1 1 + 2 + 1 2 + 3 + 1 3 + 4 = 1 2 + 1 + 1 3 + 2 + 1 4 + 3 = ( 2 - 1 ) 2 + 1 ( 2 - 1 ) + ( 3 - 2 ) 3 + 2 ( 3 - 2 ) + ( 4 - 3 ) 4 + 3 ( 4 - 3 ) = ( 2 - 1 ) ( 2 + 1 ) ( 2 - 1 ) + ( 3 - 2 ) ( 3 + 2 ) ( 3 - 2 ) + ( 4 - 3 ) ( 4 + 3 ) ( 4 - 3 ) = 2 - 1 2 2 - 1 2 + 3 - 2 3 2 - 2 2 + 4 - 3 4 2 - 3 2 [Use ( a + b ) ( a - b ) = a 2 - b 2 ] = 2 - 1 2 - 1 + 3 - 2 3 - 2 + 4 - 3 4 - 3 = 2 - 1 1 + 3 - 2 1 + 4 - 3 1 = 2 - 1 + 3 - 2 + 4 - 3 = - 1 + 4 = - 1 + 2 or - 1 - 2 [ ∵ 4 = ± 2 ] = 1 or - 3 We know that a number of the form p q , q ≠ 0 , where p and q are co-prime numbers is called a rational number. Here, we see that the number 1 can be expressed as 1 1 and the number - 3 can be expressed as - 3 1 . Hence, both 1 and - 3 are a rational numbers. Therefore, the given expression 1 1 + 2 + 1 2 + 3 + 1 3 + 4 is rational. [proved]

Show that x 2 - x y + y 2 is rational, where x = 2 - 1 2 + 1 and y = 2 + 1 2 - 1 .

We are given tp show that x 2 - x y + y 2 is rational, where x = 2 - 1 2 + 1 and y = 2 + 1 2 - 1 . We have x 2 - x y + y 2 = x 2 - 2 x y + y 2 + x y = x - y 2 + x y [Use a - b 2 = a 2 - 2 a b + b 2 ] = 2 - 1 2 + 1 - 2 + 1 2 - 1 2 + 2 - 1 2 + 1 × 2 + 1 2 - 1 = 2 - 1 2 - ( 2 + 1 ) 2 2 + 1 ( 2 - 1 ) 2 + 1 = 2 2 - 2 × 2 × 1 + 1 2 - 2 2 + 2 × 2 × 1 + 1 2 2 - 1 2 2 + 1 [Use ( a + b ) ( a - b ) = a 2 - b 2 , ( a - b ) 2 = a 2 - 2 a b + b 2 and ( a + b ) 2 = a 2 + 2 a b + b 2 ] = 2 - 2 2 + 1 - ( 2 + 2 2 + 1 ) 2 - 1 2 + 1 = 2 - 2 2 + 1 - 2 - 2 2 - 1 1 2 + 1 = - 4 2 2 + 1 = 32 + 1 = 33 We know that a number of the form p q , q ≠ 0 , where p and q are co-prime numbers is called a rational number. Here, we see that the number 33 can be expressed as 33 1 . Hence, 33 is a rational number. Therefore, the given expression is rational. [proved]

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Mathematics for Class 9>Chapter 1 - Number System>EXERCISE 1.3

Manipur Board Solutions for Chapter: Number System, Exercise 3: EXERCISE 1.3

Author:Manipur Board

Manipur Board Mathematics Solutions for Exercise - Manipur Board Solutions for Chapter: Number System, Exercise 3: EXERCISE 1.3

Attempt the practice questions on Chapter 1: Number System, Exercise 3: EXERCISE 1.3 with hints and solutions to strengthen your understanding. Mathematics for Class 9 solutions are prepared by Experienced Embibe Experts.