If x = 1 4 - x find the value of x 6 + 1 x 6

Given, x = 1 4 - x ⇒ ( 4 - x ) x = 1 ⇒ 4 x - x 2 = 1 dividing both sides by x 4 - x = 1 x ⇒ x + 1 x = 4 Now x 3 + 1 x 3 = x + 1 x 3 - 3 x + 1 x = 4 3 - 3 × 4 = 56 Again, x 6 + 1 x 6 = x 3 + 1 x 3 2 - 2 = 56 2 - 2 = 2704 - 2 = 2702 ∴ x 6 + 1 x 6 = x 3 + 1 x 3 2 - 2

If x - 1 x = 3 + 2 2 , find the value of 1 4 x 3 - 1 x 3 .

Given that, x - 1 x = 3 + 2 2 Now, x 3 - 1 x 3 = x - 1 x 3 + 3 x - 1 x = 3 + 2 2 3 + 3 3 + 2 2 = 3 3 + 2 2 3 + 3 × 3 × 2 2 ( 3 + 2 2 ) + 3 3 + 2 2 = 27 + 16 2 + 18 2 3 + 2 2 + 9 + 6 2 = 27 + 16 2 + 54 2 + 72 + 9 + 6 2 = 108 + 76 2 Therefore, 1 4 x 3 - 1 x 3 = 1 4 108 + 76 2 = 27 + 19 2 Hence, the value of 1 4 x 3 - 1 x 3 is 27 + 19 2 .

If x + 1 x = 3 1 3 , f i n d t h e v a l u e o f x 3 - 1 x 3

We know that , x - 1 x 2 = x 2 + 1 x 2 - 2 = x 2 + 1 x 2 + 2 - 4 = x + 1 x 2 - 4 G i v e n x + 1 x = 3 1 3 S o x - 1 x 2 = 3 1 3 2 - 4 = 10 3 2 - 4 = 100 - 36 9 = 64 9 ⇒ x - 1 x = ± 64 9 = ± 8 3 N o w x 3 - 1 x 3 = x - 1 x 3 + 3 × x × 1 x x - 1 x = ± 8 3 3 + 3 ± 8 3 = ± 512 27 + 8 = ± 728 27 = ± 26 26 27

If x = 2 - 3 , f i n d t h e v a l u e o f x 3 - 1 x 3

Given , x = 2 - 3 ⇒ 1 x = 1 2 - 3 = 2 + 3 2 - 3 2 + 3 = 2 + 3 2 2 - 3 2 = 2 + 3 4 - 3 = 2 + 3 Therefore , x - 1 x = 2 - 3 - 2 + 3 = - 2 3 c u b i n g b o t h s i d e s , w e g e t x - 1 x 3 = - 2 3 3 x 3 - 1 x 3 - 3 x 1 x x - 1 x = - 24 3 x 3 - 1 x 3 - 3 - 2 3 = - 24 3 x 3 - 1 x 3 = - 24 3 - 6 3 x 3 - 1 x 3 = - 30 3

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Understanding ICSE Mathematics Class 9>Algebra>Chapter 3 - Expansions>Chapter Test

M L Aggarwal Solutions for Exercise 4: Chapter Test

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 4: Chapter Test

Attempt the practice questions from Exercise 4: Chapter Test with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 9 solutions are prepared by Experienced Embibe Experts.