If a 2 - 1 a 2 = 5 evaluate a 4 + 1 a 4

Given , a 2 - 1 a 2 = 5 Now , a 2 - 1 a 2 2 = a 4 + 1 a 4 - 2 × a 2 × 1 a 2 ⇒ a 2 - 1 a 2 2 = a 4 + 1 a 4 - 2 ⇒ 5 2 = a 4 + 1 a 4 - 2 ⇒ 25 = a 4 + 1 a 4 - 2 ⇒ a 4 + 1 a 4 = 27

If a + 1 a = p and a - 1 a = q . Find the relation between p and q .

Given, a + 1 a = p & a - 1 a = q ∴ Squaring both sides we get a + 1 a 2 = p 2 ; a - 1 a 2 = q 2 we know that a + 1 a 2 - a - 1 a 2 = p 2 - q 2 ⇒ a 2 + 1 a 2 + 2 a 1 a - a 2 - 1 a 2 + 2 a 1 a = p 2 - q 2 ⇒ p 2 - q 2 = 4 ∴ p 2 - q 2 = 4 is the relation between p , q

If a 2 + 1 a = 4 , find the value of 2 a 3 + 2 a 3

Given, a 2 + 1 a = 4 ⇒ a 2 a + 1 a = 4 ⇒ a + 1 a = 4 Now , Now , 2 a 3 + 2 a 3 = 2 a 3 + 1 a 3 = 2 a + 1 a 3 - 3 a 1 a a + 1 a = 2 4 3 - 3 ( 4 ) = 2 ( 64 - 12 ) = 2 × 52 = 104 ∴ 2 a 3 + 2 a 3 = 104

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

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

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

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 ∴ x 3 + 1 x 3 = 56

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 .