Question

The number of positive integral solutions of the equation ∣∣∣∣∣x3+1xy2xz2x2yy3+1yz2x2zy2zz3+1∣∣∣∣∣=11⏐x3+1x2yx2zxy2y3+1y2zxz2yz2z3+1⏐=11​

Answer 1

Answer:

Given ⇒

x² + 1/x² = 11

Now, Using the Formula,

(a - 1/a)² = a² + 1/a² - 2

∴ (x - 1/x)² = x² + 1/x² - 2

∴ (x - 1/x)² = 11 - 2

∴ x - 1/x = √9

⇒ x - 1/x = 3

Now,

Using the Formula,

(a - 1/a)³ = a³ - 1/a³ - 3(a - 1/a)

∴ (x - 1/x)³ = x³- 1/x³ - 3(x - 1/x)

∴ (3)³ = x³ - 1/x³ - 3(3)

⇒ 27 = x³ - 1/x³ - 9

⇒ x³ - 1/x³ = 27 + 9

⇒ x³ - 1/x³ = 36

Hence, the value of the x³ - 1/x³ is 36.

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