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प्रश्न
If `(x^2 + 1)/x = 3 1/3` and x > 1; find If `x^3 - 1/x^3`
If `(x^2 + 1)/x = 3 1/3 "find" x^3 - 1/x^3`
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उत्तर
Given `(x^2 + 1)/x = 3 1/3`
`(x^2 + 1)/x = 10/3`
`x + 1/x = 10/3`
Squaring on both sides, we get
`(x + 1/x)^2 = (10/3)^2`
`x^2 + 1/x^2 + 2 = 100/9`
`x^2 + 1/x^2 = (100 - 18)/9`
`∴x^2 + 1/x^2 = 82/9`
Also,
`x - 1/x = sqrt((x+1/x)^2 - 4)`
= `sqrt(100/9 - 4)`
= `sqrt(64/9)`
∴ `x - 1/x = 8/3`
Cubing both sides, we get
`(x - 1/x)^3 = (8/3)^3`
`x^3 - 1/x^3 - 3(x - 1/x) = 512/27`
`x^3-1/x^3-3(8/3)=512/27`
`x^3 - 1/x^3 - 8 = 512/27`
`x^3 - 1/x^3 = 512/27 + 8`
`x^3 - 1/x^3 = (512 + 216)/27`
∴ `x^3 - 1/x^3 = 728/27`
Notes
Students should refer to the answer according to the question.
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