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If x=263+2, then what is the value of x+2x-2+x+3x-3? -

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Question

If `x = (2sqrt(6))/(sqrt(3) + sqrt(2))`, then what is the value of `(x + sqrt(2))/(x - sqrt(2)) + (x + sqrt(3))/(x - sqrt(3))`?

Options

  • `sqrt(2)`

  • `sqrt(3)`

  • `sqrt(6)`

  • 2

MCQ

Solution

2

Explanation:

`x = (2sqrt(3) xx sqrt(2))/(sqrt(3) + sqrt(2))`

⇒ `x/sqrt(2) = (2sqrt(3))/(sqrt(3) + sqrt(2))`

⇒ `(x + sqrt(2))/(x - sqrt(2)) = (2sqrt(3) + sqrt(3) + sqrt(2))/(2sqrt(3) - sqrt(3) - sqrt(2))`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))`

By componendo and dividendo Similarly,

`x/sqrt(3) = (2sqrt(2))/(sqrt(3) + sqrt(2))`

⇒ `(x + sqrt(3))/(x - sqrt(3)) = (2sqrt(2) + sqrt(3) + sqrt(2))/(2sqrt(2) - sqrt(3) - sqrt(2))`

= `(sqrt(3) + 3sqrt(2))/(sqrt(2) - sqrt(3))`

∴ Expression = `(x + sqrt(2))/(x - sqrt(2)) + (x + sqrt(3))/(x - sqrt(3))`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (sqrt(3) + 3sqrt(2))/(sqrt(2) - sqrt(3))`

= `(3sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2)) + (sqrt(3) + 3sqrt(2))/(sqrt(3) - sqrt(2))`

= `(3sqrt(3) + sqrt(2) - sqrt(3) - 3sqrt(2))/(sqrt(3) - sqrt(2))`

= `(2(sqrt(3) - sqrt(2)))/(sqrt(3) - sqrt(2))`

= 2

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Algebra (Entrance Exam)
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