If `sqrt(2) = 1.414, sqrt(3) = 1.732, sqrt(5) = 2.236 and sqrt(6) = 2.449`, find the value of `(2+sqrt(3))/(2-sqrt(3)) +(2-sqrt(3))/(2+sqrt(3)) +(sqrt(3) -1)/(sqrt(3) +1)`

Options

A. `13.298`
B. `14.268`
C. `16.258`
D. `12.268`

Answer

B

Explanation

We have
`(2+ sqrt(3))/(2-sqrt(3)) + (2-sqrt(3))/(2+sqrt(3)) + (sqrt(3) -1)/(sqrt(3) + 1)`
`=((2 + sqrt(3)))/((2- sqrt(3))) xx((2 + sqrt(3)))/((2+ sqrt(3))) + ((2- sqrt(3)))/((2+ sqrt(3))) xx (( 2- sqrt(3)))/( (2-sqrt(3))) + ((sqrt(3) -1))/((sqrt(3) + 1)) xx(sqrt(3) -1)/(sqrt(3) -1)`
`= ((2+ sqrt(3))^(2))/((2)^(2) - (sqrt(3))^(2))+ ((2 - sqrt(3))^(2))/((2)^(2) - (sqrt(3))^(2))+((sqrt(3) -1)^(2))/((sqrt(3))^(2) -(1)^(2))`
`=(2^(2) + (sqrt(3))^(2) + 2xx 2 xx sqrt(3))/((4-3)) + (2^(2) + (sqrt(3))^(2) - 2 xx 2xx sqrt(3))/((4-3))+ ((sqrt(3))^(2) +(1)^(2) - 2xx sqrt(3) xx1)/((3-1))`
` = ( 4+ 3 + 4 sqrt(3))+( 4+ 3-4sqrt(3))+(3+1-2sqrt(3))/(2)`
`= 7 + 4 sqrt(3) + 7 - 4sqrt(3) +(2(2-sqrt(3)))/(2) = 14 + 2 - sqrt(3) = 16 - sqrt(3)`
`= 16-1.732 = 14.268`