If `x, y, z in R` are such that they satisfy `x + y + z = 1` and `tan^(-1)x+tan^(-1)y+tan^(-1)z=(pi)/(4)`, then the value of `|x^(3)+y^(3)+z^(3)-3|` is

Options

A. `1.5`
B. 2
C. `2.5`
D. 3

Answer

B

Explanation

Let `tan^(-1)x=alpha rArr x = tan alpha`
`tan^(-1)y=beta rArr y = tan beta`
`tan^(-1)x=gamma rArr z = tan gamma`
`therefore alpha + beta + gamma = (pi)/(4)`
and x + y + z = 1 (Given)
`tan(alpha+beta+gamma)=1`
`rArr (x+y+z-xyz)/(1-xy-yz-zx)=1`
`rArr x+y+z-xyz=1-xy-yz-zx`
`rArr 1-xyz=xy-yz-zx+x+y+z-1=0`
`(because x+y+z-1=0)`
`rArr (x-1)(y-1)(z-1)=0`
If x =1, then y + z = 0
`therefore x^(3)+y^(3)+z^(3)-3=1^(3)+y^(3)-y^(3)-3=-2`