Number of real solutions of `sqrt(x)+sqrt(x-sqrt(1-x))=1` is (a) 0 (b) 1 (c) 2 (d) infinite

Options

A. `0`
B. `1`
C. `2`
D. infinite

Answer

B

Explanation

`(b)` We have `sqrt(x)+sqrt(x-sqrt(1-x))=1`
`impliessqrt(x-sqrt(1-x))=1-sqrt(x)`
Squaring
`x-sqrt(1-x)=1+x-2sqrt(x)`
`implies2sqrt(x)-sqrt(1-x)=1` ..........`(i)`
`implies (2sqrt(x)-sqrt(1-x))(2sqrt(x)+sqrt(1-x))=(2sqrt(x)+sqrt(1-x))`
`implies4x-(1-x)=2sqrt(x)+sqrt(1-x)`
`implies 2sqrt(x)+sqrt(1-x)=5x-1` ................`(ii)`
Adding `(i)` and `(ii)`,
`4sqrt(x)=5x`
`implies 16x=25x^(2)`
`implies x=0,(16)/(25)`
Clearly `x=0` does not satisfy the equation.
Putting `x=(16)/(25)` in equation
`L.H.S=(4)/(3)+sqrt((16)/(25)-(3)/(5))=(4)/(5)+(1)/(5)=1`
So `x=(16)/(25)` is the only solution.