If `S={x in N :2+(log)_2sqrt(x+1)>1-(log)_(1/2)sqrt(4-x^2)}` , then `(a)S={1}` (b) `S=Z` (c) `S=N` (d) none of these

Options

A. `S = {1}`
B. S = Z
C. S = N
D. none of these

Answer

A

Explanation

` 2+log_(2) sqrt(x+1) gt 1 - log _(1//2 ) sqrt(4- x^(2))_`
` rArr 1+ log_(2) sqrt (x+ 1 ) - log_(2) sqrt(4- x^(2) ) gt 0`
` or log_(2)2+log_(2) sqrt(x+1)- log_(2) sqrt(4-x^(2)) gt0`
` or log_(2).(2sqrt(x+1))/(sqrt(4-x^(2)))gt 0 or (2sqrt(x+1))/sqrt(4-x^(2)) gt 1`
` :. 4(x+1) gt 4 - x^(2)`
` :. x^(2) + 4x gt 0 `
` x lt -4 or x gt 0` ...(i)
Also, ` x + 1 gt 0 or x gt - 1`
` and 4 - x^(2) gt 0`
` or x in (-2, 2)`
` :. x in (0, 2)`
` rArr x = 1 ," as " x in N `