For your 1st qn, statement II is alone sufficient. Check the attachment.

3rd qn., Are x and y both positive?

A. 2(x-y) = 1
x-y = 1/2. x and y can be +ve or -ve. Try plugging in values to check this. Insufficient.

B. x/y > 1
or x > y
You don't whether x is +ve,-ve or y is +ve,-ve. Insufficient.

Combined,
you know x > y from B. To satisfy x - y = 1/2, x and y have to be +ve. You could even try out some scenarios to confirm.
Sufficient. C

4th qn., root(x^2)/x = (x^2)^1/2 / x = x/x = 1

papgust wrote:4th qn., root(x^2)/x = (x^2)^1/2 / x = x/x = 1

imo its E

sqrt(x^2)=+-x=lxl

hence rut(x^2)/x=lxl/x

papgust wrote:3rd qn., Are x and y both positive?

B. x/y > 1
or x > y

for a variable crossmultiplication is not correct ( as the signs are not)

B is insuff as x/y will be > 1 if x > y>0 or if x<y<0

hence bth can be -ve or +ve

combinng

x-y=1/2
x/y>1--->x/y - 1>0---->(x-y)/y>0---->1/2y >0----> y>0----> x>0

hence bth +ve and hence C

thephoenix wrote:

papgust wrote:4th qn., root(x^2)/x = (x^2)^1/2 / x = x/x = 1

imo its E

sqrt(x^2)=+-x=lxl

hence rut(x^2)/x=lxl/x

exactly, imo E.

Since we don't know whether x has positive or negative sign, example.. plug 4 and -4 then u can see that results will be different.