6 DS answer explanation needed / GMATPrep Test 1

1st Question:

(1) 2x - 2y = 1 implies x - y = 1/2
Now, if x = 1 and y = 1/2, then x - y = 1/2. Here, x and y are both positive.
If x = 0 and y = -1/2, then x - y = 0 - (-1/2) = 1/2. Here, x and y are not positive.
We don't get a unique answer.
So, (1) is NOT SUFFICIENT.

(2) x/y > 1 implies that either x and y are both positive or x and y are both negative. But this is NOT SUFFICIENT to answer the question again.

Combining (1) and (2), if x = 1 and y = 1/2, then x - y = 1/2. Also x/y > 1 here. If we take negative values for both x and y, then x - y = 1/2 is not satisfied. Hence, x and y can never be negative.
So, answer to the main question is yes.

The correct answer is [spoiler](C)[/spoiler].

2nd question:

(1) QR = RS implies angle RQS = angle RSQ = y, say.
But, (1) is NOT SUFFICIENT to find x.

(2) ST = TU implies angle TSU = angle SUT = z, say.
But, (2) is NOT SUFFICIENT to find x.

Combining (1) and (2), x + y + z = 180º
Also, angle PRT + angle PTR = 90º (because angle RPT = 90º)

Also, angle PRT = 180 - 2y and angle PTR = 180 - 2z
So, 180 - 2y + 180 - 2z = 90
or y + z = 135, implies x = 180 - 135 = 45º

The correct answer is [spoiler](C)[/spoiler].

3rd question:

(1) If 8k = 16 then k = 2 and k is not divisible by 4.
If 8k = 32 then k = 4, divisible by 4.
We don't get a unique answer.
So, (1) is NOT SUFFICIENT.

(2) If 9k = 12 then k is not an integer.
If 9k = 24 then also k is not an integer.
If 9k = 36 then k = 4, divisible by 4
If 9k = 48 then also k is not an integer.
If 9k = 60 then also k is not an integer.
If 9k = 72 then k = 8, divisible by 4.

Generalizing, we can say k is divisible by 4 if 9k is divisible by 12.

The correct answer is [spoiler](B)[/spoiler].