If p and q are nonzero integers, which pair of points must lie in the same quadrant?
a. (p, q) and (q, p)
b. (p, q) and (2p, 2q)
c. (p, q) and (–p, –q)
d. (p, q) and (p – 2, q – 2)

 The correct option is (B).  (p, q) and (2p, 2q).

Step-by-step explanation:  Given that p and q are non-zero integers.

We are to select the correct pair of points that must lie in the same quadrant.

FIRST PAIR :   (p, q) and (q, p)

Let, (p, q) = (2, -3), then (q, p) = (-3, 2).

Therefore, the point (p, q) lies in Quadrant IV but the point (q, p) lies in Quadrant II.

This implies that the two points does not lie in the same quadrant.

Thus, this option is not correct.

SECOND PAIR :   (p, q) and (2p, 2q)

We see that, if p is positive, then 2p will also be positive and if p is negative, then 2p will also be negative.

Similar is the case of q.

Therefore, the point (p, q) and the point (2p, 2q) lies in the same quadrant.

For example, if (p, q) = (1, 3), then (2p, 2q) = (2, 6). Both (1, 3) and (2, 6) lie in Quadrant I.

Thus, this option is CORRECT.

THIRD PAIR :  (p, q) and (-p, -q)

Let, (p, q) = (2, 3), then (-p, -q) = (-2, -3).

Therefore, the point (p, q) lies in Quadrant I but the point (-p, -q) lies in Quadrant III.

This implies that the two points does not lie in the same quadrant.

Thus, this option is not correct.

FOURTH PAIR :  (p, q) and (p-2, q-2)

Let, (p, q) = (1, -3), then (p-2, q-2) = (-1, -5).

Therefore, the point (p, q) lies in Quadrant IV but the point (p-2, q-2) lies in Quadrant III.

This implies that the two points does not lie in the same quadrant.

Thus, this option is not correct.

Hence, the correct pair of points is  (p, q) and (2p, 2q).

Option (B) is CORRECT.

i'm pretty sure it's 477,585,296, but perhaps my math is incorrect.