To determine which pair of points must lie in the same quadrant when [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are nonzero integers, let's examine each pair step-by-step:
1. Pair: [tex]\((p, q)\)[/tex] and [tex]\((q, p)\)[/tex]:
- Quadrants are determined by the signs of the coordinates.
- [tex]\((p, q)\)[/tex]: If [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are both positive, [tex]\((p, q)\)[/tex] lies in the first quadrant. If both are negative, [tex]\((p, q)\)[/tex] lies in the third quadrant. If [tex]\(p\)[/tex] is positive and [tex]\(q\)[/tex] is negative, it lies in the fourth quadrant, and if [tex]\(p\)[/tex] is negative and [tex]\(q\)[/tex] is positive, it lies in the second quadrant.
- [tex]\((q, p)\)[/tex]: The roles of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are swapped. Therefore, if [tex]\((p, q)\)[/tex] lies in the first quadrant (both [tex]\(p\)[/tex] and [tex]\(q\)[/tex] positive), [tex]\((q, p)\)[/tex] also lies in the first quadrant. Same pattern for the third quadrant.
- However, if [tex]\((p, q)\)[/tex] lies in the fourth quadrant ([tex]\(p > 0, q < 0\)[/tex]), [tex]\((q, p)\)[/tex] will lie in the second quadrant, and vice versa.
- Therefore, [tex]\((p, q)\)[/tex] and [tex]\((q, p)\)[/tex] are in the same quadrant only when [tex]\(p\)[/tex] and [tex]\(q\)[/tex] both have the same signs (either both positive or both negative).
2. Pair: [tex]\((p, q)\)[/tex] and [tex]\((2p, 2q)\)[/tex]:
- Scaling by a positive constant (here, 2) does not change the signs of the coordinates.
- If [tex]\((p, q)\)[/tex] is in the first quadrant, [tex]\((2p, 2q)\)[/tex] will also be in the first quadrant. Similarly for the other quadrants.
- Hence, [tex]\((p, q)\)[/tex] and [tex]\((2p, 2q)\)[/tex] must always lie in the same quadrant since the coordinate signs remain unchanged.
3. Pair: [tex]\((p, q)\)[/tex] and [tex]\((-p, -q)\)[/tex]:
- Negation of both coordinates changes the signs of [tex]\(p\)[/tex] and [tex]\(q\)[/tex].
- If [tex]\((p, q)\)[/tex] is in the first quadrant, [tex]\((-p, -q)\)[/tex] will be in the third quadrant; if [tex]\((p, q)\)[/tex] is in the second quadrant, [tex]\((-p, -q)\)[/tex] will lie in the fourth quadrant, and so on.
- Therefore, [tex]\((p, q)\)[/tex] and [tex]\((-p, -q)\)[/tex] are always in the opposite quadrants.
4. Pair: [tex]\((p, q)\)[/tex] and [tex]\((p-2, q-2)\)[/tex]:
- Subtraction of constants shifts the coordinates.
- Shifting by [tex]\(-2\)[/tex] along both axes does not guarantee the points will remain in the same quadrant. For instance, if [tex]\((p, q)\)[/tex] is in the first quadrant, [tex]\((p-2, q-2)\)[/tex] could move to a different quadrant depending on the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex].
- Hence, the quadrant of [tex]\((p-2, q-2)\)[/tex] is not guaranteed to be the same as [tex]\((p, q)\)[/tex].
Considering all these points, the pair that must lie in the same quadrant is:
[tex]\((p, q)\)[/tex] and [tex]\((2p, 2q)\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{(p, q) \text{ and } (2p, 2q)}
\][/tex]
