Let's analyze and solve each part of the question:

Given Information:

• $p$ and $q$ are integers.
• The range for $p$ is $2 \leq p \leq 9$.
• The range for $q$ is $-5 \leq q < 3$.

Part (a)

Find the greatest possible value of $q - p$:

To maximize $q - p$, we want $q$ to be as large as possible and $p$ to be as small as possible, as this will maximize the difference $q - p$.

• The maximum value for $q$ is $2$.
• The minimum value for $p$ is $2$.

So, $q - p = 2 - 2 = 0$

Therefore, the greatest possible value of $q - p$ is 0.

Part (b)

Find the smallest possible value of $p^2 + q^2$:

To minimize $p^2 + q^2$, we need to choose values of $p$ and $q$ that make $p^2$ and $q^2$ as small as possible, as the expression $p^2 + q^2$ involves the squares of $p$ and $q$.

• The smallest value for $p$ within its range that gives a minimum square is $p = 2$, so $p^2 = 4$.
• The smallest value for $q$ within its range that gives a minimum square is $q = 0$, so $q^2 = 0$.

Thus, $p^2 + q^2 = 4 + 0 = 4$

Therefore, the smallest possible value of $p^2 + q^2$ is 4.

Summary of Answers:

• (a) The greatest possible value of $q - p$ is $0$.
• (b) The smallest possible value of $p^2 + q^2$ is $4$.

Would you like further details or have any questions?

Here are some related questions to deepen understanding:

1. What would happen to $q - p$ if the range of $q$ was extended beyond $3$?
2. How would the minimum value of $p^2 + q^2$ change if $p$ or $q$ had additional constraints?
3. What are the effects of expanding the range of $p$ on the smallest possible value of $p^2 + q^2$?
4. Can we have a negative value for $q - p$ given the current range of $p$ and $q$?
5. How would the calculations change if $p$ and $q$ were not restricted to integers?

Tip: When maximizing or minimizing an expression involving multiple variables, consider the extremes of each variable's range.