The mid-point of the line segment joining (4p, 5) and (2, 3q) is (5, 5p - 1). The values of p and q are respectively:

1. 2, $\dfrac{8}{3}$

2. -2, $\dfrac{8}{3}$

3. 2, $\dfrac{13}{3}$

4. -2, $\dfrac{13}{3}$

Given,

Mid-point of the line segment joining (4p, 5) and (2, 3q) is (5, 5p - 1).

By mid-point formula,

(x, y) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get :

$\Rightarrow (5, 5p - 1) = \Big(\dfrac{4p + 2}{2}, \dfrac{5 + 3q}{2}\Big) \\[1em] \Rightarrow (5, 5p - 1) = \Big(\dfrac{2(2p + 1)}{2}, \dfrac{5 + 3q}{2}\Big) \\[1em] \Rightarrow (5, 5p - 1) = \Big(2p + 1, \dfrac{5 + 3q}{2}\Big)$

Comparing the x coordinates, we get :

⇒ 2p + 1 = 5

⇒ 2p = 5 - 1

⇒ 2p = 4

⇒ p = $\dfrac{4}{2}$

⇒ p = 2.

Comparing y-coordinates we get :

$\Rightarrow 5p - 1 = \dfrac{5 + 3q}{2} \\[1em] \Rightarrow 5 \times 2 - 1 = \dfrac{5 + 3q}{2} \\[1em] \Rightarrow 9 = \dfrac{5 + 3q}{2} \\[1em] \Rightarrow 18 = 5 + 3q \\[1em] \Rightarrow 3q = 18 - 5 \\[1em] \Rightarrow 3q = 13 \\[1em] \Rightarrow q = \dfrac{13}{3}.$

p = 2 and q = $\dfrac{13}{3}$.

Hence, Option 3 is the correct option.