Ajay starts from Point A and drives 20 km towards South. He then takes a left turn, drives 5 km, turns left and drives 30 km. He then takes a right turn and drives 8 km. He takes a final right turn, drives 10 km and stops at Point Q. How far (shortest distance) and towards which direction should he now drive to reach Point A again? (All turns are 90° turns only.)

To find the shortest distance and the direction Ajay needs to drive to return to Point A from Point Q, we can trace his path step-by-step using a standard two-dimensional coordinate system. Let Point A be the origin (0, 0), with the directions represented as follows:
- North corresponds to the positive y-axis direction.
- South corresponds to the negative y-axis direction.
- East corresponds to the positive x-axis direction.
- West corresponds to the negative x-axis direction.

Step 1: Start at Point A
Initial coordinates: $(x\_0,\; y\_0)\; =\; (0,\; 0)$

Step 2: Drive 20 km South
Driving South decreases the y-coordinate by 20 km.
New position: $(0,\; 0\; -\; 20)\; =\; (0,\; -20)$

Step 3: Turn left and drive 5 km
Since Ajay was facing South, a left turn changes his direction to East (increasing the x-coordinate).
New position: $(0\; +\; 5,\; -20)\; =\; (5,\; -20)$

Step 4: Turn left and drive 30 km
Since he was facing East, a left turn changes his direction to North (increasing the y-coordinate).
New position: $(5,\; -20\; +\; 30)\; =\; (5,\; 10)$

Step 5: Turn right and drive 8 km
Since he was facing North, a right turn changes his direction to East (increasing the x-coordinate).
New position: $(5\; +\; 8,\; 10)\; =\; (13,\; 10)$

Step 6: Turn right, drive 10 km and stop at Point Q
Since he was facing East, a right turn changes his direction to South (decreasing the y-coordinate).
Final position of Point Q: $(13,\; 10\; -\; 10)\; =\; (13,\; 0)$

Step 7: Find the distance and direction from Point Q back to Point A
We need to travel from Point Q (13, 0) to Point A (0, 0).
The difference in coordinates is:
$\backslash Delta\; x\; =\; 0\; -\; 13\; =\; -13$
$\backslash Delta\; y\; =\; 0\; -\; 0\; =\; 0$

The shortest distance is the straight-line distance:
$\backslash sqrt\{(-13)^2\; +\; 0^2\}\; =\; 13\backslash text\{\; km\}$

Since the change in the x-coordinate is -13 (in the negative x direction) and there is no change in the y-coordinate, Ajay must drive directly West to reach Point A.

Therefore, Ajay needs to drive 13 km towards West to return to Point A.