The diagram shows a semicircle - HSC - SSCE Mathematics Extension 1 - Question 9 - 2021 - Paper 1

To determine which parametric equations represent the given semicircle, we first recognize that the semicircle is centered at the point (3, 2) with a radius of 1. The equations need to be consistent with this position and the orientation of the semicircle which opens above the x-axis.

The general form for parametric equations for a semicircle is:

$${ x = h + r \cos t, \quad y = k + r \sin t }$$

where ((h, k)) is the center and (r) is the radius. For our semicircle:

• Center: ((3, 2))
• Radius: (1)

Thus, substituting these values in:

1. The x-coordinate would be:
   $x = 3 + \cos t$
   Since we want the semicircle to open upwards, we use (y = 2 + \sin t).

This gives us the equations as:

• $x = 3 + \cos t$
• $y = 2 + \sin t$

From this analysis, with the interval for (t) ranging from (-\frac{\pi}{2}) to (\frac{\pi}{2}), we conclude:

• The correct pair of parametric equations that represents the semicircle is found in option B:
  { x = 3 + cos t
  y = 2 + sin t
  for \frac{-\pi}{2} \leq t \leq \frac{\pi}{2} }