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

To represent the semicircle, we note that the center of the semicircle is at the point (3, 2) and the radius is 1. The parametric equations for a semicircle can be described as:

• The x-coordinate can be expressed using cosine or sine, and the y-coordinate will be based on the circle formula.

From the options provided:

1. Option A:

   • $x = 3 + ext{sin}(t)$
   • $y = 2 + ext{cos}(t)$
   • This shifts the semicircle upwards and right, but the required form is not correctly aligned.

2. Option B:

   • $x = 3 + ext{cos}(t)$
   • $y = 2 + ext{sin}(t)$
   • This represents the upper semicircle aligning well with the center and radius of the circle.

3. Option C:

   • $x = 3 - ext{sin}(t)$
   • $y = 2 + ext{cos}(t)$
   • This creates a symmetrical shift but does not align well.

4. Option D:

   • $x = 3 - ext{sin}(t)$
   • $y = 2 - ext{sin}(t)$
   • This does not represent the semicircle accurately.

Hence, the correct choice is clearly Option B.