Grade 12 Maths Paper 2 Revision Document Request
This revision guide focuses on the core components of the CAPS Grade 12
Mathematics Paper 2 curriculum for Term 2. We will cover Analytical Geometry,
Trigonometry, and Euclidean Geometry.
1. Analytical Geometry: The Circle
In Grade 12, we extend our knowledge of coordinate geometry to circles. A circle is
defined by its center (𝑎, 𝑏) and its radius 𝑟.
Key Concepts
• The Equation of a Circle:
o Centered at the origin: 𝑥 2 + 𝑦 2 = 𝑟 2
o Centered at any point (𝑎, 𝑏): (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
• Tangents to a Circle: A tangent is a line that touches the circle at exactly one
point. The radius is always perpendicular to the tangent at the point of contact.
Therefore:
𝑚radius × 𝑚tangent = −1
, Practice Problem Given a circle with the equation 𝑥 2 − 4𝑥 + 𝑦 2 + 6𝑦 = 12:
1. Determine the coordinates of the center and the length of the radius.
2. Find the equation of the tangent to the circle at the point (5,1).
Solution
1. To find the center and radius, we use the method of completing the square.
(𝑥 2 − 4𝑥) + (𝑦 2 + 6𝑦) = 12
(𝑥 2 − 4𝑥 + 4) + (𝑦 2 + 6𝑦 + 9) = 12 + 4 + 9
(𝑥 − 2)2 + (𝑦 + 3)2 = 25
o The center is (2, −3).
o The radius 𝑟 = √25 = 5 units.
2. Find the equation of the tangent at (5,1).
o First, find the gradient of the radius (𝑚𝑟 ) connecting the center (2, −3) and
the point (5,1):
𝑦2 − 𝑦1
𝑚𝑟 =
𝑥2 − 𝑥1
1 − (−3)
=
5−2
4
=
3
o Since the tangent is perpendicular to the radius:
1
𝑚tan = −
𝑚𝑟
3
=−
4
o Use the point-slope form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) with point (5,1):
3
𝑦 − 1 = − (𝑥 − 5)
4
3 15
𝑦=− 𝑥+ +1
4 4
3 19
𝑦=− 𝑥+
4 4
2. Trigonometry: Compound and Double Angles
In Grade 12, we move beyond basic ratios to identities involving multiple angles.
Key Formulae
• Compound Angles:
o cos(𝐴 − 𝐵) = cos𝐴cos𝐵 + sin𝐴sin𝐵
o cos(𝐴 + 𝐵) = cos𝐴cos𝐵 − sin𝐴sin𝐵
o sin(𝐴 ± 𝐵) = sin𝐴cos𝐵 ± cos𝐴sin𝐵
This revision guide focuses on the core components of the CAPS Grade 12
Mathematics Paper 2 curriculum for Term 2. We will cover Analytical Geometry,
Trigonometry, and Euclidean Geometry.
1. Analytical Geometry: The Circle
In Grade 12, we extend our knowledge of coordinate geometry to circles. A circle is
defined by its center (𝑎, 𝑏) and its radius 𝑟.
Key Concepts
• The Equation of a Circle:
o Centered at the origin: 𝑥 2 + 𝑦 2 = 𝑟 2
o Centered at any point (𝑎, 𝑏): (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
• Tangents to a Circle: A tangent is a line that touches the circle at exactly one
point. The radius is always perpendicular to the tangent at the point of contact.
Therefore:
𝑚radius × 𝑚tangent = −1
, Practice Problem Given a circle with the equation 𝑥 2 − 4𝑥 + 𝑦 2 + 6𝑦 = 12:
1. Determine the coordinates of the center and the length of the radius.
2. Find the equation of the tangent to the circle at the point (5,1).
Solution
1. To find the center and radius, we use the method of completing the square.
(𝑥 2 − 4𝑥) + (𝑦 2 + 6𝑦) = 12
(𝑥 2 − 4𝑥 + 4) + (𝑦 2 + 6𝑦 + 9) = 12 + 4 + 9
(𝑥 − 2)2 + (𝑦 + 3)2 = 25
o The center is (2, −3).
o The radius 𝑟 = √25 = 5 units.
2. Find the equation of the tangent at (5,1).
o First, find the gradient of the radius (𝑚𝑟 ) connecting the center (2, −3) and
the point (5,1):
𝑦2 − 𝑦1
𝑚𝑟 =
𝑥2 − 𝑥1
1 − (−3)
=
5−2
4
=
3
o Since the tangent is perpendicular to the radius:
1
𝑚tan = −
𝑚𝑟
3
=−
4
o Use the point-slope form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) with point (5,1):
3
𝑦 − 1 = − (𝑥 − 5)
4
3 15
𝑦=− 𝑥+ +1
4 4
3 19
𝑦=− 𝑥+
4 4
2. Trigonometry: Compound and Double Angles
In Grade 12, we move beyond basic ratios to identities involving multiple angles.
Key Formulae
• Compound Angles:
o cos(𝐴 − 𝐵) = cos𝐴cos𝐵 + sin𝐴sin𝐵
o cos(𝐴 + 𝐵) = cos𝐴cos𝐵 − sin𝐴sin𝐵
o sin(𝐴 ± 𝐵) = sin𝐴cos𝐵 ± cos𝐴sin𝐵
