Grade 12 Maths · Paper 2

Analytical Geometry Grade 12

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past paper questions: 38 past paper questions
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Question 3.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks

In the diagram below,

Δ SRT

is drawn where R lies on the x-axis and S lies to the left of R. T lies on the y-axis and the coordinates of S are

(m; 1)

.

The equation of RT is

2 x - y + 10 = 0

3.1 Calculate the coordinates of R.(2)

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What's in Analytical Geometry Grade 12

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The Circle38 past paper questions

More Analytical Geometry Grade 12 questions and answers

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1. Question 5.2 · March 2022 · Provincial · 3 marks
In the figure: $PQ$ is the diameter of the circle. $SP$ is the tangent to the circle at $P$ . $QT$ intersects the circle at $W$ and with $T$ on the line $SP$ . $QS$ intersects the circle at $R$ . $\hat{P}_{1} = x$ .
5.2 Prove $\hat{S} = x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Calculate angle QIn $Δ PQR$ : $P \hat{Q} R = 9 0^{\circ} - x$ (Sum of angles in $Δ$ )
  1 mark · Use angle sum of a triangle
2. ✓
  Identify tangent perpendicular to diameter $S \hat{P} Q = 9 0^{\circ}$ (Diameter $⊥$ tangent)
  1 mark · Use radius and tangent facts
3. ✓
  Calculate SIn $Δ SPQ$ : $\hat{S} + 9 0^{\circ} + (9 0^{\circ} - x) = 18 0^{\circ}$
  $\hat{S} = x$
  1 mark · Use angle sum of a triangle
2. Question 7.1 · March 2022 · Provincial · 4 marks
In the diagram, $A C$ and $CE$ are tangents to the circle with centre $O$ . $B, D, G$ and $H$ are points on the circumference. $H G$ is produced to $F$ . $B D = B H$ . $\hat{D}_{1} = \hat{D}_{4} = x$ .
7.1 Find four other angles equal to $x$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Find B3 $\hat{B}_{3} = x$ (Tangents from same point $C$ , so $CB = C D$ , $Δ CB D$ is isosceles / Tan-chord theorem implies $\hat{B}_{3} = \hat{D}_{1}$ but logic usually $CB = C D$ )
  1 mark · Use isosceles triangle properties
2. ✓
  Find H1 $\hat{H}_{1} = \hat{B}_{3} = x$ (Tan-chord theorem with tangent $A C$ and chord $B D$ - Wait, $A C$ touches at $B$ )
  1 mark · Use the tan-chord theorem
3. ✓
  Find D2 $\hat{D}_{2} = \hat{H}_{1} = x$ (Angles opposite equal sides, $B D = B H$ )
  1 mark · Use isosceles triangle properties
4. ✓
  Find H2 $\hat{H}_{2} = \hat{D}_{4} = x$ (Tan-chord theorem with tangent $CE$ and chord $D G$ - wait, tangent is at $D$ , chord $D G$ subtends $H_{2}$ . Correct)
  1 mark · Use the tan-chord theorem
3. Question 5.2.1 · March 2022 · Provincial · 2 marks
5.2 In the diagram, O is the centre of the circle and L is a point on the circumference. RLM is a tangent at L.
5.2.1 Determine with reasons the size of $\hat{O}_{2}$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Use tangent-radius theorem $M \hat{L} O = 9 0^{\circ}$ (radius $⊥$ tangent)
  1 mark · Use radius and tangent facts
2. ✓
  Calculate angle in triangle $\hat{O}_{2} = 18 0^{\circ} - (9 0^{\circ} + 2 6^{\circ}) = 6 4^{\circ}$ (sum of $∠$ s in $Δ$ )
  1 mark · Use angle sum of a triangle
4. Question 6.1 · March 2022 · Provincial · 5 marks
In the diagram below, P, Q, R and S are points on the circle. QS and PR intersect at point T. The line from V is a tangent at P. $Q \hat{R} P = 5 6^{\circ}$ and $R \hat{P} V = 7 0^{\circ}$ .
6.1 Find the size of $R \hat{S} T$ .(5)
Show the memo (3 ticks)Memo
1. ✓
  Use tan-chord theorem $R \hat{P} V = R \hat{Q} P = 7 0^{\circ}$
  2 marks · Use the tan-chord theorem
2. ✓
  Sum of angles in triangle PQR $\hat{P}_{2} = 18 0^{\circ} - (7 0^{\circ} + 5 6^{\circ}) = 5 4^{\circ}$
  1 mark · Use angle sum of a triangle
3. ✓
  Angles in same segment $R \hat{S} T = \hat{P}_{2} = 5 4^{\circ}$
  2 marks · Use angles in the same segment
5. Question 2.2.1 · March 2021 · Provincial · 3 marks
2.2 In the diagram, a circle with a tangent CD is drawn. A, B, D, and E are points on the circumference of the circle. $A E = A B$ and $A B ∥ E D$ . $\hat{A}_{1} = x$ .
2.2.1 Give, with reasons, three more angles equal to $x$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Identify first angle (D1) $\hat{D}_{1} = \hat{A}_{1} = x$ (tan chord theorem)
  1 mark · Use the tan-chord theorem
2. ✓
  Identify second angle (D3) $\hat{D}_{3} = \hat{A}_{1} = x$ (Alt. $∠$ s; $A B ∥ E D$ )
  1 mark · Use angles on parallel lines
3. ✓
  Identify third angle (D2) $\hat{D}_{2} = \hat{D}_{3} = x$ (equal chords equal angles OR angles subtended by equal chords $A E = A B$ )
  1 mark · Use angles in the same segment

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