PRG Gr.11 P2 June 2022
PAUL ROOS GYMNASIUM
MATHEMATICS PAPER 2
GRADE 11
30 MAY 2022
TIME: 2 hours
TOTAL: 100 marks
Examiner: L van Niekerk
Moderators: G Langenhoven
L Day
______________________________________________________________________________
General instructions:
1.
This paper consists of 6 questions. Answer all the questions.
2.
Do Questions 3-6 on the answer sheet and staple to the back of your folios. If you need
more writing space, use extra folios between the answer sheets.
3.
Draw a margin on the right hand side of the page and keep this margin open.
4.
If necessary, all answers must be rounded to two decimal digits.
5.
Clearly show all calculations, diagrams or graphs that you have used in determining your
answers.
6.
Full marks won’t necessarily be awarded for an answer only.
7.
Hand in your answer sheet and question paper separately.
Page 1 of 7
PRG Gr.11 P2 June 2022
QUESTION 1
16 marks
Do all the trigonometry without the use of a calculator.
1.1. If 17 cos 𝐵 + 8 = 0 and 𝐵𝜖(180°; 360°),
determine with the aid of a diagram the value of:
1.1.1 sin 𝐵
1.1.2
1
tan2 𝐵
(4)
+
1
(3)
sin2 𝐵
1.2. In the diagram 𝑃 is the point (2; −2√3) and
𝑃𝑅 ⊥ 𝑂𝑅. The reflex 𝑋𝑂̂𝑃 = 𝐴.
Calculate the following:
𝑃(2; −2√3)
1.2.1 The length of 𝑂𝑃.
(2)
1.2.2 cos(−𝐴)
(1)
1.2.3 tan(180° − 𝐴)
(2)
1.2.4
1
−√3
sin 𝐴 − 1
(3)
1.2.5 𝐴
(1)
Page 2 of 7
PRG Gr.11 P2 June 2022
QUESTION 2
39 marks
Remember to this question without the use of a calculator.
2.1. Simplify the following expressions fully:
2.1.1
2.1.2
tan(−30°).cos(370°)
(7)
tan(300°).sin 280°
sin(𝐴+180°)+2 cos(90°+𝐴).cos(𝐴−180°)
2 cos2 (180°−𝐴)−cos(−𝐴)
(8)
2.2. Prove the following identities:
2.2.1
2.2.2
2.2.3
1
tan2 𝑥
−
cos2 𝑥
1
1−2 sin 𝛼.cos 𝛼
sin 𝛼−cos 𝛼
=
=
cos4 𝑥
(4)
sin2 𝑥
sin2 𝛼−cos2 𝛼
(3)
sin 𝛼+cos 𝛼
sin2 15°+sin2 105°−cos2 250°
sin(−70°).cos 200°
=1
(7)
2.3. If cos 36° = 𝑘, write the following in terms of 𝑘.
2.3.1. cos 324°
(2)
2.3.2. sin 54°
(1)
2.3.3. tan 144°
(3)
2.3.4. cos2 216° − sin2 36°
(4)
Page 3 of 7
PRG Gr.11 P2 June 2022
QUESTION 3
17 marks
3.1. In the diagram the tangent at 𝐷 is parallel to chord 𝐴𝐶 and 𝑂 is the centre of the circle.
𝐵𝑂̂𝐶 = 60° and 𝐶𝐺̂ 𝐷 = 50°.
Determine with reasons the size of the following angles:
3.1.1. 𝐴̂1
(2)
3.1.2. 𝐴̂2
(2)
̂4
3.1.3. 𝐷
(2)
3.1.4. 𝐵𝐶̂ 𝐷
(2)
3.1.5. 𝐴𝐶̂ 𝐷
(2)
̂3
3.1.6. 𝐷
(2)
3.2 In the diagram, 𝐶, 𝐵 and 𝐴 are points on the circle with centre 𝑂. 𝐷𝐴 is a tangent to the
circle at 𝐴. Use the diagram to prove the theorem which states that 𝐷𝐴̂𝐶 = 𝐵̂.
Page 4 of 7
(5)
PRG Gr.11 P2 June 2022
QUESTION 4
8 marks
In the following sketch, 𝑂 is the centre of the circle.
𝑃𝑄 = 24 𝑐𝑚. 𝑅𝑆 = 10 𝑐𝑚 and 𝐴𝐵 = 7 𝑐𝑚.
𝐴𝑂 ⊥ 𝑃𝑄 , 𝑅𝐵 = 𝐵𝑆 and 𝑂𝐴 = 𝑥.
4.1. Give a reason why 𝑅𝐵̂ 𝑂 = 90°.
(1)
4.2. Show that 𝑅𝑂2 = 𝑥 2 + 14𝑥 + 74.
(2)
4.3. Calculate the value of 𝑥.
(5)
Page 5 of 7
PRG Gr.11 P2 June 2022
QUESTION 5
12 marks
In the diagram, circle 𝑄𝑅𝑆 has tangents 𝑃𝑄 and 𝑃𝑆.
𝑇 is the centre of the circle, with 𝑇 on 𝑄𝑅.
𝑅̂ = 𝑥
5.1. Give, with reasons, three other angles equal to 𝑥 .
(3)
5.2. Give, with reasons, three angles equal to 90°.
(3)
5.3. Give a reason why 𝑃𝑄𝑇𝑆 is a cyclic quadrilateral.
(1)
5.4. Prove that 𝑃𝑇 ∥ 𝑆𝑅.
(3)
5.5. Determine the size of 𝑃̂1+2 in terms of 𝑥.
(2)
Page 6 of 7
PRG Gr.11 P2 June 2022
QUESTION 6
8 marks
In the diagram, two circles 𝑃𝑇𝑅𝑄 and 𝑃𝑄𝐵 intersect at 𝑃 and 𝑄. 𝐴𝐵 is a tangent to the smaller
circle, with 𝑃𝑄𝐴 a straight line. 𝐵𝑄 produced meets the larger circle at 𝑇 such that 𝑃𝑇 ∥ 𝐵𝐴.
𝑇𝐴 intersects the larger circle at 𝑅.
̂ 𝟑 = 𝒙 and 𝑩
̂𝟏 = 𝒚
Let 𝑷
6.1. Determine, with a single reason, the size of 𝑄̂1 in terms of 𝑥 and 𝑦.
(2)
6.2. Prove that 𝑃𝑅𝐴𝐵 is a cyclic quadrilateral.
(3)
6.3. Prove that 𝐴𝐵 is a tangent to circle 𝑇𝑅𝐵.
(3)
TOTAL =100
Page 7 of 7