ST BENEDICT’S COLLEGE CAPS 2020 Prelim Papers/St...7 Statistics 8 8 Trigonometric Equations 4 9 3D Trigonometry 9 10 Euclidean Geometry 17 11 Euclidean Geometry 14 73 TOTAL 150 Grade

ST BENEDICT’S COLLEGE

SUBJECT Mathematics Paper 2 DATE September 2020

GRADE 12 MARKS 150

EXAMINER Ms Brown MODERATOR Gr 12 Educators

NAME DURATION 3 Hours

CLASS

QUESTION NO

DESCRIPTION MAXIMUM MARK

ACTUAL MARK

Section A

1 Statistics 13

2 Analytical Geometry 14

3 Trigonometry 23

4 Euclidean Geometry 21

5 Trigonometric Graphs 6

77

Section B

6 Analytical Geometry 21

7 Statistics 8

8 Trigonometric Equations 4

9 3D Trigonometry 9



73

TOTAL 150

Grade 12 2 of 25 Mathematics Paper 2

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This paper consists of 25 pages and an Information Sheet of 2 pages. Please check that your question paper is complete.

2. The last 3 pages can be used for additional working, if necessary. If this space is used, make sure that you indicate clearly which question is being answered.

3. Read the questions carefully.

4. Answer all the questions on the question paper.

5. Diagrams are not necessarily drawn to scale.

6. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

7. Ensure your calculator is in DEGREE mode.

8. All the necessary working details must be clearly shown. Answers only will not necessarily be awarded full marks.

9. It is in your interest to write legibly and to present your work neatly.

10. Round off to ONE decimal digits, unless otherwise stated.


SECTION A

QUESTION 1 13 MARKS

Fourteen learners who had no access to technology during lockdown attended a Geometry training course, spread over 12 Saturdays, to make up for the work they missed due to Covid19. The learners wrote a Geometry test at the end of the course. One learner was absent for the test, due to contracting the virus. The number of Saturdays attended and the mark (as a %) each learner obtained for the test are shown in the table below.

Number of Saturdays attended 12 11 10 10 9 9 7 6 5 4 12 11 6

Mark (as a %) 96 91 78 83 75 62 70 68 56 34 88 90 59

(a) Calculate the correlation coefficient. (2)

(b) Comment on the strength of the relationship between the variables. (1)

(c) Calculate the equation of the least-squares regression line. (3)

(d) The learner who was absent for the test attended the course on 8 Saturdays.

(1) Predict the mark that this learner would have scored for the test. (2)


(2) Discuss how reliable this result is. (2)

(e) Once the sick learner was granted permission by his doctor, he came in during

the next week to write the test. The class average changed to 71%.

Determine the mark that the learner attained. (3)

QUESTION 2 14 MARKS

In the figure below, two circles with centre M and P respectively are drawn.

The equation of circle M is (𝑥 + 2)2 + (𝑦 – 1)2 = 𝑟2 .

𝑆(1 ; – 2) is a point on both circles.

SR is a tangent to both circles.

P has the coordinates (a ; b).


(a) Write down the coordinates of M. (1)

(b) Determine r, the radius of circle M, in simplified surd form. (3)

(c) Determine the equation of the tangent SR, in the form 𝑦 = 𝑚𝑥 + 𝑐. (4)

(d) Given: MS : MP = 1 : 4

(1) Determine the value of a and b. (4)


(2) Determine the equation of circle P. (2)

QUESTION 3 23 MARKS

(a) Given: 𝑠𝑖𝑛40°. 𝑐𝑜𝑠22° + 𝑐𝑜𝑠40°. 𝑠𝑖𝑛22° = 𝑘

Determine, without the use of the calculator, the value of the following in terms of k:

(1) 𝑠𝑖𝑛62° (2)

(2) 𝑡𝑎𝑛118° (4)

(3) 𝑠𝑖𝑛14°. 𝑐𝑜𝑠14° (3)


(b) Simplify the following expression: (6)

𝑐𝑜𝑠(90° + 𝑥). 𝑠𝑖𝑛(𝑥 − 180°) − 𝑐𝑜𝑠2(180° − 𝑥)

cos (−2𝑥)+ 𝑠𝑖𝑛2𝑥

(c) Prove the following identity: 1−𝑐𝑜𝑠2𝜃

𝑠𝑖𝑛2𝜃.𝑡𝑎𝑛𝜃= 1 (4)

(d) Determine the general solution of: 2𝑐𝑜𝑠2𝛼 − 3𝑐𝑜𝑠𝛼 − 2 = 0 (4)


QUESTION 4 21 MARKS

(a) In the diagram, O is the centre of the circle and EC is a tangent to the circle at C.

DM = MC and OME is a straight line.

Let �̂�1 = 2𝑥.

(1) Give, with reasons, THREE angles equal to 𝑥. (6)


(2) Prove that �̂�2 = 90° − 𝑥 . (3)

(3) Prove that DOCE is a cyclic quadrilateral. (4)


(b) In the diagram below, M is the midpoint of QR in ∆PQT. T is a point on PQ,

such that PM and TR intersect at G. GH // PQ with H on QR.

PG : PM = 1 : 3.

Determine, with reasons, the numerical value of:

(1) 𝑄𝐻

𝐻𝑀 (3)

(2) 𝑅𝐺

𝑅𝑇 (5)


QUESTION 5 6 MARKS

The graph in the figure represents the curves of:

𝑓(𝑥) = 𝑐𝑜𝑠 𝑎 𝑥, and 𝑔(𝑥) = 𝑏 𝑠𝑖𝑛 𝑥 ; -180° ≤ 𝑥 ≤ 180°

(a) Determine 𝒂 and 𝒃. (2)

(b) Write down the period of 𝒈. (1)

(c) What is the minimum value of 𝒇? (1)

(d) How many values of 𝒙 will 𝒇(𝒙) +𝟏

𝟐= 𝒈(𝒙) have in the given interval? (2)


SECTION B

QUESTION 6 21 MARKS

In the diagram below, LM is a tangent to the circle with centre P(-1 ; -1) at L(-4 ; 3).

M(0 ; 6) is a point on the y-axis.

(a) Determine the inclination, α, of line PM. (3)

(b) Calculate the area of ∆PLM. (6)

y

x

M(0 ; 6)

L(-4 ; 3)

P(-1 ; -1)

α

45o


(c) Prove that the equation of the circle is given by : (4)

𝑥2 + 𝑦2 + 2𝑥 + 2𝑦 − 23 = 0

(d) (1) Determine the equations of the vertical tangents of the circle. (2)

(2) Write the coordinates of the points of contact of the vertical tangents with

the circle. (2)

(e) Another circle with equation (𝑥 − 3)2 + (𝑦 + 1)2 = 18 is given.

Show all calculations to prove that the two circles will intersect each other. (4)


QUESTION 7 8 MARKS

The speeds of 55 cars passing through a certain section of a road in Johannesburg are

monitored for one hour the day after the alcohol ban was lifted in August. The speed limit on

this section of road is 60 km per hour. A histogram is drawn to represent this data.

(a) Identify the modal class of the data. (1)

(b) Use the histogram to:

(1) Complete the table below. (2)

Class Frequency Cumulative frequency

20 < 𝑥 ≤ 30

30 < 𝑥 ≤ 40

40 < 𝑥 ≤ 50

50 < 𝑥 ≤ 60

60 < 𝑥 ≤ 70

70 < 𝑥 ≤ 80

80 < 𝑥 ≤ 90

90 < 𝑥 ≤ 100


(2) Draw an ogive (cumulative frequency graph) of the above data on the grid

provided. (3)

(c) The traffic department sends speeding fines to all motorists whose speed exceeds

66 km per hour.

Estimate the number of motorists who will receive a speeding fine. (2)


QUESTION 8 4 MARKS

Solve for a and b where:

cos(𝑎 + 𝑏) = −√2

2 if 𝑎 + 𝑏 ∈ [0° ; 180°]

cos(𝑎 − 2𝑏) = 1

2 if 𝑎 − 2𝑏 ∈ [0° ; 180°] (4)


QUESTION 9 9 MARKS

Mr Antonites is on break-duty and is monitoring social distancing of the students on the field

below. He is looking down at Mazizi and Cameron from the 1st floor corridor, who are on

the ground looking up at him.

The angle of elevation from Mazizi to Mr Antonites is (90° − 2𝜃).

AM = 𝑘 meters,

GC= 8 meters,

MĜC= 150°

MĈG = 𝜃.

(a) Give the size of MÂG in terms of 𝜃. (1)

(b) Show that MG= 𝑘 sin 2𝜃. (2)

(c) Show that MC= 𝑘 cos 𝜃. (4)


(d) Show that the area of ∆MGC can be expressed as 2𝑘𝑠𝑖𝑛2𝜃. (2)

QUESTION 10 17 MARKS

(a) In the diagram, a cyclic quadrilateral ABCD is given with centre O. (6)

Prove the theorem that states:

The sum of the opposite angles of a cyclic quadrilateral is 180°.


(b) In ∆𝐴𝐵𝐶, D is a point on AB such that AD : DB = 5 : 4.

P and E are points on AC such that

DE // BC and DP // BE.

BC is NOT the diameter of the circle.

𝐵�̂�𝐸 = 120° ,

EC = 12 units

and BC = 27 units.

Determine, with reasons:

(1) The length of AE. (3)

(2) 𝑎𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐸𝐵

𝑎𝑟𝑒𝑎 𝑜𝑓 ∆𝐸𝐶𝐵 (2)

2

2

1

1

1

1 2

2

3


(c) Hence, determine the length of DP if it is further given that ∆𝐴𝐷𝑃 /// ∆𝐴𝐵𝐸. (6)


QUESTION 11 14 MARKS

(a) Complete the following theorem statement:

“if two triangles are equiangular, then ___________________________________ (1)

(b) AP is a tangent to the circle at P. CB // DP and CB = DP.

CSA is a straight line. Let �̂� = 𝑥 and �̂�2 = 𝑦.

Prove, with reasons, that:

(1) ∆𝐴𝑃𝐶 ///∆𝐴𝐵𝑃. (4)

(2) 𝐴𝑃2 = 𝐴𝐵 . 𝐴𝐶 (1)


(3) ∆𝐴𝑃𝐶 ///∆𝐶𝐷𝑃. (4)

(4) 𝐴𝑃2 + 𝑃𝐶2 = 𝐴𝐶2. (4)


EXTRA SPACE FOR WORKING

ST BENEDICT’S COLLEGE CAPS 2020 Prelim Papers/St...7 Statistics 8 8 Trigonometric Equations 4 9 3D Trigonometry 9 10 Euclidean Geometry 17 11 Euclidean Geometry 14 73 TOTAL 150 Grade

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