ST BENEDICT’S COLLEGE - St Stithians College

ST BENEDICT’S COLLEGE

SUBJECT Mathematics Paper 2 DATE July 2019

GRADE 12 MARKS 150

EXAMINER Mrs Serafino MODERATOR Mrs Povall and Gr 12 Educators

NAME DURATION 3 Hours

CLASS

QUESTION NO

DESCRIPTION MAXIMUM MARK

ACTUAL MARK

1 Statistics 13

2 Statistics 15

3 Analytical Geometry 15

4 Trigonometry 17

5 Euclidean Geometry 15

6 Trigonometry 15

7 Trigonometry 15

8 Trigonometry 6

9 Statistics 4


11 Euclidean Geometry 15


TOTAL 150

Grade 12 2 of 21 Mathematics Paper 2

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This paper consists of 21 pages and an Information Sheet of 2 pages. Please check that your question paper is complete.

2. Read the questions carefully.

3. Answer all the questions on the question paper.

4. Diagrams are not necessarily drawn to scale.

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

6. Ensure your calculator is in DEGREE mode.

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

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

9. Round off to one decimal places unless otherwise stated.


SECTION A

QUESTION 1 13 MARKS

Mr Benecke wants to create a model whereby he can predict a learner’s final result. He decides to use his results from 2018 to create this model.

Preliminary results

(𝒙) 53 61 38 68 63 86 72 47 72 50 25 69 91 30 38 84

Final exam results

(𝒚) 64 80 59 75 76 81 75 63 81 65 33 58 96 41 51 89

a) Calculate the equation of the line of regression which best fits the data. Round your answer correct to 4 decimal places. (4)

b) Determine the correlation coefficient. (2)

c) Describe the correlation between the preliminary results and the final exam results. (3)

d) What can Mr Benecke expect a learner who achieved 50% in the preliminary exams, to achieve in the final exams? (2)

e) Can we use the same regression function to infer a preliminary result from a final result? Give a brief explanation for your answer. (2)


QUESTION 2 15 MARKS

The following table shows the absenteeism of 280 employees of a company in one year.

Number of days

absent

Frequency Midpoint Cumulative frequency

0 < d ≤ 5 32

5 < d ≤ 10 67

10 < d ≤ 15 131

15 < d ≤ 20 43

20 < d ≤ 25 7

a) Complete the table. (2)

b) On the grid below, draw an Ogive, using the information in the table above. (4)


c) Determine an estimate of the mean number of days employees are absent. (3)

d) Determine an estimate of the standard deviation of the above data set. (2)

e) Using the given information, determine the approximate number of employees absent

for 13 days or less. Indicate, using the letter A where you would read this off on the

graph. (2)

f) Determine and indicate on the graph, using the letter B,where you would read off the

median number of days absent for the given year. (2)


QUESTION 3 15 MARKS

𝐴(−3; −1) ; 𝐵(7; 2) and 𝐶(3; 9) are the vertices of ∆𝐴𝐵𝐶 in the given Cartesian plane. 𝐵𝑁 ⊥ 𝐴𝐶

and point 𝑀 is the midpoint of 𝐴𝐶. 𝐶��𝐷 = 𝜃

a) Calculate the length of 𝐴𝐶. (Leave your answer in surd form.) (2)

b) Calculate the gradient of 𝐴𝐶. (2)


c) Hence, determine the equation of 𝐵𝑁. (3)

d) Calculate the area of ∆𝐴𝐵𝐶 if 𝑁 is the point (1; 6). Give your answer correct to two decimal places. (4)

e) Calculate the size of 𝜃 (�� ) correct to 1 decimal place. (4)


QUESTION 4 17 MARKS

a) Simplify the following expression: (4)

sin(360° − 𝜃) cos(180° + 𝜃)

sin 270° sin(360° − 2𝜃)

b) If cos 4° = 𝑚, determine the value of:

1) cos 38° cos 34° + sin 34° sin 38° (2)

2) cos2 178° − cos2 272° (3)

Grade Space for your diagram and working: 9 of 21 Mathematics Paper 2

c) If 12 tan 𝑝 − 5 = 0 and cos 𝑝 < 0, using a diagram and without the use of a calculator, determine the value of:

Space for your diagram and working:

1) cos 2𝑝 (5)

2) sin(𝑝 − 45°) (3)

Grade 10 of 21 Mathematics Paper 2

QUESTION 5 15 MARKS

a)

Using the given diagram, complete the table to assist you in proving the theorem that states:

𝐴��𝐵 = 2 × 𝐴��𝐵 (5)

STATEMENT REASON

��1 = �� 1)

2)��1 = 3)

4)∴

Similarly, ��2 = 2 × ��2

5)∴

b) In the diagram AOB is a diameter of the circle AECB, with O the centre. OE ∥ BC and OE

meets AC at D. B and E are joined.


1) Prove that AD = DC. (3)

2) Prove that EB bisects ABC. (3)

3) If OEB = 𝑥, express BAC in terms of 𝑥. (4)


SECTION B

QUESTION 6 15 MARKS

a) Prove that: sin22 𝛼 ( 1

tan2 𝛼− tan2 𝛼) = 4 cos 2𝛼 (5)

b) Hence, determine the general solution for:

(sin22 𝛼

4)(

1

tan2 𝛼 − tan2 𝛼) =

√3

2 (4)


c) The identity sin(30° + 𝜃) + sin(30° − 𝜃) = cos 𝜃 is given.

1) Use the above identity to determine the value of: sin 40° + sin 20°. You may leave your answer in terms of a trigonometry ratio.

2) Use the above identity to determine the exact value of: sin 35° sin 25° − sin 85° .

(3)

(2)


QUESTION 7 15 MARKS

a) Draw on the same set of axes, sketch graphs of 𝑓(𝑥) = sin(𝑥 − 60°) and 𝑔(𝑥) = cos 2𝑥

for 𝑥 ∈ [−180°; 180°]. (6)

b) Without the use of a calculator, calculate the value(s) of 𝑥 for which:

sin(𝑥 − 60°) = cos 2𝑥 if 𝑥 ∈ [−180°; 180°]. (7)

c) Now write down the solution of sin(𝑥 − 60°) > cos 2𝑥 for 𝑥 ∈ [0°; 180°]. (2)


QUESTION 8 6 MARKS

A cyclist at training is on his way from P to A. When he reaches point C, the road forks. The road to the right leads directly to A, which is 60 km from C. P and C are 20 km apart

The road to the left leads to A via B.

C and B are 40 km apart. The angle between the roads (BC and AC), is .

The cyclist travels at a constant speed of 28 km/hour.

a) Calculate the difference between the distances of the two routes

from C to A , correct to the nearest kilometre. (4)

b) Calculate how long it will take the cyclist to reach A from P if he chooses the road

to the right leading directly to A from P. (2)

35


QUESTION 9 4 MARKS

State the skewness of the data in each case:

A: 20, 35, 40, 55, 68, 75, 92

B:

C:

D: 3 1 22Q Q Q

x

y


QUESTION 10 13 MARKS

In the diagram below, two concentric circles are drawn with centre A on the y axis. The smaller

circle cuts the y axis at the origin O and the point B. The line through B having equation

𝑦 =3

2𝑥 + 6 cuts the x- axis at R and the y- axis at B. The larger circle passes through R.

a) Determine the equation of both the circles. (7)


b) Determine the equation of the tangent to the larger circle at R. (4)

c) Determine the equation of the tangents to the smaller circle which are parallel to the y- axis. (2)


QUESTION 11 15 MARKS

In the figure TD is a tangent to the circle ABCD. AD ∥ BC. AB and DC produced meet at W. TBS

is a straight line. If B1 = B3 prove that:

1) BWTD is a cyclic quadrilateral.

2) TBS is a tangent to the circle ABCD. (6)

(5)


3) TW ∥ BC. (6)

QUESTION 12 7 MARKS

The face of a clock is drawn on a Cartesian plane with its centre at the origin and 12 o the positive y- axis. The

hour hand of the clock is represented by the equation

𝑦 = 2,145𝑥 and the minute hand by the equation

𝑦 = −0,57𝑥 where both hands lie above the x- axis.

a) Find, to the nearest degree, the angles of inclinations

of the hour hand and the minute hand. (4)


b) Hence, or otherwise, determine the time shown on the clock to the nearest minute. (3)

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