GRADE 12 PRELIMINARY EXAMINATIONS 2019 MATHEMATICS …maths.stithian.com/New CAPS 2019 Prelim...

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Bridge House College November Examinations 2019 Grade 12 Paper 2

B R I D G E H O U S E PRE-PRIMARY • PREPARATORY • COLLEGE

GRADE 12

PRELIMINARY EXAMINATIONS 2019

MATHEMATICS PAPER 2

Time: 3 hours

Total: 150

Read the following instructions carefully:

1. This question paper consists of 18 pages and 8 questions. A separate Formula

sheet is provided. Please check that your question paper is complete.

2. Read the questions carefully.

3. Number your answers exactly as the questions are numbered.

4. All the necessary working details must be clearly shown.

5. Approved non-programmable calculators may be used unless otherwise stated.

6. Answers should be rounded off to two decimal digits unless otherwise stated.

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

8. ANSWER ALL QUESTIONS ON THIS QUESTION PAPER. Additional space is provided

at the end of the paper.

NAME:__________________________________________________________

Question 1 2 3 4 5 6 7 8 Total

Actual Mark

Possible Mark

20 22 14 14 22 23 25 10 150

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B(7; -3)

D

A(-3; -3)

O

C(3; 5)

SECTION A:

Question 1:

In the accompanying figure ∆𝐴𝐵𝐶 is drawn with 𝐴(−3;−3), 𝐵(7;−3) and 𝐶(3; 5).

a. Write down the equation of 𝐴𝐵 in the form 𝑦 = 𝑚𝑥 + 𝑐.

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_________________________________________________________________ (1)

b. Write down the length of 𝐴𝐵. (2)

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c. Prove that ∆𝐴𝐵𝐶 is isosceles. (3)

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d. Calculate the co-ordinates of the midpoint 𝐷 of 𝐵𝐶. (2)

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e. The equation of straight line 𝐴𝐷 is given as 2𝑦 − 𝑥 + 3 = 0.

i. Determine the gradient of 𝐴𝐷. (1)

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ii. Find the equation of the straight line passing through point 𝐵 and parallel

to 𝐴𝐷. (3)

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f. Consider the line 𝐴𝐶.

i. Determine the gradient of 𝐴𝐶. (2)

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ii. Determine 𝛼, the angle of inclination of 𝐴𝐶. (2)

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iii. Hence, determine the size of 𝐶�̂�𝐷. Give your answer to the nearest degree.

(4)

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[20]

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−90 −45 45 90

−3

−2

−1

1

2

3

x

y

Question 2:

2.1. The graphs of 𝑓(𝑥) = 2sin(𝑥 + 45°) and 𝑔(𝑥) = 2tan𝑥 are shown for 𝑥 ∈

[−90°; 90°].

Use the graphs to answer the following questions:

a. Give the period of 𝑔(𝑥). (1)

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b. What is the amplitude of 𝑓(𝑥). (1)

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c. For which value(s) of x is 𝑓(𝑥) = 𝑔(𝑥). (1)

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d. For which value(s) of x is 𝑓(𝑥) − 𝑔(𝑥) = 2. (1)

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e. If 𝑔(𝑥) is translated 30° to the right and 2 units up, give the new equation. Call

it ℎ(𝑥). (2)

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𝑓(𝑥)

𝑔(𝑥)

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3

2

3

CD 2B

A

2.2 Simplify: cos(180°−𝜃).tan(180−𝜃).sin(90°−𝜃)

cos(𝜃−540°).sin(−𝜃) (7)

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2.3 In the diagram 𝐴𝐵𝐶 is an isosceles triangle. 𝐷 lies on 𝐵𝐶.

𝐴𝐵 = 𝐴𝐶 = 3 units and 𝐴𝐷 = 𝐷𝐶 = 2 units. �̂� = 𝜃

a. Show that the size of 𝐴�̂�𝐶 is 180° − 2𝜃. Give reasons for all steps shown.

(4)

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b. Hence, prove that: 𝑐𝑜𝑠2𝜃 =1

8 (3)

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2

1

B

32

1

D

35 F

1 E

43

2

1

CA

c. Hence, determine the value of 𝜃. (Round off to TWO decimal digits.) (2)

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[22]

Question 3:

In the diagram two circles 𝐴𝐷𝐶 and 𝐶𝐵𝐷𝐹 intersect at 𝐶 and 𝐷. 𝐵 is the centre of

circle 𝐴𝐷𝐶 and 𝐸 the centre of circle 𝐶𝐵𝐷𝐹.

𝐴𝐵𝐷 is a straight line. Let �̂� = 35°.

3.1. Determine, with reasons, the size of each of the following angles:

a. �̂�2 (2)

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b. �̂�1 (2)

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c. 𝐴�̂�𝐸 (5)

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d. �̂� (2)

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3.2. Is 𝐶𝐵𝐷𝐸 is a cyclic quadrilateral? Motivate your answer, giving all reasons. (3)

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[14]

Question 4:

Below is a set of data recording the grams of fat to the total calories in sandwiches.

SANDWICH TOTAL FAT (g)

TOTAL CALORIES

Hamburger 9 260

Cheeseburger 13 320

Quarter Pounder 21 420

Quarter Pounder with Cheese

30 530

Big Don Burger 31 560

Gill Sandwich Special 31 550

Gill Special with Avo 34 590

Crispy Chicken 25 500

Fish Fillet 28 560

Grilled Chicken 20 440

Grilled Chicken Light 5 300

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5 10 15 20 25 30 35 40

50

100

150

200

250

300

350

400

450

500

550

600

a. On the set of axes below represent the above data as a scatter plot. (4)

b. Determine the equation of the straight line that best models the data in the table ie. give the equation of the line of best fit in the form 𝑦 = 𝑚𝑥 + 𝑐. (2)

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c. Calculate the correlation coefficient and comment on the relationship between fat and calories. (4)

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d. Using the equation of the line of best fit:

i. Calculate the how many calories a sandwich with 32 g of fat will contain. (2)

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x

y

ii. Do you think the value obtained in (i) is realistic? Explain your answer. (2)

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[14]

TOTAL FOR SECTION A: 70 MARKS

SECTION B:

Question 5:

5.1. 𝐾(𝑥; 𝑦), 𝐿(0;−3);𝑀(8;−9) and 𝑁(9;−6) are the vertices of a quadrilateral.

The equation of the line 𝐿𝐾 is 𝑘𝑦 + 21 = 𝑥 and

The gradient of the line K𝑀 is −7.

a. Calculate the value of 𝑘. (2)

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𝐿(0;−3)

𝑀(8;−9)

𝑁(9;−6)

𝐾(𝑥; 𝑦)

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b. Prove that 𝐿𝐾⏊𝐾𝑀if 𝑘 = 7. (3)

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c. If 𝐿�̂�𝑀 = 90°, prove that 𝐾𝐿𝑀𝑁 is a cyclic quadrilateral. (3)

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d. Determine the equation of the circle 𝐾𝐿𝑀𝑁 in the form

(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟2. (5)

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5.2 The equation of the circle 𝑥2 + 6𝑥 + 𝑦2 + 4𝑦 = 4 is given.

a. Determine the centre of the circle. (4)

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12

D

E

321

C

21

B

F

32

1

A

b. Determine the equation of the tangent to at the point 𝐻(−4; 2). (5)

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[22]

Question 6:

6.1. In the diagram, 𝐹𝐴 is a tangent to circle 𝐴𝐵𝐶𝐷 at 𝐴.

𝐴𝐷𝐸 and 𝐹𝐵𝐶𝐸 are straight lines.

a. Prove that ∆𝐷𝐸𝐶 ∣∣∣ ∆𝐵𝐸𝐴 (4)

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b. Prove that ∆𝐹𝐴𝐵 ∣∣∣ ∆𝐹𝐶𝐴 (4)

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D E

CB

A

c. Hence show that 𝐹𝐴. 𝐶𝐴 = 𝐹𝐶. 𝐴𝐵 (2)

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6.2 Complete the missing information (including any constructions) to prove the

following theorem:

A line drawn parallel to one side of a triangle divides the other two sides in

proportion.

Given: ∆𝐴𝐵𝐶, 𝐷 on 𝐴𝐵 and 𝐸 on 𝐴𝐶 with 𝐷𝐸 ∣∣ 𝐵𝐶

Required to prove: 𝐴𝐷

𝐷𝐵=

𝐴𝐸

𝐸𝐶

Proof: Draw altitudes ℎ and 𝑘 of ∆𝐴𝐷𝐸 from 𝐸 and 𝐷 to the bases 𝐴𝐷 and 𝐴𝐸

respectively. Draw 𝐷𝐶 and 𝐵𝐸.

𝐴𝑟𝑒𝑎∆𝐴𝐷𝐸

(𝑎)=

(𝑏)1

2𝐷𝐵.ℎ

=𝐴𝐷

𝐷𝐵 (same height ℎ)


(𝑐)=

(𝑑)1

2𝐸𝐶.𝑘

=𝐴𝐸

𝐸𝐶 (same height 𝑘)

But 𝐴𝑟𝑒𝑎∆𝐵𝐷𝐸 = 𝐴𝑟𝑒𝑎∆(𝑒) (same base 𝐷𝐸, same height; 𝐷𝐸 ∣∣ 𝐵𝐶)

∴𝐴𝑟𝑒𝑎∆𝐴𝐷𝐸

(𝑓)=


𝐴𝑟𝑒𝑎∆𝐶𝐸𝐷

∴𝐴𝐷

𝐷𝐵=

𝐴𝐸

𝐸𝐶 (6)

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F

EDC

T

B

A

𝑎 = _____________________________________________________

𝑏 = _____________________________________________________

𝑐 = _____________________________________________________

𝑑 = _____________________________________________________

𝑒 = _____________________________________________________

𝑓 = _____________________________________________________

6.3 In the diagram ∆𝐴𝐵𝐶 has 𝐷 and 𝐸 on 𝐵𝐶.

𝐵𝐷 = 18𝑐𝑚 and 𝐷𝐶 = 27𝑐𝑚.

𝐴𝑇: 𝑇𝐶 = 2: 1 and 𝐴𝐷 ∣∣ 𝑇𝐸.

a. Write down the value of 𝐶𝐸

𝐸𝐷 (1)

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b. Show that 𝐷 is the midpoint of 𝐵𝐸. (2)

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c. If 𝐹𝐷 = 2𝑐𝑚, calculate the length of 𝑇𝐸. (2)

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d. Calculate the value of 𝐴𝑟𝑒𝑎∆𝐴𝐷𝐶

𝐴𝑟𝑒𝑎∆𝐴𝐵𝐶 (2)

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[23]

Question 7:

7.1. If sin24° = 𝑝, express, without using a calculator, the following in terms of 𝑝.

a. cos24° (3)

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b. sin168°. sin(−78°) (5)

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7.2 Show that cos𝜃+cos3𝜃

cos2𝜃= 2cos𝜃 (5)

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7.3 Determine the general solution of the equation

1 + 4sin2𝑥 − 5sin𝑥 + cos2𝑥 = 0 (6)

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7.4 The front page of a greeting card has a length 𝑥 and a breadth 𝑦, as shown in

the accompanying sketch. The card is opened so that the front page makes an

angle of 90° with the back page.

Show that cos(𝑄�̂�𝑆) =𝑥2

𝑥2+𝑦2 (6)

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Question 8:

In the diagram below 𝐴𝐶, 𝐵𝐷 and 𝐶𝐷 are tangents to the circle 𝑂 with diameter

𝐴𝐵 = 4𝑐𝑚. 𝐶𝐷 touches the circle at point 𝐸.

If 𝐴𝐶 = 𝑎 and 𝐵𝐷 = 𝑏, prove that 𝑎𝑏 = 4. (HINT: Join 𝑂𝐶;𝑂𝐸 and 𝑂𝐷).

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[10]

TOTAL FOR SECTION B: 80 MARKS

TOTAL FOR PAPER: 150 MARKS

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GRADE 12 PRELIMINARY EXAMINATIONS 2019 MATHEMATICS …maths.stithian.com/New CAPS 2019 Prelim...

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