TECHNICAL MATHEMATICS PAPER 2
GRADE 12
NATIONAL SENIOR CERTIFICATE
JUNE 2018

INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.

INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.

1. This question paper consists of 10 questions. Answer ALL the questions.
2. Clearly show ALL calculations, diagrams, graphs, et cetera, that you have used in determining your answers.
3. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.
4. Where necessary, ALL answers should be rounded off to TWO decimal places, unless stated otherwise.
5. Number the answers correctly according to the numbering system used in this question paper.
6. Diagrams are NOT necessarily drawn to scale.
7. It is in your own interest to write legibly and to present your work neatly.
8. An information sheet with formulae is attached.
9. Diagram sheets are attached for QUESTION 3.1.2, QUESTION 6.2 and QUESTION 8.5. Write your name in the spaces provided and then hand in the diagram sheets with
   your ANSWER BOOK.

QUESTION 1
In the diagram below A(-4; 4) , B(0; -6) and C(8; 0) are the vertices of ∆ABC with D(2; 2) and E on AC and AB respectively such that BC || DE.

Determine: 
1.1 The length of BC (3)
1.2 The coordinates of E, the midpoint of AB (2)
1.3 The gradient of BC (2)
1.4 The equation of the line passing through points D and E (4)
1.5 The size of ABC(6)
[17]

QUESTION 2
The diagram below shows a circle with centre at the origin O with the tangent passing through point P(-5; -12) .

Determine: 
2.1 The equation of the (3)
2.1.2 The equation of the tangent to the circle at point P, in the form y = mx + c(4)
2.2 Sketch the graph defined by 4x² + 9y² = 36 . Clearly show all intercepts with (5)
[12]

QUESTION 3
3.1 In the diagram below, it is given that 13sinθ = 12 and θ∈[90°;180°] .
Use the diagram as provided in the SPECIAL ANSWER BOOK and answer the questions that follows.

3.1.1Determine the coordinates of P(4)
3.1.2Determine the numerical value of tanθ + secθ . (3) 
3.1.3Determine the size of θ, rounded off to ONE decimal (2)
3.2Simplify the following:
      tan (180°- θ).√1- sin² θ       
cos² (180°+ θ) + sin² (360° - θ) (7)
3.3Prove, using basic trigonometry identities, that:

1+ cos a +     sin a    = 2cosec a       (6)
   sina         1+ cosa

[22]

QUESTION 4
In the figure below, PS and QR represent two poles perpendicular to SR ground level. Furthermore, the angle of elevation, from S to Q is θ. The height of the one pole, PS = 4,95 m and the other pole QR = 9,48 m. The poles are connected to each other with two cables, namely PQ and SQ, with SQ = 15,64 m.

4.1Determine the size of the angle of elevation, θ. (2) 
4.2Determine the area of ∆PSQ. (4)
4.3Calculate the distance between P and Q, rounded off to TWO decimal (5)
[11]

QUESTION 5
Given f (x) = 2sin x and x ∈[0°;360°]

5.1Use the figure in the SPECIAL ANSWER BOOK and draw on the same set of axes, the graph of g(x) = cos( x - 30°). (4)
5.2Use your graphs to answer the following
5.2.1Write down the amplitude of f .(1)
5.2.2Write down the range of g . (2)
5.2.3Which values of x are f (x) - g(x) = 1, 5 ? (1)
5.2.4Which values of x are f (x) ≤ g(x) ? (3)
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Give reasons for ALL your statements in QUESTION 6, 7 AND 8.
QUESTION 6
6.1Complete the following statement:
A line parallel to one side of a triangle, divides the other two sides ..(1)
6.2In ∆ABC, AP = 3, PB = 6, AQ = 2 and QC = 4 PQ || BC and RS || BA.

If CR : RB = 1 : 3, calculate QS. (7) 
[8]

QUESTION 7
7.1 In the right-angled triangle ABC, AC = 11,82 cm, BC = 8,36 cm and ACˆB = 45°. In ∆DCB, CD = 5,91 cm and  DCˆ B = 45°.

7.1.1 Write down with reasons the lengths of AB and (3) 
7.1.2 Determine the following ratios: AB , BC and AC   (3)
                                                          BD   DC       BC       
7.1.3 Use your answers obtained in QUESTION 1.2 to make a conclusion about ∆ABC and ∆BDC.(1)
7.2 In the diagram below, DRS is a tangent to the circle, ARTM. Chords AT and RM intersect at P.

7.2.1 Determine the size of the following angles:

1. Rˆ 3(2)
2. Rˆ 4(2)

7.2.2Write down two angles that are equal to Aˆ   (2)
7.2.3Given ∆APR ||| ∆MPT with PR = 11,11 cm, AP = 8,29 cm and AT = 12,01 cm, calculate the length of MP. (5)
[18]

QUESTION 8
8.1 In the circle below, O is the centre, PO is a radius extended to bisect WR at T. PR = 9,47 units and TP = 8,43 units

Calculate the length of the chord WR. (6) 
8.2 In the diagram below, O is the centre of the circle and BAˆO = 40°.

Calculate the size of the angle  BCˆA .(6) 
[12]

QUESTION 9
9.1 In the diagram below, O is the centre of the circle AOˆB = 108,5° . The diameter of the circle is 10,84 cm in length.

9.1.1Convert 108,5° to (2) 
9.1.2Determine the radius. (1)
9.1.3Determine, d, the length of the arc of the shaded (3)
9.1.4Determine the area of the shaded (3)
9.2 In the diagram below the length of chord, CD = 8,76 units. The height of the smaller segment, FG = 3,15 units

Calculate EF, the length of the diameter of the circle.(4)
9.3 The picture below represents a piece of off-cut metal. To determine the area of the metal, it is divided into equal sections on the horizontal with different heights as illustrated in the diagram.

Calculate the approximated area of the piece of metal by using the mid-ordinate rule. Measurements are in mm. Give your answer in cm².(5) 
[18]

QUESTION 10
10.1 A wheel of diameter 570 mm rotates at 15 revolutions per second

Calculate: 
10.1.1The angular velocity (3)
10.1.2The circumferential velocity in m/s (4)
10.1.3The distance that a point on the circumference of the wheel will cover in 2 min, to the nearest meter (4)
10.1.4The angular displacement during 0,3 s (3)
10.2 A shape as shown below is made from a cylinder, 50 mm in diameter and 100 mm long with a hemisphere on one end

Determine the volume of the shape. (7)
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TOTAL: 150