TECHNICAL MATHEMATICS PAPER 2
GRADE 12
NATIONAL SENIOR CERTIFICATE
NOVEMBER 2019

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 11 questions.
2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided.
3. Clearly show ALL calculations, diagrams, graphs, etc. that you have used to determine your answers.
4. Answers only will NOT necessarily be awarded full marks.
5. If necessary, round off answers to TWO decimal places, unless stated otherwise, Diagrams are NOT necessarily drawn to scale.
6. You may use an approved scientific non-graphical), unless stated otherwise. calculator (non-programmable and
7. An information sheet with formulae is included at the end of the question paper,
8. Write neatly and legibly.

QUESTION 1
The building shown in the picture below, has sides in the form of quadrilaterals. Quadrilateral ABDF models a side view of the building, in the Cartesian plane, with vertices A(-2:8), B, D and F(-8;-4).
The acute angle formed by the x-axis and BD is 76 and D = a
FD is parallel to the x-axis.
(The diagram is NOT drawn to scale.)

Determine:
1.1 The size of a
1.2 The length of AF (leave answer in simplified surd form)
1.3 The gradient of BD (rounded off to the nearest integer)
1.4 The coordinates of the midpoint of AF
1.5 Hence, the equation of the perpendicular bisector of AF in the form y

QUESTION 2
2.1 In the diagram below, O (0:0) is the centre of circle ABC with A(-4; -3) and C(4; -3). Tangents PQ and MN touch the circle at B and A respectively. The equation of tangent MN is given by y= -⁴/₃x - ²⁵/₃ MN II PQ

2.1.1 Determine the equation of the circle.
2.1.2 Write down:

1. The coordinates of B
2. The gradient of PQ

2.1.3 Hence, determine the equation of tangent PQ in the form y=...
2.2 Given the graph defined by the equation: x² + 8y² - 32 = 0
2.2.1 Express the equation in the form: x ² + y  ² = 1
                                                              a²     b²
Leave your answer in simplified surd form.
2.2.2 Hence, sketch the graph on the set of axes provided. Clearly show ALL the intercepts with the axes.

QUESTION 3
3.1 Given: θ = 20° and a = 32°
Calculate the numerical value of:
3.1.1 sin 3a
3.1.2 sec θ -1
           tan a
3.2 Given the sketch below:

Determine the following in terms of m:
3.2.1 sin35º
3.3 Given: 2cosθ
3.2.2
3.3.1 Show that the above equation can be expressed as tanθ = - 2
3.3.2 Hence, or otherwise, determine the value(s) of θ.

QUESTION 4
4.1 Simplify:
4.1.1 cot²2β - cosec²2β
4.1.2 tan²A cosec²A - cos 2π
4.2 Given the identity:
cosec (180° + θ). sin (360°- θ) - [sin (180° + θ))^{sec 60º} = cos θ
4.2.1 Write down the numerical value of sec 60°.
4.2.2 Hence, prove the identity.

QUESTION 5
The graphs below represent the curves of functions f and g defined by f(x) = asinx and g(x)=-cos bx respectively for x ∈ [0° ; 180°].
R(21,5°; -0,7) and T are the points of intersection of the curves of and g.

Use the graphs above to answer the following questions.
5.1 Give the period of f.
5.2 Determine the numerical values of a and b.
5.3 Write down the coordinates of T.
5.4 Determine the value(s) of x for which:
5.4.1 g(x) . f(x) > 0 for x ∈ [ 90°; 180°]
5.4.2 f(x)  will be undefined
        g(x)

QUESTION 6
In the diagram below, AB represents a vertical tower. Mpolokeng is standing at point C which is 150 m away from base B of the tower. The angle of elevation of A from C is 50º. Mpolokeng then walks 300 m up an inclined road to point D. BC is produced to E such that BE ⊥ DE. Points A, B, C, D and E lie in the same vertical plane. ACD = β > 90°

6.1 Calculate the distance of AC.
6.2 Hence, determine the size of β, if the area of AACD= 3,3648 x 10⁴ m².
6.3 Hence, determine the distance of AD.

QUESTION 7
7.1 Complete the following theorem:
The line drawn from the centre of a circle to the midpoint of a chord is ...
7.2 In the diagram below, O is the centre of circle CADB.
CD is the diameter of the circle.
CED⊥AB with E on AB.
AB 8 units, OC = 5 units, AC = 4√5 units, EO = x units and DBE = 26,6º

7.2.1 Determine, with reasons, the size of each of the following angles:

1. C.
2. A
3. B₁

7.2.2 Write down, without giving reasons, the length of:

1. AE
2. ED in terms of x 

7.2.3 Hence, or otherwise, determine the numerical value of x.

QUESTION 8
8.1 Complete the following theorem: (1)
The angle subtended by the diameter at the circumference of the circle is ...
8.2 In the diagram below, O is the centre of circle BCDE with diameter BE.
AB is a tangent to the circle at B.
BC and ED are produced to meet at F.
CO || FE
ABF = 30°

8.2.1 Determine, with reasons, the size of each of the following angles:

1. CBO
2. D₂
3. O₁
4. O₂

8.2.2 Show, with reasons, that FC = FD

QUESTION 9
9.1 Complete the following theorem:
If a line divides two sides of a triangle in the same proportion, then the line is... (1)
9.2 The diagram below shows circle DGF with centre O having a radius of 6 units.
OD and OF are produced to B and C respectively.
BC is a tangent to the circle at G.
OG and DF intersect at E such that OG⊥DF.
OD: OB = 3:5

9.2.1 Show, with reasons, that DF || BC.
9.2.2 Determine:

1. The ratio of BC: DF
2. The length of EG
3. The numerical value of Area Δ OBG
                                         Area Δ ODE

9.3 Show, with reasons, that ADOE IA BOG.

QUESTION 10
10.1 The pictures below show a wheelbarrow and an enlargement of the wheel of the wheelbarrow. The wheel consists of a tyre and a rim with a circular hole in the centre of the rim for the axle.

The diagram alongside shows the wheel of the wheelbarrow. O is the common centre of the hole, the rim and the tyre. Chord MN of the rim is also a tangent to the hole at point B. Chord KL of the tyre is the tangent of the rim at point A. OBAC is a straight line with C a point on the outer circle.

The diameter of the wheel is 40 cm. The angular velocity of the rotating wheel, when the wheelbarrow is pushed, is 64π revolutions per minute. The length of the radius of the hole is 1,5 cm.
10.1.1 Give the length of BC.
10.1.2 Hence, determine the length of AB if the length of chord KL is 32 cm. (5)
10.1.3Calculate the rotational frequency (1) of the rotating wheel
10.1.4Hence, determine the circumferential velocity of the rotating wheel.
10.2 The picture below shows a wheel consisting of a rubber tyre and a rim with equally spaced spokes. The diagram models the wheel with the centre. OA and OB are two radii (spokes) of the rim. The length of a spoke is 5,2 cm.

10.2.1 Determine the size, in radians, of acute angle AOB. Determine the
10.2.2 Hence, or otherwise, determine (correct to ONE decimal place) the length of minor are AB.
10.2.3 Hence, or otherwise, determine the area (to the nearest cm) of minor sector AOB.

QUESTION 11
11.1 An irregularly shaped portion, as shown in the picture below, was cut from a rectangular iron sheet which has an area of 19,125, m. The diagram alongside the picture models the irregular shaped portion which has one straight side, 7,5 m long, which is divided into 5 equal parts.

The six ordinates dividing the equal parts: 1,8 m: 2 m:b: 1,3 m; 1,8 m and q
11.1.1 Write down the length (correct to ONE decimal place) of b, if b is the square root of the length of the second ordinate from the left. (1)
11.1.2 Hence, determine the length (correct to ONE decimal place) of 4, if the area of the irregular shaped portion is two thirds of the area of the iron sheet.
11.2 A greenhouse structure, as shown in the diagram below, has a shape made up of a combination of a rectangular right prism and half of a right cylinder. The length, breadth and height of the rectangular part of the greenhouse is 3,5 m, 10 m and h respectively. The volume of the rectangular part of the greenhouse is 70 m³.

The following formulae may be used:
Total surface area of a right cylinder = 2πr² + 2xπh
Area of a rectangle = length breadth
Volume of a right prism = (area of the base) height
11.2.1 Determine the height (h) of the rectangular prism. 
11.2.2 Write down the length of the radius of the half cylinder.
11.2.3 Show whether the total external surface area of the greenhouse is less than 120 m²
TOTAL: 150