Technical Mathematics Grade 12 Term 3 September Past Papers and Memos

1. Q1 1.1.1 x ( 3x −1) = 0 (2) 22 marks
2. Q2 2.1 Given: x2 + 2x + 1 = 0 9 marks
3. Q3 3.1 Simplify the following, showing ALL calculations, where applicable: 22 marks
4. Q4 4.1 Given functions g and h defined by g ( x ) = − 3 and h ( x ) = 2− x − 3 respectively. 28 marks
5. Q5 5.1 A lump sum of R50 000 is invested at 10,5% per annum compounded monthly. 15 marks
6. Q6 6.1 Given: f ( x ) = 1 − x 17 marks
7. Q7 Given: g ( x ) = x3 + ax2 − 5x − 3a 17 marks
8. Q8 The voltage across a capacitator in an electrical circuit is modelled by the function 8 marks
9. Q9  ( 7 − 2 x + 9 x ) dx 12 marks

Instructions to candidates

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

1. Q1 In the diagram below line AB is inclined at an angle θ to the x-axis. A(- 4 ; - 2) and B(4 ; 4). 15 marks
2. Q2 2.1 In the diagram below, O(0 ; 0) is the centre of the circle defined by 𝑥 2 + 𝑦 2 = 𝑟 2 . 12 marks
3. Q3 3.1 Given that: X = 27° and Y = 123,57° 13 marks
4. Q4 4.1 Simplify the following: 12 marks
5. Q5 The following functions are defined by 𝑓(𝑥) = acos 2𝑥 and 𝑔(𝑥) = b sin (𝑥 + p) for 7 marks
6. Q6 The picture below shows a triangular roof top for a dog house. Below it is a diagram giving the 9 marks
7. Q7 7.1 Complete the following theorem: 12 marks
8. Q8 8.1 Complete the following theorem: 23 marks
9. Q9 9.1 Complete the following statement: 12 marks
10. Q10 10.1 A wheel of a motor bike rotates at 320 rev / min. 21 marks
11. Q11 11.1 The diagram below shows an irregular figure with one straight side divided into five 14 marks

Instructions to candidates

1. This question paper consists of ELEVEN questions.
2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided.
3. Clearly show ALL calculations, diagrams, graphs, et cetera which you have used in determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non- graphical) unless stated otherwise.
6. If necessary, round off your answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. An information sheet with formulae is included at the end of the question paper.
9. Write neatly and legibly. EC/SEPTEMBER 2026) TECHNICAL MATHEMATICS P2 3

1. Q1 1.1.1 2xx1−=0  x = 0 A 23 marks
2. Q2 12 – 3q < 0  q > 4 CA 6 marks
3. Q3 2xx  exponential property/ 24 marks
4. Q4 4.1.2 ������ = 0−2  substitution/vervanging A 33 marks
5. Q5 5.1 i m  substitution/vervanging A 16 marks
6. Q6 6.1 f (x +−h) f (x) 16 marks
7. Q7 7.1 f (x) = (x −2)(x−2)(x+3) (xx−+2)( 3) A 16 marks
8. Q8 8.1 v (0) = 9 m/s  9 m/s A 6 marks
9. Q9 9.1.1 −+4 x  dx  −2x2 A 10 marks

1. Q1 1.1.2 2 (Correct to TWO decimal places). 23 marks
2. Q2 The solutions to a quadratic equation are: 6 marks
3. Q3 3.1 Simplify the following WITHOUT using a calculator: 24 marks
4. Q4 4.1 The sketched graphs below represent the functions defined by: −x and 33 marks
5. Q5 5.1 A city’s population is growing at an annual rate of 3% compounded every 3 months. 16 marks
6. Q6 6.1 Determine  by using FIRST PRINCIPLES if 16 marks
7. Q7 Given: 2 . The graph has 3 roots defined by 16 marks
8. Q8 A car moves along a straight road with its velocity given by the function: 6 marks
9. Q9 9.1 Determine the following integrals: 10 marks

Instructions to candidates

1. This question paper consists of NINE questions.
2. Answer ALL the questions.
3. Answer QUESTION 4.2.5 and QUESTION 7.4 on the ANSWER SHEETS provided. Write your name, surname and class in the spaces provided on the ANSWER SHEETS and hand in the ANSWER SHEETS with your ANSWER BOOK.
4. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers.
5. Answers only will NOT necessarily be awarded full marks.
6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
7. If necessary, round off answers to TWO decimal places, unless stated otherwise.
8. Diagrams are NOT necessarily drawn to scale.
9. An information sheet with formulae is included at the end of the question paper.
10. Write neatly and legibly.

1. Q1 Die diagram hieronder het hoekpunte R −5 ; 3 ; S 4 ; 5 ,T(x ; y) en U(1 ; −6). Waar 17 marks
2. Q2 2.1 In diagram hieronder is O(0 ; 0), die middelpunt van die sirkel gedefinieer deur 11 marks
3. Q3 3.1 Gegee dat: ������������ = 1,271 rad 15 marks
4. Q4 4.1 Vereenvoudig die volgende: 11 marks
5. Q5 Gegee die funksies en vir x∈[0 ; 360 ]. 12 marks
6. Q6 ʼn Skerpskieter staan by punt S en kyk op na punt T, die bopunt vaʼn n boom waar sy teiken, ʼn 14 marks
7. Q7 In die diagram hieronder is, TA die raaklyn aan die sirkel met middelpunt O. B, D en T is punte 11 marks
8. Q8 In die diagram hieronder is, sirkel PQRSmet middelpunt O geteken. Dit word verder gegee dat 15 marks
9. Q9 9.1 Voltooi die volgende stelling: 10 marks
10. Q10 10.1 ʼn Huishoudelike waaier het lemme wat 25 cm lank is en draai teen 200 revolusies per 22 marks
11. Q11 11.1 ʼn Onreëlmatige figuur met een reguit sy met ʼn lengte van 28 m wat in sewe gelyke 12 marks

Instructions to candidates

1. Hierdie vraestel bestaan uit ELF vrae.
2. Beantwoord AL die vrae in die SPESIALE ANTWOORDEBOEK wat verskaf word.
3. Dui ALLE berekeninge, diagramme, grafieke, ensovoorts wat jy gebruik het om die antwoorde te bepaal, duidelik aan.
4. Volpunte sal NIE noodwendig aan slegs antwoorde toegeken word NIE.
5. Jy mag ʼn goedgekeurde wetenskaplike sakrekenaar (nieprogrammeerbaar en niegrafies) gebruik, tensy anders vermeld.
6. Indien nodig, rond jou antwoorde tot TWEE desimale plekke af, tensy anders vermeld.
7. Diagramme is NIE noodwendig volgens skaal geteken NIE.
8. ʼn Inligtingsblad met formules is aan die einde van die vraestel ingesluit.
9. Skryf netjies en leesbaar.

1. Q1 x + 17 x – 23 = 0 (2) 24 marks
2. Q2 Given: f (x) = ax – 3x + 2 7 marks
3. Q3 3.1 Simplify the following without using a calculator: 21 marks
4. Q4 Given the function f, defined by fx= − 2 −1 28 marks
5. Q5 5.1 An asset valued at R15 000 depreciates at a rate of 3% per annum compounded 15 marks
6. Q6 Determine by using FIRST PRINCIPLES if 17 marks
7. Q7 Given: f (x) = – x – 4x – 3x 18 marks
8. Q8 The second Newton’s Law of Motion of a body falling due to gravity is given by: 8 marks
9. Q9 9.1 Given f (x) = x – 8 12 marks

Instructions to candidates

1. This question paper consists of NINE questions.
2. Answer ALL the questions.
3. Answer QUESTIONS 4.1.3 and 7.4 on the ANSWER SHEETS provided. Write your name and school’s name in the spaces provided on the ANSWER SHEETS and hand in the ANSWER SHEETS with your ANSWER BOOK.
4. Number the answers correctly according to the numbering system used in this question paper.
5. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers.
6. Answers only will NOT necessarily be awarded full marks.
7. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise.
8. If necessary, round off answers to TWO decimal places, unless stated otherwise.
9. Diagrams are NOT necessarily drawn to scale.
10. An information sheet with formulae is included at the end of the question paper.
11. Write neatly and legibly.

1. Q1 In the diagram below ABCD is a parallelogram with vertices A(− 1 ; 4), B(3 ; 6), C and 17 marks
2. Q2 2.1 The diagram below shows a circle with equation ������2 + ������2 = 13 and a tangent line 8 marks
3. Q3 3.1 Given: P = 128,2˚ and S = 204,7˚ 15 marks
4. Q4 sin(180° − θ).cot (180°+θ).sin270° 11 marks
5. Q5 Given ������(������) = tan������ and ������(������) = sin������ − 1; ������ ∈ (0°;360°) 12 marks
6. Q6 AB is a tower, anchored at point C, forming an angle of elevation 35°. 10 marks
7. Q7 7.1 Complete the following theorem statement: 16 marks
8. Q8 8.1 Complete the following theorem statement: 9 marks
9. Q9 9.1 Complete the following theorem statement: 18 marks
10. Q10 10.1 A train moving on a circular track with a diameter of 14 6425 km, takes 50 minutes to 21 marks
11. Q11 11.1 The irregular shape, with area 149,38 cm2, below has a straight side of length x cm 13 marks

Instructions to candidates

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

1. Q1 1.1.1 0= (3������−2)(������+1) (2) 21 marks
2. Q2 The nature of the roots of: (3) 7 marks
3. Q3 3.1 Simplify without the use of a calculator (show all steps): 24 marks
4. Q4 4.1 Given the function f (x)   6 2 and g x  x2 x6 27 marks
5. Q5 5.1 An asset worth R800 000 depreciated once off at a rate of 8,8%. Determine the current 13 marks
6. Q6 6.1 Determine the derivative of f (x) = 1 x+ by using FIRST PRINCIPLES. 16 marks
7. Q7 Given : h (x) = 2x + x – 5x + 2: 20 marks
8. Q8 The picture below shows water ejection from a stationary fire extinguishing truck. Ignoring 10 marks
9. Q9 9.1 Determine the following integrals: 12 marks

Instructions to candidates

1. This question paper consists of NINE questions.
2. Answer ALL the questions.
3. Answer QUESTIONS 4.1.5 and QUESTION 7.5 on the ANSWER SHEETS provided. Write your name and school name in the spaces provided on the ANSWER SHEETS and hand in the ANSWER SHEETS with your ANSWER BOOK.
4. Number the answers correctly according to the numbering system used in this question paper.
5. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers.
6. Answers only will NOT necessarily be awarded full marks.
7. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
8. If necessary, round off answers to TWO decimal places, unless stated otherwise.
9. Diagrams are NOT necessarily drawn to scale.
10. An information sheet with formulae is attached at the back of the question paper.
11. Write neatly and legibly.

1. Q1 In the diagram below A(3; 2), B 6; 3 and C are the vertices of ABC. 13 marks
2. Q2 In the diagram below, the equation of the circle is given by 13 marks
3. Q3 If and , calculate the values of the following (rounded off to TWO 22 marks
4. Q4 Given f (x) sin3x and g(x)cosx for x[0;180] 14 marks
5. Q5 In the diagram below ABC is given with AB = 8 units, BD = 6 units. 11 marks
6. Q6 6.1 Complete the following statement: 16 marks
7. Q7 7.1 Complete the following statement: 10 marks
8. Q8 8.1 Complete the following statement: 18 marks
9. Q9 In the diagram below, O is the centre of the circle with and OD = 5 cm. 15 marks
10. Q10 10.1 A petrol lawn mower has a pull chord to enable the engine to start. In order for the 12 marks
11. Q11 The area of an irregular metal sheet, with one side a straight side, is 256 m . The ordinates 6 marks

Instructions to candidates

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, et cetera which you have used in determining the answers.
4. Answers only will NOT necessarily be awarded full marks.
5. You may use an approved scientific calculator (non-programmable and non-graphical) unless stated otherwise.
6. If necessary, round off your answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. An information sheet with formulae is included at the end of the question paper.
9. Write neatly and legibly.