Mathematics Grade 12 Term 3 September Past Papers and Memos

1. Q1 1.1.1 2 x ( x − 3) = 0 (2) 25 marks
2. Q2 Sequences and Series2.1 Given the arithmetic series: 7 +16 + 25 + 34 + ... 20 marks
3. Q3 AlgebraGiven the quadratic number pattern: −3 ; − 6 ; −11 ; ... 6 marks
4. Q4 Analytical GeometryThe sketch below shows the graphs of f ( x ) = −2x + k and g ( x ) = +q 15 marks
5. Q5 5.1 The graphs of f ( x ) = ax2 + bx + c and g ( x ) =; x  0 are drawn below. 20 marks
6. Q6 Finance6.1 A father invests money for his daughter on her fifth birthday. On her twenty-first birthday 13 marks
7. Q7 7.1 Given the function: f ( x ) = 3x 2 − 1 13 marks
8. Q8 Functions and Graphs8.1 Given the function: f ( x ) = − x 3 + 3x 2 + 9 x + 5 17 marks
9. Q9 A piece of wire, three metres (3 m) long, is bent into the frame work of a rectangular prism with 7 marks
10. Q10 Probability• The probability that event A will occur is 0,3 9 marks
11. Q11 Three representatives from four different departments of a company will be featured on a photo 5 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 that you have used in determining your answer.
4. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
5. Answers only will not necessarily be awarded full marks.
6. If necessary, round off answers to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Number the answers correctly according to the numbering system used in this question paper.
9. An information sheet with formulae is included at the end of the question paper.
10. Write neatly and legibly.

1. Q1 StatisticsThe ogive (cumulative frequency curve) below shows the number of social media messages sent 11 marks
2. Q2 StatisticsFrom a school sports team, eight young athletes were chosen. The table below shows their 10 marks
3. Q3 Analytical GeometryIn the diagram below, S (5 ; 2 ), Q (-9 ; 0 ) and R are vertices of ∆ SQR. Point T is the 19 marks
4. Q4 Analytical GeometryA circle with centre E passes through the origin. The circle intersects the x-axis at G and the 20 marks
5. Q5 Trigonometry5.1 Given: sin 49° = q, determine the following in terms of q: 17 marks
6. Q6 Trigonometry6.1 Given: cos (A − B) = cos AcosB + sinAsinB 14 marks
7. Q7 TrigonometryGiven: f (x ) = a cos(x + p ) where x  − 180 0 ;180 0  10 marks
8. Q8 TrigonometryIn the diagram, Q, P and R are points on the same horizontal plane. S is the top of a 8 marks
9. Q9 Euclidean Geometry9.1 Complete the theorem below: 11 marks
10. Q10 Euclidean Geometry10.1 In the diagram below, ∆PQW is drawn with R on side PQ and S on side PW. RS is joined 12 marks
11. Q11 In the diagram below, A, R, T and C are points on the circumference of the circle. B is centre 18 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, etc. 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 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 Algebra, equations and inequalities1.1.2 2 (korrek tot TWEE desimale plekke) (3) 25 marks
2. Q2 Number patterns, sequences and series2.1 Twee leerders, Amanda en Leroy was ʼn taak gegee om ʼn gemeenskaplike verhouding (r) 11 marks
3. Q3 Number patterns, sequences and series3.1 Die volgende inligting van ʼn kwadratiese getalpatroon word gegee: 13 marks
4. Q4 Functions and graphsGegee die funksie: , vir 16 marks
5. Q5 Functions and graphsDie grafieke van en is hieronder geteken. Die 20 marks
6. Q6 Finance, growth and decay6.1 Lester belê R8 000 teen 13% per jaar, kwartaalliks saamgestel. Aan die einde van die 16 marks
7. Q7 Differential calculusBepaal vanuit eerste beginsels as (4) 10 marks
8. Q8 Functions and graphs8.1 Gegee die funksie: f x =−x +5x +8x−12 16 marks
9. Q9 Differential calculusDie som van twee getalle is 25. Bepaal die twee getalle, sodat die kwadraat van die een getal 7 marks
10. Q10 Probability and counting10.1 Twee onderling uitsluitende gebeurtenisse, A en B is sodat en 11 marks
11. Q11 Probability and countingKodes van 3 simbole word gevorm deur die ses-en-twintig letters van die alfabet (A–Z) en die 5 marks

Instructions to candidates

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

1. Q1 StatisticsDie ouderdomme (in jare) van besoekers wat ʼn gastehuis (B&O) op ʼn spesifieke dag 10 marks
2. Q2 Statisticsʼn Atleet se eindtye en die daaglikse temperature was gemonitor gedurende 10 jaar se 9 marks
3. Q3 Analytical geometryA(‒3; ‒8), B, C(3; 4) en D(3; ‒ 6) is hoekpunte van ʼn vierhoek. Hoeklyne BD en AC sny by 23 marks
4. Q4 Analytical geometryIn die diagram hieronder is, x2 + y2 −2x+4y −3= 0 die vergelyking van die groter sirkel 19 marks
5. Q5 TrigonometryGegee: tan = −5 , waar 900  2700. Met behulp van ʼn skets en sonder die 26 marks
6. Q6 TrigonometryGeskets hieronder is die grafiek van f (x)= cos x − 2 in die interval van x −120 ;240 12 marks
7. Q7 TrigonometryIn die diagram hieronder, is PS ʼn vertikale toring wat op ʼn horisontale vlak QRS staan. 10 marks
8. Q8 Euclidean geometry8.1 In die diagram hieronder, is O die middelpunt van die sirkel. Die sirkel gaan deur 11 marks
9. Q9 Euclidean geometryIn die diagram hieronder is, A, B en E punte op die omtrek van die sirkel. AB is verleng 19 marks
10. Q10 Euclidean geometryIn die diagram hieronder is, PQRS ʼn reghoek. PR sny XY by W. XY ⊥ PS. 11 marks

Instructions to candidates

1. Hierdie vraestel bestaan uit 10 vrae.
2. Beantwoord AL die vrae in die SPESIALE ANTWOORDEBOEK wat voorsien word.
3. Dui ALLE berekeninge, diagramme, grafieke, ens. wat jy in die beantwoording van die vrae gebruik het, duidelik aan.
4. Slegs antwoorde sal NIE noodwendig volpunte verdien NIE.
5. Jy mag ʼn goedgekeurde wetenskaplike sakrekenaar gebruik (nieprogrammeerbaar en niegrafies), tensy anders vermeld.
6. Indien nodig, rond 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 Statistics1.1 A school’s hockey team recorded the number of push-ups each player completed in a 11 marks
2. Q2 StatisticsThe table shows the percentages scored by a sample of 15 candidates in the third term and final 9 marks
3. Q3 Analytical geometryIn the diagram below, D(4; 5), R(-2; 2), T and S form a quadrilateral. RD cuts the y-axis at N 18 marks
4. Q4 Analytical geometryIn the diagram below, a circle, centered at ( ), touches the x-axis at S and the line OA 22 marks
5. Q5 Trigonometry5.1 If sin 54 = p , express each of the following in terms of p, without the use of a 30 marks
6. Q6 TrigonometryIn the figure below, B, C and D are points on the same horizontal plane. AB is a vertical 9 marks
7. Q7 TrigonometrySketched below is a graph of f (x) =cos(x−45) for −45° ≤ ������ ≤ 180°. Use the graph to 11 marks
8. Q8 Analytical geometryIn the diagram below, PR is a diameter of circle PQRS with centre O. PR intersects with 9 marks
9. Q9 Euclidean geometryIn the diagram below, O is the centre of the circle and KP is a tangent to the circle. LN, the 11 marks
10. Q10 Euclidean geometryIn the diagram below, ∆PQR and ∆MNO are given with P = M, Q = N and R = O. 20 marks

Instructions to candidates

1. This question paper consists of 10 questions.
2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided.
3. Clearly show ALL calculations, diagrams, graphs, etc. 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 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 Algebra, equations and inequalities1.1.1 xx+4 −21=0 (2) 25 marks
2. Q2 Number patterns, sequences and series2.1 An arithmetic series has a common difference of 4. (3x−1) and (2x+8)are the 16 marks
3. Q3 Number patterns, sequences and series3.1 Given that p = 0,7 =0,777777... 10 marks
4. Q4 Functions and graphsIn the diagram below the graph of a hyperbolic function, fx( ) = , where k is a 10 marks
5. Q5 Functions and graphsGiven the function: fx( ) = −3x +1 10 marks
6. Q6 Functions and graphsThe diagram below shows the graphs of f (x) = x2 −4x−11 and g(x) = f /(x). A and B 13 marks
7. Q7 Finance, growth and decay7.1 Corniel bought an ice cream machine that depreciated(become less) at 17% p.a. on the 15 marks
8. Q8 Differential calculusDetermine from first principles if f (x) =−x 3x . 11 marks
9. Q9 Functions and graphs9.1 Calculate the coordinates of the turning points of f. (4) 18 marks
10. Q10 Differential calculusThe diagram below shows a rectangle OABC, where B lies on the straight line 6 marks
11. Q11 Probability and counting11.1 During a survey(study) at a certain school, 900 learners were asked to indicate(show) 16 marks

Instructions to candidates

1. This question paper consists of ELEVEN questions. Answer ALL the questions.
2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.
3. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise.
4. Answers only will not necessarily be awarded full marks.
5. If necessary, round off answers to TWO decimal places, unless stated otherwise.
6. Diagrams are NOT necessarily drawn to scale.
7. Number the answers correctly according to the numbering system used in this question paper.
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 (3xx 1)(  2)  0 23 marks
2. Q2 Sequences and SeriesS 3(10) 2(10)  substitution 16 marks
3. Q3 Sequences and Series3.1 Black: 5; 6; 7  sequence 10 marks
4. Q4 4.1 xxR; 0 x  R x  0 23 marks
5. Q5 5.1 f : y 2x ; y 0 9 marks
6. Q6 1,138032482  simplification 15 marks
7. Q7 7.1 f (x)  5x  x  3 13 marks
8. Q8 Functions and GraphsThe graph is concave up when > 0 18 marks
9. Q9 Trigonometry9.1 h(x)  x  4x  3 7 marks
10. Q10 7 values need to be placed 8 marks
11. Q11 11.1 10 9! (9)  32 659 200 8 marks

1. Q1 StatisticsA hundred athletes took part in a long jump competition. The distance, in centimetres, of 10 marks
2. Q2 StatisticsThe following table shows a comparison of the distances (centimetres) jumped by 6 long 10 marks
3. Q3 Analytical geometryIn the diagram below, E(p; 8), F(–2; –3), G and H(8; 7) are vertices of rhombus EFGH. 17 marks
4. Q4 Analytical geometryIn the diagram below, a circle centred at R(4; 2) passes through the origin O, S and 23 marks
5. Q5 Trigonometry5.1 Given that: cos26°= p 30 marks
6. Q6 TrigonometrySketched below is the graph of f(x) = cos x , in the interval ������ ∈ [–180°; 180°]. 12 marks
7. Q7 TrigonometryIn the diagram below, TP represents the height of a building. The foot of the building P and 8 marks
8. Q8 Euclidean geometryIn the diagram, PRQT is a cyclic quadrilateral in the circle with QR = QT. Chord RT is 7 marks
9. Q9 Euclidean geometryIn the diagram, the two circles touch internally at M. MK is a common tangent to the circles. 12 marks
10. Q10 Euclidean geometryIn the diagram below, ΔABC and ΔDEF are given such that A = D, B = E and C = F. 21 marks

Instructions to candidates

1. This question paper has 10 questions.
2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided(given).
3. Show ALL calculations, diagrams, graphs, etc. which you have used in determining your 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. Round off answers correct 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.

1. Q1 Algebra, equations and inequalities1.1.1 xx+2 −15=0 (3) 24 marks
2. Q2 Number patterns, sequences and series2.1 Given the quadratic number pattern: −16 ; −16 ; −12 ; −4 ; ... 15 marks
3. Q3 Number patterns, sequences and series3.1 Given that in a geometric sequence T = 768 and T =12288. Determine the 9 marks
4. Q4 Functions and graphsThe diagram below shows the graphs of −3 and g(x) =bx,where b > 0. 16 marks
5. Q5 Functions and graphsThe diagram below shows the graph of f (x) = −x2 −2x+8. 17 marks
6. Q6 Finance, growth and decay6.1 Eli bought a laptop 4 years ago. The value of the laptop depreciates from 17 marks
7. Q7 Differential calculus7.1 Determine from first principles if f (x) =−5 2x2 . (5) 12 marks
8. Q8 Differential calculus8.1 The diagram below shows the graph of f (x) = 2x +bx +cx+d. 16 marks
9. Q9 Differential calculusThe diagram below shows a solid triangular prism. The triangle is right-angled with a 8 marks
10. Q10 Probability and counting10.1 A and B are two independent events such that P(A) = 0,2 and P(not B) = 0,45. 8 marks
11. Q11 Probability and countingIn a certain province vehicle number plate codes have the following format: 8 marks

Instructions to candidates

1. This question paper consists of ELEVEN questions. Answer ALL the questions.
2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answer.
3. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise.
4. Answers only will not necessarily be awarded full marks.
5. If necessary, round off answers to TWO decimal places, unless stated otherwise.
6. Diagrams are NOT necessarily drawn to scale.
7. Number the answers correctly according to the numbering system used in this question paper.
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, where x is a rational number. (2) 24 marks
2. Q2 Sequences and SeriesThe first three terms of the first differences of a quadratic sequence are 102 ; 108 ; 114 ; … 7 marks
3. Q3 Sequences and Series3.1 The first four terms of an arithmetic sequence are: 2 ; 3 ; 0,8 ; 1 ;… 10 marks
4. Q4 Sequences and SeriesThe sum of the first two terms of a geometric series with positive terms, ������ ≠ −1, is four 6 marks
5. Q5 Functions and GraphsGiven ������(������) = −1 9 marks
6. Q6 Analytical GeometryIn the diagram, the graphs of ������(������) = −������ + ������ + 2 and ������(������) = ������ −������ are drawn below. 19 marks
7. Q7 Functions and GraphsIn the diagram, the graph of ������(������) = ������������������ ������ is drawn. 7 marks
8. Q8 Finance8.1 An investor indicates that he will be able to treble the value of the investment at the end 12 marks
9. Q9 9.1 Determine ������/(������) from first principles given ������(������) = ������2 − ������������, where b is a (5) 12 marks
10. Q10 Analytical GeometryThe graph of ������/ , the derivative of f, is drawn below. ������(������) = ������������3 + ������������2 + ������������ + ������ ;������ ≠ 20 marks
11. Q11 The rectangular milk carton has a square base which holds 1 litre of milk. It has a specially designed 9 marks
12. Q12 ProbabilityA printing company uses 3 machines, A,B and C, to produce banners. 8 marks
13. Q13 In a survey conducted by 220 grade 12 learners in a school, the following data were collected. 7 marks

Instructions to candidates

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

1. Q1 Statisticslearners’ School Based Assessment (SBA) marks for the year. 10 marks
2. Q2 StatisticsThe speeds, in kilometres per hour, of cyclists that passed a point on the route of the Ironman 10 marks
3. Q3 Analytical geometryIn the diagram, P, Q (0 ; −2) and R (6 ; −4) are the vertices of triangle PQR. The equation 22 marks
4. Q4 Analytical geometryIn the diagram below, C is the centre of the circle defined by ������2 − 6������ + ������2 − 4������ + 9 = 0. 18 marks
5. Q5 Trigonometry5.1 If cos22° = p; determine the following in terms of p: 25 marks
6. Q6 Analytical geometryIn the diagram below, the graphs of f (x)  tanbx and ������(x) =cos x − 30° are drawn on the 8 marks
7. Q7 Euclidean geometryPoints L, M and N are in the same horizontal plane. KL is a vertical tower. The angle of 7 marks
8. Q8 Analytical geometry8.1 Complete the following theorem statement: 15 marks
9. Q9 Euclidean geometry9.1 Given that PC is a tangent to the circle ACB; BAP and BCQ are straight lines. 16 marks
10. Q10 Euclidean geometry10.1 In the diagram, ΔMON is drawn. S is a point on MO and T is a point on MN such 19 marks

Instructions to candidates

1. This question paper consists of 10 questions.
2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided.
3. Clearly show ALL calculations, diagrams, graphs, et cetera that 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 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 Statistics1. The following information represents the amount of maize exported to other 13 marks
2. Q2 StatisticsThe following information (in %) represent contributions made by the Agricultural and 7 marks
3. Q3 Analytical geometryIn the diagram, K (–1 ; 4) ; L(3 ; 6); M and N(4 ; 1) are vertices of a parallelogram. R is the 20 marks
4. Q4 Analytical geometryIn the diagram, T is the centre of two concentric circles. The larger circle has equation 20 marks
5. Q5 Trigonometry5.1 Use the diagram below to calculate, without the use of a calculator, the following 24 marks
6. Q6 TrigonometryIn the diagram below, the graph of f (x) = 2sinx is drawn for the interval x[−120 ; 180]. 11 marks
7. Q7 Euclidean geometryIn the diagram, PQRS is a cyclic quadrilateral with QS = QR and PQ = PS = d units. QPS = ������. 5 marks
8. Q8 Euclidean geometry8.1 In the diagram, ABCD is a cyclic quadrilateral in the circle centered at O. ED is a tangent to 14 marks
9. Q9 Euclidean geometryIn the diagram, PV and VQ are tangents to the circle at P and Q. PQ is produced to S and 12 marks
10. Q10 Euclidean geometry10.1 In the diagram below, ∆ABC and ∆DEF are drawn with ˆ ˆ ˆ ˆ ˆ ˆ . 24 marks

Instructions to candidates

1. This question paper consists of 10 questions.
2. Answer ALL questions.
3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.
4. Answers only will not necessarily be awarded full marks.
5. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.
6. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise.
7. Diagrams are NOT necessarily drawn to scale.
8. Number the answers correctly according to the numbering system used in this question paper. Write neatly and legibly. Mathematics P2 3 September 2019 Preparatory Examination NSC