Mathematics Grade 11 P2 North West November 2024 QP (Eng)

PROVINCIAL ASSESSMENT GRADE 11 MATHEMATICS P2 NOVEMBER 2024 MARKS: 150 TIME: 3 hours This question paper consists of 12 pages and 1 information sheet. Copyright reserved Please turn over

Mathematics/P2 2 NW/November 2024 Grade 11 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. 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. If necessary, round off answers correct to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. An information sheet with formulas is included at the end of the paper. 9. Write neatly and legibly. Copyright reserved Please turn over

Mathematics/P2 3 NW/November 2024 Grade 11 QUESTION 1 The traffic department set up a roadblock on the N12 between Potchefstroom and Klerksdorp. During the first hour, they stopped 11 minibus taxis. The following dataset is a summary of the number of passengers per minibus taxi. 18, 26, 25, 18, 16, 12 ,10, 8, 18, 17, 8 1.1 Calculate the mean number of passengers per taxi. (2) 1.2 Calculate the standard deviation for this dataset. (1) 1.3 Taxis in which the number of passengers is more than one standard deviation above the average can be considered overloaded. How many taxis were overloaded? (2) 1.4 During the second hour of the roadblock, a further 5 minibus taxis were stopped. The number of passengers in these 5 taxis are: x, 13, 18, y, 20. These 5 numbers are already arranged from smallest to largest. The entire dataset for the first and second hours is collectively represented on a box-and-whisker diagram. The new average number of passengers for the 16 taxis is 16. The five-number summary can be seen on the diagram below. 1.4.1 Discuss the skewness of the data. (1) 1.4.2 What percentage of the minibus taxis had more than 11 passengers? (1) 1.4.3 Calculate the value of x and y. (4) [11] Copyright reserved Please turn over

Mathematics/P2 4 NW/November 2024 Grade 11 QUESTION 2 The traffic department has also conducted further investigation into where the best locations would be on the N12 to install speed cameras. Part of the investigation involved a survey of the different speeds of vehicles on the N12 between Potchefstroom and Klerksdorp. The following table shows the results of the investigation: SPEED (d) FREQUENCY CUMULATIVE (in km/h) (number of vehicles) FREQUENCY 40 < 𝑑 ≤ 60 25 60 < 𝑑 ≤ 80 46 80 < 𝑑 ≤ 100 67 100 < 𝑑 ≤ 120 79 120 < 𝑑 ≤ 140 24 140 < 𝑑 ≤ 160 8 160 < 𝑑 ≤ 180 1 2.1 Use the table in the answer book and complete the cumulative frequency column. (2) 2.2 How many vehicles were observed in the survey? (1) 2.3 Illustrate the information in the table by drawing a ogive (cumulative frequency curve). (4) 2.4 Use the ogive to determine the median of the data. Write down only the median speed. (2) [9] Copyright reserved Please turn over

Mathematics/P2 5 NW/November 2024 Grade 11 QUESTION 3 In the figure quadrilateral KLMN with vertices K (7 ; -2), L (0 ; -3), M (8 ; -9) and N (9 ; -6) are given. Extend NK to intersect the x-axis and form the angle β. Extend LK to intersect the x-axis and form the angle θ. 3.1 Calculate the gradient of KN. (2) 3.2 Hence, determine the equation of the straight line KN. (2) 3.3 Calculate 𝛽, the inclination angle of KN. (3) 3.4 Calculate the size of α. (4) 3.5 Show that KL is perpendicular to KM. (2) 3.6 Prove that KLMN is a cyclic quadrilateral. (5) [18] Copyright reserved Please turn over

Mathematics/P2 6 NW/November 2024 Grade 11 QUESTION 4 In the diagram below, P(-1 ; 3), R(3 ; -1) and S(a ; b) are the vertices of ∆𝑃𝑅𝑆. Point T lies on the perpendicular bisector of PR. 4.1 Calculate the coordinates of T, the midpoint of PR. (2) 4.2 Calculate the length of PR. (Leave the answer in simplest surd form.) (2) 4.3 Determine the equation of ST, if ST⊥PR and 𝑚𝑃𝑅 = −1. (3) 4.4 Hence prove that a = b. (1) 4.5 If the area of ∆𝑃𝑅𝑆 = 12 square units and a, b < 0, determine the coordinates of S. (5) [13] Copyright reserved Please turn over

Mathematics/P2 7 NW/November 2024 Grade 11 QUESTION 5 Answer this question without the use of a calculator: 5.1 If 13𝑐𝑜𝑠𝜃 − 12 = 0 and 𝑡𝑎𝑛𝜃 is positive, using a sketch, determine the value of: 3 cos(360° − 𝜃) + 2sin(360° − 𝜃) 1 tan(180° + 𝜃) (7) 5 5.2 Simplify: tan 135° . 𝑠𝑖𝑛210° cos660° + 𝑐𝑜𝑠180° (6) 5.3 If 𝑐𝑜𝑠42° = 𝑝, determine the following in terms of p: 5.3.1 𝑐𝑜𝑠138° (2) 5.3.2 3𝑠𝑖𝑛48° (2) 5.3.3 𝑠𝑖𝑛(−2202°) (3) 5.4 Prove the following identity: sin 𝑥 1 1 + = (6) 1 + 𝑐𝑜𝑠𝑥 𝑡𝑎𝑛𝑥 𝑠𝑖𝑛𝑥 5.5 Given: (𝑎2 + 𝑏 2 )𝑡𝑎𝑛2 𝜃 = 2𝑎𝑏 with 𝑎 > 0, 𝑏 > 0 and 𝜃𝜖(0° ; 90°). Calculate the value of 𝑠𝑖𝑛𝜃 in terms of a and b. (5) [31] Copyright reserved Please turn over

Mathematics/P2 8 NW/November 2024 Grade 11 QUESTION 6 6.1 Solve the following equation for 𝑥 ∈ [−180° ; 180°]. cos(𝑥 − 30°) = 𝑠𝑖𝑛𝑥 (5) 6.2 Draw the graphs of 𝑓(𝑥) = cos(𝑥 − 30°) and 𝑔(𝑥) = sin 𝑥 for the interval 𝑥 ∈ [−180° ; 180°] on the same set of axes. Clearly indicate the intersections with the axes as well as the turning points on the graph. (5) 6.3 For which values of x would 𝑓(𝑥) − 𝑔(𝑥) ≤ 0; 𝑥 ∈ [0° ; 180°]? (2) [12] QUESTION 7 In the diagram below, circle ACD is given where AC is a diameter. DC is produced to B and AB = 8 units. 𝐵̂ = 30° and BC = 5 units. 7.1 Prove that AC = 4,44 units (rounded off to TWO decimal places.) (2) 7.2 Calculate the following: 7.2.1 𝐴̂1 (2) 7.2.2 𝐶̂2 (1) 7.2.3 AD (3) [8] Copyright reserved Please turn over

Mathematics/P2 9 NW/November 2024 Grade 11 QUESTION 8 A mine dumps its waste at a landfill site where it takes the shape of a cone shown in the diagram below. The radius of the circular base is r meters, the height is h meters and the angle between the slant height and the base of the cone is 50°. The circumference of the landfill site is 245 meters. 1 Volume of cone = 3 𝜋𝑟 2 ℎ Total surface area of cone = 𝜋𝑟 2 + 𝜋𝑟𝑠 8.1 Determine the height of the mine waste, correct to ONE decimal place. (3) 8.2 Determine the volume of the mine waste, rounded to the nearest cubic meter. (3) [6] Copyright reserved Please turn over

Mathematics/P2 10 NW/November 2024 Grade 11 QUESTION 9 9.1 In the diagram, AB is a chord of a circle with center O. Prove the theorem that states: If M is the midpoint of AB, then OM⊥AB. (5) 9.2 In the diagram below O is the center of the circle with AD = BD. OB = 20 mm and AB = 24 mm. Calculate the length of: 9.2.1 DE (4) 9.2.2 FB (3) [12] Copyright reserved Please turn over

Mathematics/P2 11 NW/November 2024 Grade 11 QUESTION 10 The circle is given below with TQA as a tangent. PS = SR and PQ = PR. PQ SR . SR is extended to A. Q ̂ 1 = 𝑥 and Q ̂ 3 = 𝑦. 10.1 Give, with reasons, THREE angles equal to y. (6) 10.2 Prove that ̂ R3 = 𝑥 (4) 10.3 Prove that 𝑆̂ = 180° − 𝑥. (1) 10.4 Calculate the value of x and y. (6) [17] Copyright reserved Please turn over

Mathematics/P2 12 NW/November 2024 Grade 11 QUESTION 11 The circle ASBK with center O is given below. Furthermore, a semicircle is drawn through points A, O, B, T and R. P lies on AR. 𝐴𝑂̂ 𝐵 = 140°. 11.1 Calculate the size of 𝑆̂. (2) 11.2 Prove that AS KB. (5) 11.3 Calculate the size of 𝐴𝑇̂𝑅. (2) 11.4 Prove that TP will be a tangent to a circle passing through points B, K, and T if 𝑅̂ = 80° and TR = TP. (4) [13] TOTAL: 150 Copyright reserved

Mathematics/P2 13 NW/November 2024 Grade 11 INFORMATION SHEET: MATHEMATICS GR11  b  b 2  4ac x 2a A  P(1  ni) A  P(1  ni) A  P(1  i) n A  P(1  i ) n  x  x2 y1  y 2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y 2  y1 y  mx  c y  y1  m( x  x1 ) m m  tan  x 2  x1 a b c In ABC:   sin A sin B sin C a 2  b 2  c 2  2bc. cos A 1 area ABC  ab. sin C 2   xi  x  n 2 x x  2  i 1 n n n( A) P ( A)  P(A or B) = P(A) + P(B) – P(A and B) n S  Copyright reserved