Mathematics Grade 12 P2 Western Cape Prelim 2024 QP

Downloaded from hlayiso.com MATHEMATICS PREPARATORY EXAMINATION GRADE 12 MATHEMATICS P2 SEPTEMBER 2024 MARKS: 150 TIME: 3 hours This question paper consists of 12 pages, 1 information sheet and an answer book of 22 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 September 2024 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 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 to determine 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 formulae is included at the end of the question paper. 9. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 September 2024 QUESTION 1 During a congress that took place in July, the ages of the 100 members who attended, were recorded in the following table: AGE (𝒙) NUMBER OF (IN YEARS) PEOPLE 15 < 𝑥 ≤ 25 8 25 < 𝑥 ≤ 35 14 35 < 𝑥 ≤ 45 22 45 < 𝑥 ≤ 55 37 55 < 𝑥 ≤ 65 𝑎 65 < 𝑥 ≤ 75 3 1.1 Calculate the value of 𝑎. (1) 1.2 Write down the modal class of the given data. (1) 1.3 Complete the cumulative frequency column in the table given in the ANSWER BOOK. (2) 1.4 Draw a cumulative frequency graph (ogive) to represent the data on the grid provided in the ANSWER BOOK. (3) 1.5 The congress organisers discovered that a mistake was made, and that 3 people counted in the range 45 < 𝑥 ≤ 55 should have been counted in the range 35 < 𝑥 ≤ 45. Calculate the estimated average age of the people who attended the congress after the mistake was corrected. (2) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 4 September 2024 QUESTION 2 The scatter plot below shows the number of heaters a company sold per month and the average temperature of that month in a certain year. Both the temperature in ℃ and the number of heaters sold, are given in the table below. MONTH J F M A M J J A S O N D TEMPERATURE (℃ ) 2 7 8 10 18 22 28 25 16 11 2 3 NUMBER OF HEATERS SOLD 98 100 75 67 24 18 6 11 40 55 88 95 120 100 NUMBER OF HEATERS SOLD 80 60 40 20 0 0 5 10 15 20 25 30 TEMPERATURE (°C) 2.1 Describe the correlation between the number of heaters sold and the average temperature per month. Verify your answer by referring to the correlation coefficient. (2) 2.2 Determine the equation of the least squares regression line for the data. (3) 2.3 Predict the number of heaters sold for a month where the average temperature is 20℃. (2) 2.4 Draw the least squares regression line on the grid given in the ANSWER BOOK. (2) 2.5 Calculate the standard deviation of the number of heaters sold. (2) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 September 2024 QUESTION 3 In the diagram below HS and TB are the diagonals of parallelogram HTSB. • HS and TB intersect at M(1; 2). • TB intersect the 𝑥- and 𝑦-axes at the origin. • T(𝑥; 6) and S(4; 1) are vertices of HTSB. • MT ̂S = 𝛼 and the inclination angle of TB is 𝜃. 3.1 Determine: 3.1.1 the equation of the line TB in the form 𝑦 = 𝑚𝑥 + 𝑐. (2) 3.1.2 the size of 𝜃. (2) 3.1.3 the coordinates of H. (3) 3.2 Show that the 𝑥-coordinate of T is 3 if it is further given that TS = √26. (3) 3.3 ̂S. Calculate 𝛼, the size of MT (4) 3.4 Calculate the area of ∆BTS. (4) 3.5 Determine the perpendicular height of the parallelogram if TS is the base. (3) [21] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 September 2024 QUESTION 4 In the diagram, the equation of the circle centred at B is given as: 𝑥 2 − 8𝑥 + 𝑦 2 + 6𝑦 = 0 • DS and PA are tangents to the circle at D and A respectively. • A is the 𝑦-intercept of the circle. • DS is parallel to the 𝑦 − axis. • C(1; 1) is a point on the circumference of the circle. • CD and DE are chords of the circle. • BE, if drawn, is parallel to DS. • The smaller segment of the circle on chord DE is shaded. 4.1 Show that the coordinates of B is (4 ; −3). Show all your calculations. (3) 4.2 Determine: 4.2.1 the equation of the tangent DS. (2) 4.2.2 the length of chord CD. (3) 4.2.3 the equation of the tangent PA in the form 𝑦 = 𝑚𝑥 + 𝑐. (5) 4.2.4 the area of the shaded segment. (3) 4.3 Another circle with center M and equation (𝑥 + 1)2 + (𝑦 − 2)2 = 8 is drawn. Determine, with an explanation, whether the two circles will intersect or not. (3) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 September 2024 QUESTION 5 5.1 Without using a calculator, simplify the following expression completely: cos(−𝜃) × sin(90° − 𝜃) × (1 + tan2 𝜃) (5) 5.2 Given: cos(A + B) = cos A. cos B − sin A. sin B 5.2.1 Use the formula for cos(A + B) to derive a formula for sin(A − B). (4) 5.2.2 Without using a calculator, show that: 1 sin(𝑥 + 63°). cos(𝑥 + 378°) + cos(𝑥 + 63°). cos(𝑥 + 108°) = (5) √2 5.3 Given that sin 10° = √𝑘 Without using a calculator, write each of the following in terms of 𝑘: 5.3.1 sin 190° (2) 5.3.2 cos 20° (3) 5.3.3 cos 50° (4) 5.4 Given: 1 (cos 𝜃 + sin 𝜃) − (cos 𝜃 + sin 𝜃)(cos 𝜃 − sin 𝜃) (cos 𝜃 − sin 𝜃) 5.4.1 Simplify the expression above to a single trigonometry ratio. (5) 5.4.2 Determine the general solution of 𝜃 for which this expression will be (3) undefined? [31] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 September 2024 QUESTION 6 In the diagram, the graph of 𝑓(𝑥) = tan( 𝑥 + 𝑝°) is drawn for the interval 𝑥 ∈ [−135°; 180°] with asymptotes at 𝑥 = −45° and 𝑥 = 135°. 𝑦 𝑥 −135° −90° −45° 0 45° 90° 135° 180° -1 𝑓 6.1 Write down the value of 𝑝. (1) 6.2 Draw the graph of 𝑔(𝑥) = sin 2𝑥 for the interval 𝑥 ∈ [−135° ; 180°] on the grid given in the ANSWER BOOK. Show ALL intercepts with axes, as well as the minimum and maximum points on the graph. (3) 6.3 Write down the period of 𝑔. (1) 6.4 The graph of 𝑔 is shifted 45° to the left to form the graph of ℎ. Determine the equation of ℎ in the simplest form. (2) 6.5 Use the graph to determine the values of 𝑥 in the interval 𝑥 ∈ [−135° ; 0°] for which: 6.5.1 𝑓(𝑥) ≤ −1 (2) 6.5.2 sin 𝑥 cos 𝑥 + 2 < 2 (3) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 September 2024 QUESTION 7 In the diagram below, ABC is a vertical triangular wall on the floor CBD. CA = 13 meters, CD = BD = 𝑘 meters, AĈB = 𝛼 and BD ̂ 𝐶 = 2𝛼 7.1 Show that: 7.1.1 CB = 13 cos 𝛼 (1) 13 (4) 7.1.2 𝑘= 2 tan 𝛼 7.2 Calculate the area of the floor ΔBCD if 𝛼 = 26° (2) [7] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 September 2024 QUESTION 8 8.1 Complete the following theorem: The angle at the centre of a circle is twice the angle ______________________ subtended by _________________. . (2) 8.2 In the diagram below, O is the centre of the circle. K, L, M and N are points on the circumference of the circle. KS is a tangent to the circle at K and L̂ = 100° 8.2.1 Determine, with reasons: (a) the size of ̂ N. (2) (b) the size of ̂ K2. (3) 8.2.2 Prove, giving reasons, that: MN ∥ KS (4) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 11 September 2024 QUESTION 9 In the diagram below, O is the centre of the circle. P, R, S and A are points on the circumference of the circle. AOBC is a straight line such that B lies on PS with PB = BS and ̂ A1 = 𝑥 9.1 Determine, with reasons, ̂ P1 in terms of 𝑥. (4) 9.2 Prove, giving reasons, that: ̂ S. 9.2.1 AC bisects PA (4) 9.2.2 COSR is a cyclic quadrilateral. (4) [12] QUESTION 10 10.1 ̂ = ̂ In the diagram, ∆MVT and ∆AKF are drawn such that M A, ̂ V = ̂ K and ̂ T = F̂. Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, that is prove MV MT that = . AK AF (7) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 September 2024 10.2 In the diagram, HLKF is a cyclic quadrilateral. The chords HL and FK are produced to intersect at M. The line FG parallel to KL, meets MH produced at G. MK 1 = and LH = HG. KF 2 10.2.1 Give a reason why GF̂M = LK ̂ M. (1) 10.2.2 Prove, giving reasons, that: (a) ∆MFH ||| ∆MGF (4) (b) MF = √6 (5) ML [17] TOTAL: 150 Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 SEPTEMBER 2024 INFORMATION SHEET − b  b 2 − 4ac x= 2a A = P(1 + ni ) A = P(1 − ni ) A = P(1 − i) n A = P(1 + i) n Tn = a + (n − 1)d Sn = n 2a + (n − 1)d  2 Tn = ar n−1 Sn = ( a r n −1 ) ; r 1 S = a ; −1  r  1 r −1 1− r F=  x (1 + i ) − 1 n  P=  x 1 − (1 + i ) −n  i i f ( x + h) − f ( x ) f ' ( x) = lim h→ 0 h  x + x 2 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 (x − a )2 + ( y − b)2 = r 2 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 sin( +  ) = sin  cos  + cos  sin  sin( −  ) = sin  cos  − cos  sin  cos( +  ) = cos  cos  − sin  sin  cos( −  ) = cos  cos  + sin  sin  cos 2  − sin 2   𝑠𝑖𝑛 2 𝛼 = 2 𝑠𝑖𝑛 𝛼 . 𝑐𝑜𝑠 𝛼 cos 2 = 1 − 2 sin 2  2 cos 2  − 1  n 2 x  (x − x ) i x=  = i =1 2 n n n( A ) P(A) = P(A or B) = P(A) + P(B) – P(A and B) n(S) yˆ = a + bx b=  (x − x )( y − y ) (x − x) 2 Copyright reserved