Mathematics Grade 12 P1 North West June 2024 QP (Eng)

PROVINCIAL ASSESSMENT GRADE 12 MATHEMATICS P1 JUNE 2024 MARKS: 150 TIME: 3 hours This question paper consists of 8 pages and 1 information sheet. Copyright reserved Please turn over

Mathematics/P1 2 NW/June 2024 Grade 12 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. 3. Number the answers correctly according to the numbering system used in this question paper. 4. Clearly show ALL calculations, diagrams, graphs, et cetera 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. Write neatly and legibly. Copyright reserved Please turn over

Mathematics/P1 3 NW/June 2024 Grade 12 QUESTION 1 1.1 Solve for x: 1.1.1 2𝑥(4 − 𝑥 ) = 0 (2) 1.1.2 𝑥(3𝑥 − 7) = 4 (correct to TWO decimal places) (4) 1.1.3 𝑥 + √𝑥 − 3 = 15 (5) 1.1.4 𝑥 2 + 𝑥 − 12 > 0 (4) 1.2 Solve for x and y simultaneously: 3𝑥 − 𝑦 = 1 and 𝑥 2 + 2𝑥𝑦 = 3𝑦 2 − 7 (6) 𝑎 𝑏 𝑐 1.3 If √2 = 3; √3 = 5 and √5 = 8, determine the value of 𝑎 × 𝑏 × 𝑐. (4) [25] QUESTION 2 2.1 Given a quadratic number pattern: 19; 8; −1; −8; . . . 2.1.1 Determine the next term of the pattern. (1) 2.1.2 Determine the general term of the quadratic number pattern. (4) 2.1.3 Determine the difference between the 25th and the 26th terms of the quadratic number pattern. (3) 2.1.4 Calculate the value of the smallest term in the quadratic pattern. (3) 2.2 Given the arithmetic sequence: 3; 𝑏; 13; 18; . . . 2.2.1 Write down the value of 𝑏. (2) 2.2.2 Determine the 𝑛𝑡ℎ term of the sequence. (2) 2.2.3 Calculate the sum of the first 30 terms in the sequence. (2) 2.3 Given: 2𝑥 + 2𝑥+1 + 3.2𝑥 + 2𝑥+2 + . . . (15 𝑡𝑒𝑟𝑚𝑠) = 𝑘. 2𝑥 + 𝑝 Determine the values of 𝑘 and 𝑝 for which 𝑘, 𝑝 ∈ ℤ and 𝑘 is an odd number. (4) [21] Copyright reserved Please turn over

Mathematics/P1 4 NW/June 2024 Grade 12 QUESTION 3 3.1 Given the geometric series: 1 −4 + 2 − 1 + . . . + 32 3.1.1 Write down the general term of this series. (2) 3.1.2 Write the series in sigma notation. (2) 3.1.3 Calculate the sum to infinity of this series. (2) 1 𝑛 3.2 Given: 𝑆𝑛 = 32 − 32 (2) . 3.2.1 Determine 𝑆5 . (1) 3.2.2 How many terms must be added for the sum to be equal to 255 ? (3) 8 3.2.3 If 2𝑛 = 𝑝, determine the value of 𝑆5−𝑛 − 𝑆5+𝑛 in terms of 𝑝. (3) [13] QUESTION 4 −6 Given: ℎ(𝑥 ) = − 2 𝑥+3 4.1 Write down the equations of the asymptotes of ℎ. (2) 4.2 Sketch the graph of ℎ. Clearly label ALL the intercepts with the axes and the asymptotes on your graph. (4) 4.3 Determine the equation of the line of symmetry of ℎ for 𝑚 > 0. (2) 4.4 Write down the range of ℎ(𝑥). (2) 4.4 For which values of 𝑥 will ℎ(𝑥) ≥ 0 ? (2) [12] Copyright reserved Please turn over

Mathematics/P1 5 NW/June 2024 Grade 12 QUESTION 5 27 Given: 𝑔(𝑥) = 𝑎 𝑥 . A (−3; 8 ) is a point on 𝑔 and B is the 𝑦-intercept. 5.1 Write down the coordinates of B. (2) 5.2 Determine the value of 𝑎. (2) 5.3 Determine the equation of ℎ if ℎ is the reflection of 𝑔 in the 𝑦-axis. (2) 5.4 Write down the equation of ℎ−1 (𝑥 ), the inverse of ℎ, in the form 𝑦 = . . . (2) 5.5 For which values of 𝑥 will ℎ−1 (𝑥) ≤ 1 ? (2) 5.6 Write down the domain of ℎ−1 (𝑥 − 3). (2) [12] Copyright reserved Please turn over

Mathematics/P1 6 NW/June 2024 Grade 12 QUESTION 6 The graphs of 𝑓 (𝑥 ) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥 ) = 𝑚𝑥 + 𝑞 are drawn below. The graphs 1 intersect at B and H. B is also the 𝑥-intercept of 𝑓 and 𝑔. C (2 ; 0) is also a 𝑥-intercept of 𝑓. 3 D(− 2 ; 4) is a point on 𝑓. A is the turning point of 𝑓. E(0; −2) is the 𝑦-intercept of 𝑔. The angle of inclination of 𝑔 with the positive 𝑥-axis is 135°. 6.1 Show that the equation of 𝑔(𝑥 ) = −𝑥 − 2. (2) 6.2 Write down the coordinate of B. (2) 6.3 Show that the equation of 𝑓(𝑥) = −4𝑥 2 − 6𝑥 + 4. (4) 6.4 Determine the coordinate of A, the turning point of 𝑓. (3) 6.5 Determine the coordinate of H. (4) 6.6 For which values of 𝑥 will 𝑓 ′ (𝑥 ). 𝑔′ (𝑥 ) < 0 ? (2) 6.7 Use the graph of 𝑓 to determine the value(s) of 𝑘 for which 𝑓 (𝑥 ) = 𝑘 will have two unequal negative roots. (2) 6.8 After 𝑓 is reflected about the 𝑦-axis and then moved downward so that the 𝑥-axis is a tangent to ℎ, ℎ(𝑥) is formed. Write down the new coordinate for D. (2) [21] Copyright reserved Please turn over

Mathematics/P1 7 NW/June 2024 Grade 12 QUESTION 7 7.1 Determine 𝑓′(𝑥) from first principles if 𝑓(𝑥 ) = 𝑥 2 + 4𝑥 + 1. (5) 7.2 Determine: 7.2.1 𝑓′(𝑥) if 𝑓 (𝑥) = −3𝑥 4 + 5𝑥 2 (2) 7.2.2 𝑑𝑦 2 if 𝑦 = √𝑥 − 𝑥 𝑑𝑥 (4) 7.3 Determine the equation of the tangent to the curve of ℎ(𝑥) = −𝑥 3 + 2𝑥 − 5 at 𝑥 = −2. (4) 7.4 The graph 𝑦 = 𝑔′(𝑥) has a minimum turning point at (1; −4). Determine the values of 𝑥 for which 𝑔 is concave up. (2) [17] QUESTION 8 Given: 𝑓 (𝑥 ) = 𝑥 3 − 8𝑥 2 + 5𝑥 + 14 8.1 Write down the y-intercept of 𝑓. (1) 8.2 Show that (𝑥 − 7) is a factor of 𝑓. (2) 8.3 Hence, factorise 𝑓(𝑥) completely. (4) 8.4 Determine the coordinates of the turning points A and B. (6) 8.5 Find the coordinates of the point where the concavity of the graph changes. (3) [16] Copyright reserved Please turn over

Mathematics/P1 8 NW/June 2024 Grade 12 QUESTION 9 Draw a graph of 𝑓 if 𝑓 (𝑥 ) = (𝑥 + 𝑝)2 (𝑥 − 𝑞 ) and  𝑓 (6) = 0  𝑓 ′ (0) = 𝑓 ′ (4) = 0  𝑓 ′ (𝑥 ) < 0 for 0 < 𝑥 < 4  𝑓 ′ (𝑥 ) > 0 for 𝑥 < 0 𝑜𝑟 𝑥 > 4  𝑓 (2) = −16 [4] QUESTION 10 A stone is thrown upward and its height after 𝑡 seconds is given by equation ℎ = 18𝑡 − 4𝑡 2 metres. 10.1 Determine the height of the stone after 2 seconds. (2) 10.2 1 (3) Determine the velocity of the stone after 1 2 seconds. 10.3 What is the maximum height that the stone will reach ? (4) [9] TOTAL: 150 Copyright reserved

Mathematics/P1 NW/June 2024 Grade 12 INFORMATION SHEET: MATHEMATICS  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  x2 y1  y 2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y  y1 y  mx  c y  y1  m( x  x1 ) m 2 m  tan  x 2  x1 x  a2   y  b2  r 2 a b c In ABC:   a 2  b 2  c 2  2bc. cos A sin A sin B sin C 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  cos2   sin 2   cos 2  1  2 sin 2  sin 2  2 sin  . cos 2 cos2   1  n 2  fx  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