Mathematics Grade 12 P2 NSC November 2025 QP (Eng)

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE MATHEMATICS P2 NOVEMBER 2025 MARKS: 150 TIME: 3 hours This question paper consists of 14 pages, 1 information sheet and an answer book of 27 pages. Please turn over Copyright reserved

Mathematics/P2 2 DBE/November 2025 NSC Confidential INSTRUCTIONS AND INFORMATION Read the following instructions and information 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 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

Mathematics/P2 QUESTION 1. 3 NSC Confidential DBE/November 2025 There were 11 cars of the same model for sale at a car dealership. The age (in years) of each car and its corresponding selling price (in rands) is provided in the table below. AGE OF CAR (IN YEARS) SELLING PRICE OF CAR (IN RAND) 293 000 265 000 256 000 219 000 241 000 246 000 226 000 176 000 154 000 180 000 |W AIA |B LB) B| Ld / 42 | bo 148 000 11 Determine the equation of the least squares regression line. 1.2 Predict the selling price of a similar car at this car dealership that is 5 years old. 13 Use the correlation coefficient to show whether the prediction made in QUESTION 1.2 is valid or not. 1.4 Use the answer to QUESTION 1.1 to write down the estimated average yearly decrease in the selling price of these 11 cars. Copyright reserved Please turn over (3) Q) (2) (1) 18]

Mathematics/P2 4 DBE/November 2025 NSC Confidential QUESTION 2 2.1 The cumulative frequency table below shows the amount of time that people spent on a particular website on a certain day. TIME, ¢ CUMULATIVE (IN MINUTES) FREQUENCY 0<¢<20 16 0<#<40 40 0<t<60 59 0<1<80 67 0<t<100 70 2.1.1 How many people visited this website on that day? 2.1.2 How many people spent more than 40 and up to 80 minutes on the website? 2.1.3 Draw a histogram to represent the information provided in the cumulative frequency table. 2.1.4 Comment on the skewness of the data. 2.2 There are 9 players in a basketball team. The coach calculated that on average, each player scored 12 points during a game. The points scored by 8 of the 9 players from the team is given below: 11 14 19 20 10 14 How many players' points score was outside ONE standard deviation of the mean points score? Copyright reserved Please turn over Qa) (2) (3) Q) (6) [12]

Mathematics/P2 5 DBE/November 2025 NSC Confidential QUESTION 3 In the diagram, P(-1 ; 8), Q(-4;-6) and R(12 ; 2) are the vertices of APQR. The angle of inclination of QR is @. y P(-1 ; 8) R(12 ; 2) 0 LL QE; -6) 3.1 Calculate the length of QR. Leave your answer in simplified surd form. 3.2 Calculate the gradient of QR. 3.3 Calculate the size of 0. 3.4 Determine the equation of QR. 3.5 PQRS, in that order, is a parallelogram. Write down the coordinates of S. 3.6 T isa point on QR such that PT 1 QR. Calculate the coordinates of T. 3.7 Calculate the area of parallelogram PQRS. Copyright reserved Please turn over (2) (2) (2) (2) (2) (5) (3) [18]

Mathematics/P2 6 DBE/November 2025 NSC Confidential QUESTION 4 In the diagram, M(p ; q) is the centre of the circle that intersects the x-axis at A and B. C is a point such that the line drawn from M to C is parallel to the y-axis and intersects the x-axis at D. MC intersects the circle at E(—6 ; -1). Tangents drawn from C touch the circle at A and B. AD = (g-1) units. y A ID B fe} x E Cc 4.1 Write down the value of p. (1), 4.2 Show that g = 4. (4) 43 Determine the equation of the circle in the form (x-a) +(y -by =r’. Q) 44 If the circle is translated 2 units to the left, determine the minimum distance between the circle and the y-axis. qd) 4.5 Calculate the coordinates of A and B. (3) 46 Determine the equation of tangent BC. (4) 47 Write down the coordinates of C. (2) 48 Calculate the size of ACB. (4) [21] Copyright reserved Please turn over

Mathematics/P2 7 DBE/November 2025 NSC Confidential QUESTION 5 5.1 Itis given that tan50°=k. Express EACH of the following in terms of &: S.1.1 cos 40° (2) 5.12 2 sin 25°.cos 25° (5) —2+4sin? 25° 5.1.3 sin 10° (4) 52 Given: sin (540 iz).con(09 +x) sin(-x) 5.2.1 Simplify the expression above fully to a single trigonometric ratio. (4) 5.2.2 Hence, determine the values of x in the interval x <[0°;360°] for which in (540° 7 90° sin(5 =) gos( 0+) will be real. Q) sin(-x) [17] QUESTION 6 a 2 ___ (sinx—cosx)(1+sin x.cos.x) 6.1 Prove that: [tan (180°— xx) ](1-cos x)+cos x= (6) —cos x 6.2 It is given that sin? x ; cos’ x and 5sin 2x are the first three terms of an arithmetic sequence. The constant difference of the arithmetic sequence is NOT zero. Determine the general solution for x. (7) [13] Copyright reserved Please turn over

Mathematics/P2 8 NSC Confidential QUESTION 7 In the diagram, the graph of f(x)=cos2x is drawn for x e[-135° ; 135°]. DBE/November 2025 7A Write down the period of f 7.2 On the set of axes provided in the ANSWER BOOK, draw the graph of g(x)=tan2x-1 for xe[-135°; 135°]. 73 Graph / is translated 45° to the left to form graph h. Determine the equation of h in its simplest form. 74 Write down the range of h. 715 Determine the values of x for which (1-tan2x)(cos2x)>0 x [0° ; 135°]. Copyright reserved in the interval Please turn over Q) (3) () () (4) [16]

Mathematics/P2 9 DBE/November 2025 NSC Confidential QUESTION 8 As part of a school project, learners are required to design a portable stage for a puppet show, as shown in the diagram below. The design must fulfil the following requirements: 8.1 8.2 8.3 8.4 BKLC is a horizontal base having KBC =104° and BKC =52,6°. The rectangular backdrop, ABCD, is vertical to the horizontal base and must have an area of 648 cm?. The sides of ABCD must be in the ratio AB: BC=1:2. The stage must be partly enclosed with triangular sides ABK and DCL. Area = 648 om B Cc 104° 52,6° K L Show that AB = 18 cm. (2) Calculate the length of AC. (2) Calculate the length of diagonal KC. (2) if AB=BK, calculate the size of KAC. (4) [10] Copyright reserved Please turn over

Mathematics/P2 10 DBE/November 2025 NSC Confidential Provide reasons for your statements in QUESTIONS 9, 10 and 11. QUESTION 9 9.1 In the diagram, O is the centre of the circle. K, T and P lie on the circle. K Use the diagram above to prove the theorem which states that the angle subtended by a chord (or arc) at the centre of the circle is equal to twice the angle subtended by the same chord (or arc) at the circumference, that is prove that TOP =2TKP. 6) Copyright reserved Please turn over

Mathematics/P2 ll DBE/November 2025 NSC Confidential 9.2 In the diagram, O is the centre of the circle. POF is the diameter of the circle and MF is a tangent to the circle at F. OM cuts the circle at L. p=32°. P 32° 1 4] O L M F Calculate, with reasons, the size of: 9.2.1 (oy (2) 9.2.2 M (3) [10] Copyright reserved Please turn over

Mathematics/P2 12 NSC Confidential QUESTION 10 DBE/November 2025 In the diagram, PQRS is a cyclic quadrilateral. T is a point on the circle such that QT is perpendicular to SR at V. PT and ST are drawn. Q =35° and R=§,. 10.1 Calculate, with reasons, the size of QTS. 10.2 Prove that PQ || SR. 10.3 Prove that PT is a diameter of the circle. Copyright reserved Please turn over (3) (3) Q) [8]

Mathematics/P2 13 NSC Confidential DBE/November 2025 QUESTION 11 11.1 In the diagram, A ABC is drawn. BA is produced to E. F and D are points on BC such that AD || EF. AC and EF intersect at G. po id eS an =>. 5 GA 2 Calculate, with reasons, the value of: FD IL.1.1 — 2 CF (2) BA al. — ‘4 W412 Fe 4) Area of AGCF 11.1.3 ——————— 4 Area of GFDA (4) Copyright reserved Please turn over

Mathematics/P2 14 DBE/November 2025 NSC Confidential 11.2 In the diagram, WVZR isa cyclic quadrilateral. RZ is produced to Q. A tangent is drawn from Q to touch the circle at S. WV is produced to E, a point on ZQ. RW produced meets ZV produced in P. PQ || WE. RS and ZS are drawn. P Q Prove, with reasons, that: 1121 pp =PW.QR (2) QE 11.2.2 If APQZ ||| ARQP, then PQ’ =RQ.QZ 65) 11.2.3. AQSZ ||| AQRS (3) 11.2.4 PQ=QS QB) QE.PZ 11.2.5 W = 4 JQR.QZ @ [23] TOTAL: 150 Copyright reserved

Mathematics/P2 DBE/November 2025 NSC Confidential INFORMATION SHEET ya bt vb" ~4ac 2a A=P(l+ni) A=P(-ni) 4=P(-i" A= P(+i)’ T, =a+(n-1)d 8, =F Pat (n—1d] T, =ar™" s _alr* -1) yr¥l S, = 4s -ler<l 7 r-l l-r Pe (i+) -1 pa2lo(l+i) i L 104) = tim LOA D-LO) Le) nod h d=, Ft, w(A7,22%) 2 2 yomxtc y-y, =m(x-x,) m= m= tan 0 x —X, (x-al +(y—-by =r? In ABC; 2 == © snA4 sinB sinc a’? =b? +c? —2be.cos A 1 : area AABC = | ab.sin C sin (@ +B) =sina.cos f+ cosa.sin B cos(a + 8) =cosa.cos B-sina.sin B cos’ a —sin’ a cos2a =41—2sin’ a 2cos? a-1 Copyright reserved sin(a — 8) =sina.cos f—cosa.sin B cos(a - 8) = cosa@.cos B+ sina.sin B sin 2a = 2sin a.cos a Y&,-¥) oa n P(A ot B) = P(A) + P(B) — P(A and B) Le-x)v-7)