Mathematics Grade 12 P2 NSC November 2021 QP (Eng)

__Downloaded from hlayiso.com__ basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 Fi ai Ra ake eA a I = RD TIA AISI I IIR : MATHEMATICS P2 7 u ri ” a " NOVEMBER 2021 " Sooo o sooo nooo énppAonsncnenpsasgsrgrsnce MARKS: 150 TIME: 3 hours This question paper consists of 13 pages and 1 information sheet. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/November 2021 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 11 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. If necessary, round off answers correct to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 DBE/November 2021 NSC QUESTION 1 A bakery kept a record of the number of loaves of bread a tuck-shop ordered daily over the last 18 days. The information is shown in the table below. 10 ll 13 14 14 15 16 18 18 19 19 20 21 35 35 37 40 41 1.1 Calculate the: Ll Mean number of loaves of bread ordered daily (2) 1.1.2 Standard deviation of the data (1) 1.1.3 Number of days on which the number of loaves of bread ordered was more than one standard deviation above the mean (2) 1.2 The tuck-shop owner was not able to sell all the loaves of bread delivered daily. He calculated the mean number of loaves sold over the 18 days to be 20. Calculate the number of loaves of bread which were NOT sold over the 18 days. (2) 1.3 One of the two box and whisker diagrams drawn below represents the data given in the table above. Graph A: e—___ — T T T T T T 8 10 12 14 16 Ii T T 8 20 22 24 26 28 30 32 34 36 38 40 42 Graph B: e— |__» 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 1.3.1 Which ONE of the two box and whisker diagrams, drawn above, correctly represents the data? Write down a reason for your answer. (2) 1.3.2 Describe the skewness of the data. (1) [10} Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 4 DBE/November 2021 NSC QUESTION 2 A farm stall sells milk in 5-litre containers to the local community. The price varies according to the availability of milk at the farm stall. The price of milk, in rands per 5-litre container, and the number of 5-litre containers of milk sold, are recorded in the table below. Price of milk in rands per 26 32 36 28 | 40 33 29 34 | 27 | 30 5-litre container (x) Number of S-litre containers of | 4g | 39 | 26 | 44 | 23 | 32 | 39 | 29 | 42 | 33 milk sold Y) 21 On the grid provided in the ANSWER BOOK, draw the scatter plot to represent the data. 2.2 Determine the equation of the least squares regression line for the data. 23 If the farmer sells a 5-litre container of milk for R38, predict the number of 5-litre containers of milk he will sell. 2.4 Refer to the correlation between the price of 5-litre containers of milk and the number of 5-litre containers of milk sold, and comment on the accuracy of your answer to QUESTION 2.3. Copyright reserved Please turn over (3) (3) @) (2) [10]

Downloaded from hlayiso.com Mathematics/P2 QUESTION 3 5 DBE/November 2021 NSC In the diagram, A(-2 ; 10), B(k; 4) and C(4 ;~2) are the vertices of AABC. Line BC is produced to H and cuts the x-axis at E(12; 0). AB and AC intersects the x-axis at F and G respectively. The angle of inclination of line AB is 81,87°. 3.1 3.2 3.3 3.4 A(-2;10) ¥ 81,87° H F pL 0) x C(4 ;-2) Bk; 4 Calculate the gradient of: 3.1.1 BE (2) 3.1.2 AB (2) Determine the equation of BE inthe form y = mx +c (2) Calculate the: 3.3.1 Coordinates of B, where k<0 (2) 3.3.2 Size of A (4) 3.3.3 Coordinates of the point of intersection of the diagonals of parallelogram ACES, where S is a point in the first quadrant (2) Another point T(p ; p), where p> 0, is plotted such that ET = BE=4-17 units. 3.4.1 Calculate the coordinates of T. (5) 3.4.2 Determine the equation of the: (a) Circle with centre at E and passing through B and T in the form (x-a)? +(y—-b) =r? (2) (b) Tangent to the circle at point B(k ; &) te Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 DBE/November 2021 NSC QUESTION 4 In the diagram, the circle centred at N(—1 ; 3) passes through A(-1 ;-1) and C. B(-4; 2), C, D and E are joined to form a parallelogram such that BE is parallel to the x-axis. CD isa tangent to the circle at C and CD =6 units. y c 6 D NC1; 3) BC4 ;2) - E [e} x AG1;-1) 4.1 Write down the length of the radius of the circle. qd) 4.2 Calculate the: 4.2.1 Coordinates of C (2) 42.2 Coordinates of D (2) 4.2.3 Area of ABCD @) 4.3 The circle, centred at N, is reflected about the line y= x. M is the centre of the new circle which is formed. The two circles intersect at A and F. Calculate the: 4.3.1 Length of NM (3) 4.3.2 Midpoint of AF (4) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 DBE/November 2021 NSC QUESTION 5 5.1 Without using a calculator, simplify the following expression to ONE trigonometric ratio: sin 140°. sin(360° — x) 5 cos 50°. tan(—x) (6) +2 5.2 Prove the identity: =2ain_ttcogx tl _ 2cosx-1 (4) 1~cos(540°- x) 5.3 Given: sin36°= fl -p? Without using a calculator, determine EACH of the following in terms of p: 5.3.1 tan36° 3) 5.3.2 cos108° (4) [17] QUESTION 6 6.1 Given: cos(a — 8) = cosa cos $ +sinasin B 6.1.1 Use the given identity to derive a formula for cos(@ + £) (3) 6.1.2 Simplify completely: 2cos6xcos4x—cos10x+2sin? x (5) 6.2 Determine the general solution of tanx=2sin2x where cos x <0. (7) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/November 2021 NSC QUESTION 7 In the diagram below, the graph of f(x)=-2sinx is drawn for the interval x €[-180° ; 180°]. y 7 = - x -180° 90° 0° | 90° 180° - ——— pape = 7A On the grid provided in the ANSWER BOOK, draw the graph of g(x) =cos(x~60°) for xe€[-180°; 180°]. Clearly show ALL intercepts with the axes and turning points of the graph. @) 72 Write down the period of f(3x). (2) 73 Use the graphs to determine the value of x in the interval x €[-180°; 180°] for which f(x)- g(x)=1. Q) 74 Write down the range of k, if k(x) = ; g(x)+1. 2) (8] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 DBE/November 2021 NSC QUESTION 8 In the diagram below, T is a hook on the ceiling of an art gallery. Points Q, S and R are on the same horizontal plane from where three people are observing the hook T. The angle of elevation from Q to T is x. QSR =90°+x, QRS=x, QR=5 units and TS =SQ. T 90° +x x Q 5 R 8.1 Prove that QS =5 tan x QB) 8.2 Prove that the length of QT=10sin x (5) 8.3 Calculate the area of ATQR if TQR=70° and x=25°, (2) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 NSC QUESTION 9 DBE/November 2021 In the diagram, PQRS is a cyclic quadrilateral. PS is produced to W. TR and TS are tangents to the circle at R and S respectively. T=78° and Q=93°. Q 91 Give areason why ST=TR. 9.2 Calculate, giving reasons, the size of: 9.2.1 8, 9.2.2 8, Copyright reserved Please turn over (1) (2) (2) 5]

Downloaded from hlayiso.com Mathematics/P2 11 DBE/November 2021 NSC QUESTION 10 In the diagram, BE and CD are diameters of a circle having M as centre. Chord AE is drawn to cut CD at F. AELCD. Let C=x. 10.1 Give a reason why AF = FE. (1) 10.2 Determine, giving reasons, the size of M, in terms of x. (3) 10.3 Prove, giving reasons, that AD is a tangent to the circle passing through A, C and F. (4) 10.4 Given that CF =6 units and AB = 24 units, calculate, giving reasons, the length of AE. (5) {13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/November 2021 NSC QUESTION 11 11.1 In the diagram, chords DE, EF and DF are drawn in the circle with centre O. KFC is a tangent to the circle at F. Prove the theorem which states that DFK =E. () Copyright reserved Please turn over

Mathematics/P2 11.2 In the diagram, PK is a tangent to the circle at K. Chord LS is produced to P. N and M are points on KP and SP Downloaded from hlayiso.com 13 DBE/November 2021 NSC Chord KS and LN intersect at T. 11.2.2 11.2.3 11.2.4 Copyright reserved Prove, giving reasons, that: (@) K,=NML (b) KLMN is acyclic quadrilateral Prove, giving reasons, that ALKN ||| AKSM. If LK =12 units and 3KN=4SM, determine the length of KS. If it is further given that NL = 16 units, LS = 13 units and KN = 8 units, determine, with reasons, the length of LT. TOTAL: respectively such that MN || SK. (4) (1) (5) (4) 12 150

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2021 NSC INFORMATION SHEET: MATHEMATICS - —b+b* —4ac x 2a A =P(l+ni) A = P(l=ni) A=P(l-i)" A=P(1+i)" T, =a+(n-Nd S, =FRa+ a] T, =ar"" 5, = 2 =). et 8, =; -1erel r-1 l-r pad+)" 1 pa2lod+)" i i ly) = tm FOAMS) f'(%) = tim k x tx, WtVz a= V5) +0.) M2 2%) y=mxe+re y-y, =m(x-x,) m=22—M m=tan@ % —%y (x-a)’ +(y—byY =r? In AABC: a. § _<¢ sind sinB sinc a’ =b? +c" —2be.cos A area AABC = jab sin c sin(a + B)=sin acos B +cos asin B sin(a — B)= sin acos 8 —cos asin B cos(a@ + 8) = cosa cos § -sin asin B cos(a — 8) =cosacos 8 +sinasin B cos’ asin? a cos2a=| 1-2sin’a sin2a = 2sinacosa 2cos’a—-1 -_ De, -%/) x= ot == n n(A) P(A) (8) P(A or B)= P(A) + P(B)- P(A and B) p=atbhx j= a FY-5) ¥ @-3x/ Copyright reserved