Mathematics Grade 12 Paper 1 MP June 2023 Paper & Memo

BF, education MPUMALANGA PROVINCE REPUBLIC OF SOUTH AFRICA MARKS: 150 TIME: 3 hours This question paper consists of 8 pages and a one-page information sheet. Copyright reserved Please Turn Over

owntstided from Stanmorephysics.com MDE/lune'2023 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. l. This question paper consists of 10 questions. Answer ALDtt An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. Answers only will not necessarily be awarded full marks. Number the answers correctly according to the numbering system used in this question paper. Diagrams and graphs 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

Dawalaaded from Stanmorephysics.com QUESTION 1 1.1 1.2 1.3 QUESTION 2 Solve for x. 1.1.1 1.1.2 1.1. 1.1 (2+x)(-x+4)=0 ? 2x44 (Correct to 2 decimal places) -2V¥x-1=4 x-12 > -x? Solve the following equations simultaneously: x? —xy—Sy? =-5 and x+2y=1 Determine the values of ¢ for which the equation 5* =2-¢ will have real solutions, Given the arithmetic series 3+ 8+ 13 +...... (to 253 terms) Write down the 4" term of the series. Calculate the 253 term of the series. Express the series in sigma notation. Calculate the sum of the series. How many terms of the series are divisible by 4? Given the quadratic sequence: 1; 3,7; p; ... Calculate the value of p. Determine the n'" term of the sequence. The first difference between two consecutive terms is 62. Calculate the values of these terms. 2.1 2.2 2.3 2.4 2.5 QUESTION 3 . 3.1 3.1.1 3.1.2 3.1.3 3.2 Copyright reserved Determine the largest integer value of m if 2.2") < 20. kel MDE/June 2023 (2) 4) (6) (4) (6) (3) [25] (1) (3) (2) (2) (4) [12] G3) (4) (3) 6) [15] Please Turn Over

iawulaaded from Stanmorephysics.com MDE/Sune 2023 QUESTION 4 The following sketch shows the graphs of f(x) =x? +bx+c¢ and g(x) =ax+q. The graph of fintersects the x-axis at A(-2 ; 0) and at B. C is the y-intercept of both fand g, axis of symmetry of f is x=1. ay 1 { | g f { i N . A i B * H H H ! ! ! Cc 4.1 Determine 4.1.1 The coordinates of B (2) 4.1.2 The equation of fin the form y=x? +bx+c (2) 4.1.3. The values of a and q. (4) 4.1.4 The coordinates of the turning point of f. (2) 4.2 M is the reflection of C in the axis of symmetry off, Write down the coordinates of M. (2) 43 For which values of x is f(x) > g(x)? (2) 4.4 Write down the coordinates of the turning point of y = —f(x = (2) 45 N is a point on the x-axis and T is a point on f such that TN perpendicular to the x-axis. TN intersects g at P. Calculate ximum (4) length of TP. [20] Copyright reserved Please Turn Over

Mawnteaded from Stanmorephysics. com MDE/June 2023 QUESTION 5 Sketched below is the graph of f(x) =—log, x y4 T >x f 5.1 Write down the domain of f (i) 5.2 Write down the equation of f~ in the form y=.... (2) 5.3. Describe the transformation from f~' to hif A(x)=3* -5 (2) 5.4. Use the graph of f to solve for x if A(x) is the reflection of f about the x-axis and k({x)23. (4) [9] QUESTION 6 The function f defined by f(x) = 4 +q has the following properties. x+p The range of fis ye R,y #2 ¢ The axis of symmetry with a positive gradient is y = x +1 e The graph of fpasses through (0;—4) 6.1 Write down the value of q. (1) 6.2 Calculate the values of a and p . (4) 6.3 Sketch a neat graph of this function. Your graph must include tercepts with the axes and asymptotes if any. (5) [10] Copyright reserved Please Turn Over

Meuwnleaded from Stanmorephysics.com MDE/Iune 2023 QUESTION 7 7.1 Thelma purchased a new car. The bank offered her a loan at an effective interest rate of 10,4% p.a. Deter: he nominal rate compounded monthly that she is required to pay. (4) 7.2 Ani r bought shares in a certain company, but found that his money had become half of its original value after a period of 4 years. What was the annual rate of decay, on a reducing balance method, for the shares he had bought? (4) 7.3 Thini borrowed R80 000, exactly five years ago, at 21% interest p.a. She paid back R25 000 exactly three years ago and R55 000 exactly a year ago. She wants to settle her remaining debt today. 7.3.1 Calculate the value of the loan to date. (2) 7.3.2 How much must Thini pay to settle the debt today? (5) [15] QUESTION 8 8.1 Given f(x) =4x—2x’, determine /"(x) using FIRST PRINCIPLES. (5) 8.2 Determine: 1 8.2.1 pi» -4] 6) dy. _ y2 8.2.2 x if xJy =x* —4x (4) 8.3 Find the equation of the tangent to the graph of f(x) = x? —6x? ich the gradient of the tangent to fis equal to 36 and x < 0. ; (6) [17] Copyright reserved Please Turn Over

Mawnleaded from Stanmorephysics.com MDE/June 2023 iC QUESTION 9 The graph below represents the functions fand g with f(x) = ax’ +cex+2and g(x) =—xt+ and (1; 0) are the x -intercepts of f The graph of fand g intersect at Cc, ay B A y ci; [9 a 9.1 Determine the coordinates of C. (1) 9.2 Show by calculation that a=-1 and c= 3, (4) 9.3. Determine the coordinates of B, the turning point of 7. (3) 9.4 Find the x-coordinate of the point of inflection of /. (2) 9.5 Write down the values of & for which f(x) = will have only one root. (2) 9.6 Write down the values of x for which f'(x) < 0. (2) [14] Copyright reserved Please Turn Over

Boweleaded from Stanmorephysics.com MDE/June 2023 QUESTION 10 10.1 Given that A and B are independent events, P(B only) = 0,4 P(A and B) = 0,1 ; P(A only) = x and P(not A or B) = y. Calcul 10.1. e values of x and y. (4) 10.1 e probability that at least one of A or B occurs. @) 10.2 Inacity 10% of all crimes are violent. 95% of all violent crimes are reported, buy only 45% of non- violent crimes are reported. 10.2.1 Draw a tree diagram showing all the possible outcomes. (3) 10.2.2 What is the probability that a random crime will be reported? (3) [13] TOTAL: 150 Copyright reserved Please Turn Over

Downloaded from Stanmorephysics.com MDE/June 2023 INFORMATION SHEET ea wet 6? -4ac ~ 2a A=P(1+ in) A=P(\-in) A=P(1-i)' A=P(\+i)" T,=a+(n S,= Sba+(r -1)d] r, = ar"! 5-2 V ral S.= 74-3 -1<rcl - - Tr Pe xf(.+ iy" - 1] pate G+)" ee) = lim f(x +h)- f(x) i i hoo h d= (x) - 4) +Q2 -nF M222, 4 #2) yamxte y-y= mx —x) m = 2M m=tan@ * (x-a? +(y-b¥ =P? a’ =b? +c? 2be.cosA Area of AABC = Sabin c sin(a + B)= sina.cos B + cosa.sin B sin(a - Bp) = sina.cos B — cosa.sin B cosa - Bp) =cosa@.cosf + sina.sinB cosa + Bp) =cosa.cos# — sina.sinB cos’ A — sin? A cos2A =41—2sin? A sin2A = 2sin A.cosA 2cos? A-1 ae X<, -x/ =o ea n P(4)= en P(A or B)= P(A) )— P(A and B) D=atbx _ of 9) Copyright reserved

Downloaded from Stanmorephysics.com education Ri s MPUMALANGA PROVINCE RS) REPUBLICOFSOUTH AFRICA NATIONAL SENIOR CERTIFICATE ' GRADE 12 MATHEMATICS P1 JUNE 2023 MARKING GUIDELINES This marking guideline consists of 12 pages. Please turn over

‘powntsdded from Stanmecephysics.qam MDE/une 2023 QUESTION 1 T11] @-2Cx+4)=0 x-2=0 or-x+4=0 yea? = =4 x=4 11.2] 3 ix+4 ¥ standard form 3. x-4=0 -(2)+, (27 -4(3)(-4) ¥ substitution in correct “= 2(3) ea 2452 6 vv values of x x=-0,87 or x=1,54 (4) 1.1.3 | y-2/¢-1 34 —2yx-L=4~x Visolate the surd . ovx—1} =(4—x)? ¥ square both sides 4x-4=16-8x+x" x? —12x+20=0 Y standard form ¥ factors (x—10Xx—2)=0 x-values x=10 or x#2 ¥ x=10 only (6) 1.1.4] x?-x-12>0 y : standard (x-4)(x +3) >0 V factors Y critical values 3 4 OR —HO OT 3 4 x<-3 or x>4 ¥ solution (4) Copyright reserved Please turn over

yobwpritegded from Stanmeveshysics.cam MDE/June 2023 1.2 |x+2y=1 x=1-2 ¥ subject of formulae 1- 1-2y)-Sy? =-5 ( ¢ ¥) 2 y , Y substitution 1- y’ —yt+2y* —Sy* =-5S ¥ standard form y? —Sy+6=0 (y-3)(y-2) =0 ¥ factors y=3ory=2 ¥ y=3 or y=2 x =1-2(3) or x = 1-2(2) x=-5S orx = -3 ¥x=-Sor x=-3 13 | 5*>0 ¥ condition © 2-t>0 roa wt<2 (3) [25] Copyright reserved : Please turn over

Yewprifsdded from StanmecedhysiGse cam MDE/June 2023 QUESTION 2 21 [18 V answer (1) 2.2 | T,=5n-2 ¥nth term Toss 53)-2 ¥ substitution Ths 3 ¥ answer (3) 23 |S Y limits Dbn=2) mi ¥5n-2 (2) 2 Fae ‘7 Sy, = 2531 +1263] substitution 2 v answer Soo = 160149 OR Sys3 = #120) +(253-1)5] ¥ substitution ¥ answer (2) Sy5; = 160149 2.5 | 8428448 +,..+ 1248 ¥ terms divisible by 4 a=8,d=20 T, =20n-12 ¥ general term 20n—12 =1248 n=63 ¥ equating 4) v value of 7. [12] Copyright reserved Please turn over

‘pewrilsdded from Stanmeceshysics.com QUESTION 3 MDE/June 2023 3.1.1 ¥ first differences 2 p-T ¥ equating p-7-4 ¥p=13 @) p-7-4=2 p=13 3.1.2 | 2a=2 3a+b=2 atb+c=l a=l 3(l) +b =2 l-l+c=1 |“a=l b=-1 c=! Yb=-1 v _ te c=! (4) T, =n? -n+1 VT, =n n+l 3.1312,4, 6... a=2 d=2 T, =2n ¥ general term on = 62 ¥ equating n=31 Term number 31 and term number 32 Y both 31 and 32 @) 3.2 2.2"? < 2046 ka? 2.29 +2.2' 42.2? +2.2° +..+2.2"? < 2046 Y expansion 24+44+84+164...4 a=2, r=2, n=m-(-2)41 at Od =m+3 m+3 . 2(2"* -1) < 2046 ¥ substitution into -1 correct formulae 2" 1 <1023 2" < 1024 ques <2! m+3<10 ¥m<7 5 m<7 vm=6 ” “m=6 faYaY {15} Copyright reserved Please turn over

. 4 MDE/June 2023 '‘Downtedded from Stanmereshysics.com QUESTION 4 4.1.1 =2+x | ¥ method 2 v¥x=4 x=4 B(4: Answer only full marks 2) 4.1.21 yet Qer- ¥ method y 2x-8 ¥ correct equation (2) 4.1.3 | C(0;-8) ¥ C(0;-8) a= 8-0 ¥ substitution 0-4 Ya-2 a= : =2 ¥q=-8 (4) g(x) =2x-8 a=2 q=-8 4.1.4} y=(1)? -2()-8 v value of y = v ye (;-9) (13-9) 2) 4.2 | O+x_ 1 v¥x=2 > =2 * YMQ2;-8) (2) M(2;-8) 4.3 | x<0 or x>4 v¥x<0 Vx>4 (2) 44 |@;9) Vx=3 vy=9 (2) 45 | 7p =2x—-8-(x* -2x-8) ¥ top function —bottom TP =-x? +4x function TP' =-2x+4 v derivative -2x+4=0 ¥ equating t x=2 Y answer TP =-(2)° +4(2) TP =4 Copyright reserved Please turn over

i, MDE/June 2023 Downtedded from Stanmore sics.com une Y factoring a negative ¥ completing the square ¥ “maximum 4 when x=2 (4) There is a maximum of 4 when x = 2 [20] QUESTION 5 5.1 |x>0 OR xe (0;0) ¥ domain () 5.2 | y=-log,x y=log, x" ¥ interchanging x and y x=log, y" 3" = y? _{1y -* Y\3) FYE 3 ¥ correct equation Answer only full marks (2) 5.3 | fis shifted 5 units vertically down v5 units ¥ down (2) 5.4 | log,x=3 %x=27 xa3 ¥ —log, x 2-3 x27 VV 0<xS27 “. -log, x 2-3 (4) 0<x<27 ia i e Copyright reserved ES Please turn over

Dowrtydded from Stanmeceshysicsscgn MDE/June 2023 QUESTION 6 61 | q=2 Y answer (4) 62 Tpt2 vp =-] y=xtl 2=x+l1 x=l OR ¥ sub (0;-4) vane ) 6.3 ya ------------- waa dq-------------- y=? -2 vx intercept vy intercept ¥ asymptote ‘ ¥ shape of graph (5) [10] [aYaYey Copyright reserved Please turn over

Downisaded from Stanmerephysics.com MDE/June 2023 ig, Guidelines QUESTION 7 TA bei -(1- 2304) ¥n=12 oe 12 iy, £49409103376 — 1 j= 2104 v 12 ig 24O191% Y substitution into the correct formula ¥10,91% (4) 7.2 0,5P = P(i-i)* ¥ subs in correct formula 0,5=(1-1* ¥ divide by P 40,5 =1-i ‘ v4" root i=1- 4 0,5 i=0,1591 ¥i=15,91% i =15,91% (4) 7.3.1 | Loan with interest to date = 80000(1 + 0,21)° ¥ subs in correct formula = R207499,40 ¥ R20 7499,40 (2) 7.3.2 | Repayments with interest to date ¥ 25000(1 +.0,21)° _25000(1+ 0,21)? +55000(1 + 0,21)’ ¥55000(1 40,21)! = R101673,20 . a. ¥ R101673,20 Outstanding balance =Loan with interest to date- “method . Repayments with interest to date ¥105826,20 =R207499,40 - R10 673,20 =R105826,20 (5) 15 — [iS] Copyright reserved Please turn over

Downtsaded from Stanmor Sics.cOomM MDE/June 2023 jarking Guidelines QUESTION 8 8.1 P(x h) =4(x +h) 2x +h)? =4x+4h-2x? -4xh—2h? fG mn LEH FH) Y f(x+h) 4.0 h r oy AetaN= 28? — ach 20? —(4x-2x?) 0 A ¥ subt. into correct 2 F'@)= tim Vivdxhn 2h formula hoo h ¥ 4h 2 r'@) = lim A(4—4x—2h) 4h—4xh-2h hoo h S'(x) =4-4x ¥ factoring A v answer (5) 8.2.1 | p jx! -x?] Vx = 4x9 42x74 v4 v2xt [G) 8.2.2 x? 4x ¥ dividing by x vy = x x vy =x-4 ¥ squaring y=(x-4)? y=x? -8x+16 @ 2-8 v¥ answer (4) 83 | m=36 Vv f'(x) = 36 f'(x) = 3x? -12x 3x* -12x =36 3x? -12x-36=0 Y factors (x-6)(x +2) =0 Vx=-2 x #6 or x=-2 f(-2) = (-2)° - 6(-2)’ v¥ y=-32 f(-2)=-32 y—(-32) = 36(x -(-2)) 5 y=36x+40 Y equation? ©) cr Copyright reserved EE Please turn over

Wobwritsaled from Stanmerephysics.com MDE/June 2023 QUESTION 9 91 1CQ@:0) V answer @ 9.2 ¥ substitute (—1 ; 0) ¥ subject of formula ¥ substitute (2 ; 0) ¥ sub in second equation (4) 9.3 | f(x) =-3x? +3 ¥ equating derivative 0=-3x? +3 to zero 0=-3(x-D(x+1 vx-value x=l £0) =-()’ +30) +2 =4 vy-value BU: 4) GB) 94 | -1+1_ 0 ¥ method 2 OR f'() =-3x? +3 f"(x) = -6x ~6x=0 ¥ answer (2) x=0 95 | k<Oork>4 v¥k<O0 Vk>4 e) 9.6 | x<-lor x>1 ¥ both critical values Vx< Bing >1 (2) | Copyright reserved Please turn over

Dowriksddeer from Stanmoreph¥sics.com MDE/June 2023 QUESTION 10 10.1.1 | PB 1+ 0,4 ¥ P(B) =0,5 5 Pi (B) = 0,1 ¥ P(A) x P(B) = 0,1 (*¥0,1(0,5) = 0,1 x=0, ¥x=0,1 y=l-0,1 = 4 y=0.9 ¥y=0,9 @ 10.1.2 | P(AorB) = P(A) + P(B) - P(A and B) ¥ formulae = 0,2+0,5-0,1 v P(A) = 0,2 = 0,6 ¥ answer (3) 10.2.1 R VR o% ¥ 0,1 0,95 Vv 0,1 0,05 ~ NR VNR v 0,1 x 0,05 R Py 0,45 NVR NV ¥ non violent 0355S NR NVNR (3) 10.2.2 | P(Violent crime rep) or P(Non -violent crime rep) | ¥ 0,1x0,95 = 0,1« 0,95+0,9 x 0,45 ¥ 0,9x 0,45 =0,5 ¥ answel (3) [aYaYay [13] TOTAL MARKS: 150 Copyright reserved