Mathematics Grade 12 GP March 2023 Paper & Memo

Downloaded from Stanmorephysics.com GAUTENG PROVINCE EDUCATION REPUBLIC OF SOUTH AFRICA JOHANNESBURG WEST DISTRICT MARKS: 50 DURATION: 1 HOUR This question paper consists of 6 pages including the formula sheet.

BDownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 rade 12 INSTRUCTIONS AND INFORMATION 10. This question paper consists of 6 questions. Answer ALL the questions in your answer book. Use the appropriate and correct numbering system as it is used on this paper. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. 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. Diagrams are NOT necessarily drawn to scale. An information sheet with formulae is included at the end of this paper. It is in your own interest to write legibly and to present your work neatly. Copyright reserved Please turn over

BDownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 rade 12 QUESTION 1 1.1 Solve for x. 11.1 -x(2x-3)=0 (2) 11.2 0=3x?-5x-11 (3) 1.1.3. x(8x + 4) —2(8x+ 4) <0 (3) 1.2 The roots of a quadratic equation, in terms of p, are given as: 4+/8—p3 ya we Determine the value(s) of p for the roots fo be real. (3) [11] QUESTION 2 The first four (4) terms of a quadratic pattern are: 11; 20; 33; 50;... 2.1 Determine the general term of this pattern in the form T,, = an? +bn+c. (4) 2.2 Prove that the sum of the first n first-differences of this quadratic pattern can be given by S, = 2n?+7n. (2) [6] QUESTION 3 A convergent geometric series is given by: 4 + Mev + a 3.1 Calculate the values of x. (3) 3.2 If x =1, calculate the sum to infinity, 5, . (2) [5] Copyright reserved Please turn over

BDownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 rade 12 QUESTION 4 (1—x) ; (x+2) and (2x —5) are the first three (3) terms of an arithmetic sequence. 4.1 Determine the value of x . (3) 4.2 If the first three (3) terms of this pattern are: 9 ; —6 ; —21; ..., calculate the numerical value of the sum of the first 100 terms, Syqq . (2) 4.3 Hence, or otherwise, calculate m if: m-1 yy (24 — 15n) = Stop + 73 320 (4) n=0 [9] QUESTION 5 The graph of f(x) = ao +q_is sketched below: Ay | \f i < 30-1 Xx i 1 | —______s aaa --- eee poo fe { -1 \ | v 5.1 Write down the values of p and q (2) 5.2 The straight line g(x) = —x +k is one of the axes of symmetry of the graph of f . Determine the value of k. (2) 5.3 If h(x) = —2[g(x)], determine the equation of the inverse of A, A, in the form h-4(x)=mxt+c. (3) 5.4 Draw aneat sketch of the graphs of h and h~1 on the same set of axes. Clearly show all intercepts with axes, point of intersection and the axis of (4) symmetry. [11] Copyright reserved Please turn over

BDownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 rade 12 QUESTION 6 The graphs of f(x) = ax? and g(x) = b* are sketched on the same set of axes. Points A(—1;2) and B are points of intersection of f and g . The graph of f has the turning point at the origin: A(-1;2) A 6.1 Calculate the values of a and b. (2) 6.2 The inverse of f is NOT a function. Write down at least one condition which can be used to restrict the domain of f such that its inverse will bea function. (1) 6.3. For which value(s) of x , where x € (—00;0], will g(x) S f(x)? (2) 6.4 If h(x) = g(x + 3) , write down the coordinates of ... 6.4.1 A’ , the new coordinates of A on the graph of h. (1) 6.4.2 A" , the new coordinates of A onthe graph of h-?, the inverse of h (2) . [3] TOTAL = 50 MARKS Copyright reserved Please turn over

Downloaded from Stanmorephysics.com INFORMATION SHEET _-b4Vb? -4a0 2a A=P(l+ni) A=P(-ni) A=P(-i)" A=P(+i)” T, =a+(n—Nd Ss, =F a+ (nnd) T, =ar"" s, = sea!) gr#l S.=jSs-ler<l r- -r patsy -1) pa2t-a+)"] i i ‘¢¢) = Jim LAtM-FO) i rr y27-Y y=mxte Y-yY =m(x—-x,) n= m=tan@ ae | (x-a) +(y-by =r? fading sinA sinB sinC a? =b? +c? —2be.cosA area AABC = Sab sinc sin(a + £)=sina.cosf+cosa.sinB sin(a—B)=sina.cosB —cosa.sinB coda + £)=cosa.cos#—sina.sin cos(a — 8) =cosa.cosf# +sina.sinB cos’ a —sin’ a cos2e@ =41-2sin’? a sin2a = 2sina.cosa 2cos? a -1 : S (x, x)? __>s# - 2 %) x= —— n n n(A) = P(A) = (5) P(A or B) = P(A) + P(B) - P(A and B) n pratbx p= 2-2) De-% Copyright reserved

Downloaded from Stanmorephysics.com GAUTENG PROVINCE EDUCATION REPUBLIC OF SOUTH AFRICA JOHANNESBURG WEST DISTRICT GRADE 12 eee eee eee eee eee eee eee eee eee eee cee eee eee eee cece MARKS: 50 This Marking Guidelines consists of 6 pages including this cover page.

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 IMPORTANT NOTES AND INFORMATION + Ifa candidate answers a question TWICE, only mark the FIRST attempt. ¢ Ifa candidate has crossed out an attempt of a question and not redone the question, mark the crossed out version. * Accept any other Mathematically valid attempt which yields a correct answer and credit full marks. * Consistent accuracy applies in ALL aspects of the marking memorandum. « Assuming answers/values in order to solve a problem is NOT acceptable. QUESTION 1 Q# Suggested Solutions Descriptors 11.1 x(2x — 3) =0 v¥x=0 x=O0orx=2 vx=2 2 2 (2) 1.1.2 0 = 3x*-—5x-11 Y correct substitution __ -btvb?=4ac ¥x=-1,25 x= 2a ¥x=2,92 -(-5)tVC5)2-4@) C1) oo) or OCD (3) x =-1,25 or x = 2,92 11.3 x(3x + 4) — 2(3x +4) <0 ¥ factors/ method (x-2)@x+4) <0 4 Y critical values Critical values: x = 2; x = -3 ‘ ¥ solution -f<xs<2 “3 3 (3) 1.2 For real roots: ¥8-p?>0 8-p>=0; p#0 ¥p#0 —p? =>-8 Yps2 piss8 (3) ps2 Subtotal [11] QUESTION 2 Q# Suggested Solutions Descriptors 21 11; 20 ; 33; 50 Ya=2 1% differences: 9; 13; 17 ¥b=3 2™ differences: 4; 4 ¥c=6 . 2a=4 3a+b=9 atb+c=11 |” In=2n?+3n+6 a=2 3(2)+b=9 24+34c=11 b=3 c=6 (4) Ty = 2n? + 3n+6 2.2 1 differences: 9; 13; 17 Y correct substitution a=9:d=4 ¥ simplification S, = 5a + (n= 1d] (2) = 51209) + (n~ 1)4] = 5 {14 + 4n] = 7n + 2n? Subtotal [6] Copyright reserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 QUESTION 3 Q# Suggested Solutions Descriptors 3.1 = = Y-1< = <1 -1<*P<1 -3Se+1<3 yises? @) —4<x<2, but x#—-1 3.2 Sa ye Y correct substitution _ sey v answer 141 rae (2) = 10 Subtotal [5] QUESTION 4 Q# Suggested Solutions Descriptors (2x —5) — («+ 2) = (x+2)-(1-x) v method 4.1 x-7=2x+1 ¥ simplification -8=x ¥ x=-8 (3) 9;-6;-21 ¥ correct substitution 4.2 a=9,d =—15,n=100 y n Sy = 5 [2a + (n= 1d] answer °) 100 St00 = = BO) + (100 — 1)(—15)] = —73 350 43 m=1 v RHS = —30 (24 — 15n) = Si99 + 73 320 ¥ F224) + (m= 1)(-15)] n=0 ¥ standard form = —73 350 + 73 320 ¥ m= 5 and rejection (4) = [2(24) + (m — 1)(—15)] = —30 m(48 — 15m +15) = —60 63m — 15m? = -60 0 = 15m? — 63m — 60 0 = 5m? — 21m — 20 0 = (5m+4)(m—-5) m=5 0rm# -: Subtotal [9] QUESTION 5 Q# Suggested Solutions Descriptors 5.1 p=3 Yp=3 q=-1 ¥q=-l (2) 5.2 gix)=-x+k ¥ correct substitution -1=-(-3)+k _...at point (—3;-1) y¥ -4=k —4=k (2) Copyright r eserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 5.3 h(x) = —2(—x - 4) ¥ h(x) = 2x +8 =2x+8 v swapping x and y For inverse: Vv yan 4 x=2y+8 a) x—8=2y (3) x-85 “Fan -yss-4 shl(x)=2+-4 5.4 y Y correct graph shapes v x- and y-intercepts interchanging ¥ correct point of intersection ¥ axis of symmetry passes through the correct point of < intersection (4) Subtotal [11] QUESTION 6 Q# _| Suggested Solutions Descriptors 6.1 | f(x) = ax? g(x) = b* Ya=2 2 =a(-1)? 2=b? il na=2 b= 1 vb= 2 ~2 (2) 6.2 |x<0 orx>0 ¥ any one of the correct conditions (1) 63 |x<-1 OR x€(—o;-1] OR -w<x<-1 Y correct critical value Y correct notation (2) 6.4.1 | A’ (—4; 2) ¥ (-4;2) (1) 6.4.2 | A" (2; -4) ¥¥(2;-4) (2) Subtotal [8] TOTAL = 50 MARKS Copyright reserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 QUESTION 1 1.1 Solve for x. 11.1 x(2x-—3)=0 (2) 11.2 0=3x?-5x-11 (3) 113 x(3x+4)-2@x+4 <0 (3) 1.2 The roots of a quadratic equation, in terms of p, are given as: 4+/8-p3 x=—~Y fe (3) Determine the value(s) of p for the roots to be real. [11] QUESTION 2 The first four (4) terms of a quadratic pattern are: 11; 20; 33; 50;... 2.1 Determine the general term of this pattern in the form T,, = an? + bn+c. (4) 2.2 Prove that the sum of the first n first-differences of this quadratic pattern can be given by S, = 2n?+7n. (2) [6] QUESTION 3 a S(xt+1 A convergent geometric series is given by: ~ dy 5(x+1)? + 5(x+1)3 9 27 Copyright reserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 3.1 Calculate the values of x. (3) 3.2. If x =1, calculate the sum to infinity, 5, . (2) [5] QUESTION 4 (1—x) ; (« +2) and (2x —5) are the first three (3) terms of an arithmetic sequence. 4.1 Determine the value of x. (3) 4.2 If the first three (3) terms of this pattern are: 9 ; —6 ; —21; ..., calculate the numerical value of the sum of the first 100 terms, Sio9 . (2) 4.3 Hence, or otherwise, calculate m if: > (24 — 15n) = Sip + 73 320 (4) n=0 [9] QUESTION 5 2 The graph of f(x) = vp +q_ is sketched below: A | ay xv Copyright reserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 5.1 Write down the values of p and q (2) 5.2 The straight line g(x) = —x + k is one of the axes of symmetry of the graph of f . Determine the value of k. (2) 5.3 If h(x) = —2[g(x)] , determine the equation of the inverse of A, h~?, in the form h-4(x) =mxt+c. (3) 5.4 Draw aneat sketch of the graphs of h and h~1 on the same set of axes. Clearly show all intercepts with axes, point of intersection and the axis of (4) symmetry. [11] Copyright reserved Please turn over

Bownleadedtfrom Stanmorephysics.com JW-GDE/ March 2023 Marking Guidelines rade 12 QUESTION 6 The graphs of f(x) = ax? and g(x) = b* are sketched on the same set of axes. Points A(—1;2) and B are points of intersection of f and g . The graph of f has the turning point at the origin: A(-1;2) A 6.1 Calculate the values of a and b. (2) 6.2 The inverse of f is NOT a function. Write down at least one condition which can be used to restrict the domain of f such that its inverse will bea function. (1) 6.3. For which value(s) of x , where x € (—00;0], will g(x) S f(x)? (2) 6.4 If h(x) = g(x + 3) , write down the coordinates of ... 6.4.1 A’ , the new coordinates of A on the graph of h. (1) 6.4.2 A" , the new coordinates of A onthe graph of h-?, the inverse of h (2) . [3] TOTAL = 50 MARKS Copyright reserved Please turn over

Downloaded from Stanmorephysics.com INFORMATION SHEET _-b4Vb? -4a0 2a A=P(l+ni) A=P(-ni) A=P(-i)" A=P(+i)” T, =a+(n—Nd Ss, =F a+ (nnd) T, =ar"" s, = sea!) gr#l S.=jSs-ler<l r- -r patsy -1) pa2t-a+)"] i i ‘¢¢) = Jim LAtM-FO) i rr y27-Y y=mxte Y-yY =m(x—-x,) n= m=tan@ ae | (x-a) +(y-by =r? fading sinA sinB sinC a? =b? +c? —2be.cosA area AABC = Sab sinc sin(a + £)=sina.cosf+cosa.sinB sin(a—B)=sina.cosB —cosa.sinB coda + £)=cosa.cos#—sina.sin cos(a — 8) =cosa.cosf# +sina.sinB cos’ a —sin’ a cos2e@ =41-2sin’? a sin2a = 2sina.cosa 2cos? a -1 : S (x, x)? __>s# - 2 %) x= —— n n n(A) = P(A) = (5) P(A or B) = P(A) + P(B) - P(A and B) n pratbx p= 2-2) De-% Copyright reserved