Maths-Gr10-June-2022-QP-_-Memo_hlayiso.com_.pdf

Downloaded from hlayiso.com KWAZULU�NATAL PROVINCE EDUCATION REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE [ GRADE 10 ] � �----------------------------------------�� :: MATHEMA ICS !'i II II II II II II II II II II II II II II MARKS: TIME: 2 hours This question paper consists of 10 pages. Copyright reserved Please tum over

Downloaded from hlayiso.com INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Read the questions carefully. 3. Answer ALL the questions. 4. Number your answers exactly as the questions are numbered. 5. Clearly show ALL calculations, diagrams, graphs, etc. which you have used ip.,._�rmining your answers. e;v 6. Answers only will NOT necessarily be awarded full marks. • VC,• 7. You may use an approved scientific calculator (non-programmabi).�� non-graphical), unless stated otherwise. 8. � unless If necessary, round off answers correct to TWO decimal'�es, ►J stated otherwise. 9. () Diagrams are NOT necessarily drawn to scale. �� 10. Write neatly and legibly. Copyright Reserved Please tum over

Downloaded from 3 hlayiso.com Mathe�tics Common Test June 2022 QUESTION 1 1.1 Simplify the following expression fully: ( 2) 1.2 Factorise the following expression fully: (2) 1.3 Solve for x: 1.3.1 (x+m)(x-n)=O (2) 1.3.2 (3) 1.4 Simplify the following: (4 ) [13) QUESTION2 1 2.1 Consider the general term: �� _2n + 5 \ 2.1.1 Write dowb�t TWO tenns of the sequence. (2) *� 2.1.2 Dettq.?e which term in the sequence has a value of (2) \.� 395 2.2 r'1 7 13 Co�der the following sequence: 5 · - · a · b · - · c ' 4' ' ' 9 ' . 2n+3 The general term for the sequence 1s T,, = --2- n Determine the values of a , b and c. (3) Copyright Reserved Please tum over

Downloaded from hlayiso.com 2.3 The first four patterns in a sequence are shown below. Each pattern is made from dots and one-centimetre lines. The area of each small square is l cm 2 . Pattern I Pattern 2 Pattern 3 Pattern 4 B 2.3. l Use the table below and pattern above to determine the values of a, b, c, and d. Pattern 1 2 3 4 5 Area (cm 2) 2 6 12 20 a Number of dots 6 b 20 30 C Number of one centimetre lines 7 17 31 49 d (4) 2.3.2 The area of pattern n can be written as T,, = n(n + 1) cm2 • Find the area of pattern 50. (1) [12) Copyright Reserved Please tum over

Downloaded from hlayiso.com Common Test June 2022 QUESTION3 3.1 The graphs of /(x) = a:c2 + q and g(x) = mx + c are sketched below. The graph off intersects the x-axis at (- 4; 0) and B, and the y-axis at (0 ; 4) which is also the turning point off One ofthe points of intersection of/ and g is A. y 4 g B X r Use the graphs and the information given above to determine: 3.1.1 the equation of g. (2) 3.1.2 the values of a and q. (3) 3.1.3 the coordinates of B. (1) 3.1.4 the domain and range off (3) 3.1.5 the equation of k (x), if k is the graph off reflected about the x-axis. (2) 3.1.6 one value of x for which f(x) = 0. (I ) 3.1.7 the value(s) of x for which f(x) > g (x), given that A (2 ; 3). (2) 3.2 The function h(x) = kx + q, is described with the following properties: k> 0;k:;t:l h(0) =-7 h(3) =0 horizontal asymptote : y = -8 (3) Using the information provided, draw a neat sketch graph of h(x) = kx + q [17] Copyright Reserved Please tum over

Downloaded from hlayiso.com NSC - Gl�.DE 10 QUESTION 4 12 Given: f(x)=-+3 4 .1 Write down the equations of the asymptotes. (2) 4 .2 Determine the equation of g , the axes of symmetry off, with a positive gradient. (2) 4 .3 Sketch the graph off and g on the same system of axes, indicating all intercepts with the axes and asymptotes. (4) (8) QUESTIONS 5-1 In MFG, E = 90° and fr= 0. EG = 2 units, EF = 3units and FG=-✓13 units. E Write down the values of: 5.1.1 tan0 ( 1) 5.1.2 cosec0 (1) 5.1.3 cos(9O° -0) (2) Copyright Reserved Please tum over

Downloaded from hlayiso.com Common Test June 2022 5.2 In the diagram below, P(x ; y) is a point in the fourth quadrant. BOP=0 and 17 cosB-15 = 0. y X B ,. P(x; y) Make use of the information provided and the diagram to: 5.2.1 determine the length of OP. 5.2.2 calculate the value of tan 0. (2) 2 (2) 5.2.3 prove that cos 0+ sin2 0 = l (9) QUESTION 6 6.1 If x = 15° and y =22,5° , use you calculator and determine ( correct to TWO decimal places) the following: 6.1.1 cos3 x-sin2 x (2) 5sec2y 6.1.2 (2) cotx 6.1.3 ✓cosec(y-x) (2) 6.2 Solve for x, correct to ONE decimal place, where 0 ° s; x s; 90 ° : 6.2.1 tan2x = 1,01 (2) sin(2x-20 °) 6.2.2 -�------'-=0,099 (3) 3 6.3 Without the use of a calculator, showing all working, determine the value of: sin 60 °.sec 30 ° + tan2 60 ° (5) 2 tan45° (16) Copyright Reserved Please tum over

Downloaded from hlayiso.com QUESTION 7 The graphs of f(x) = atanx and g(x)=2cosx+ p for x E [0 °;360°] are sketched below. -1 -2 g --4 Use the graphs and the information provided to answer the following questions. 7.1 Write down the values of a and p. ( 2) 7.2 Write down the following: 7.2.1 the amplitude of g. (1) 7.2.2 the range off (I) 7.2.3 the period of f (1) 7.3 Use the graphs to determine the number of solutions to f(x) = g(x) in the interval XE [0°;180°] (1) [6] Copyright Reserved Please turn over

Downloaded from hlayiso.com th Common Test June 2022 Ma Give reasons for ALL geometry statements in QUESTIONS 8 and 9. QUESTION 8 8.1 In the diagram below, the diagonals of a rhombus ABCD intersect at 0. .Bl = X and c\ = 3x -16° . D � 8.1.1 State the value of 03 . ( 1) 8.1.2 Calculate the value of x. ( 3) 8.1.3 Hence, find the value of A 1 • (2 ) 8.2 � � In the quadrilateral PQRS below, Q1 = 58 ° and PSR = 110° . QS .l_ PR, QR= RS and PQ = PS. s Use the diagram to calculate, with reasons, the size of: 8.2.1 (2) 8.2.2 ( 3) [11) Copyright Reserved Please tum over

Downloaded from hlayiso.com .- • • • 7 ••• - -••-. -. -. QUESTION9 9.1 In the diagram below, EFGH is a square and P is a point on HG such that EPH = 39° . EP and FH intersect at D. p (4) Calculate the size of EDF. 9.2 In the diagram below LP II SN,LM II PN and PQ = MN. Let LQS = x. s N Use the diagram to prove, with reasons, that QSN = 2LQS (5) (9) TOTAL: 100 Copyright Reserved

Downloaded from hlayiso.com KWAZULU-NATAL PROVINCE EDUCATION REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE I GRADE 10 j � �-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --��� # ,; MATHEMATICS •�, COMMON TEST JUNE 2022 MARKING GUIDELI E It � � � � • MARKS: 100 TIME: 2 hours NOTE: • If a candidate answered a QUESTION TWICE, mark only the FIRST attempt. • If a candidate crossed out an answer and did not redo it, mark the crossed-out answer. • Consistent accuracy applies to ALL aspects of the marking guidelines. • Assuming values/answer in order to solve a problem is unacceptable. This marking guideline consists of 9 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline QUESTION 1 I. I 1 2 9 x y( 6x- x-2y2) 3 2 3 3 =2x3y--y3 ✓ 2x3 y ✓ --y3 2 (2) 2 1.2 (a+l) 2 -4b2 ✓ (a+1-2b) = [( a+1) -2b] [( a+l)+2b] ✓ (a+1 + 2b) = ( a + 1-2b)(a+l +2b) (2) 1.3.1 (x+m)(x- n)=0 ✓ x=-m ✓ x=n x=-m or x=n (2) 1.3.2 4 x -4° -255=0 ✓ simplification 4 x -1-255 =0 4 x = 256 ✓ prime base 2 2x = 28 :. 2x = 8 ✓ answer (3) :. x=4 1.4 y+2_7 4 y+ _ 6.2 x +I ✓ Y.22 .7 = Y.2 2.7 ✓ 2x.24 - 6.2 x.i 2-'.24 -6.2 x.i Y.22.7 = x 4 ✓ common factor: 2x 2 (2 - 6.2) 28 =- 4 (4) ✓ answer =7 (13] QUESTION2 2.1.1 I !3 .' 1 ✓ - 3 ✓ I (2) 2.1.2 1 1 = -- -2n+5 395 ✓ substitution -2n +5 =-395 -2n =-400 ✓ answer (2) n=200 Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline 2.2 2(3) +3 ✓ a =1 ½= = a =l (3)2 2(4) +3 ✓ b=.!..!_ T- 4- 2 = b=.!..!_ 16 (4) 16 2(5)+3 13 13 Tr,= = c=- ✓ C=- (5) 2 25 25 (3) 2.3.1 2 ; 6 ; 12 ; 20 ; 30 ✓ a=30 ✓ b=l2 ✓ c=42 6 ; 12 ; 20 ; 30 ; 42 ✓ d =71 7 ; 17 ; 31 ; 49 ; 71 (4) 2.3.2 T,, =n(n+l) ½o = 50(50 +1) ✓ answer ½o = 2550cm2 (1) [12) QUESTION 3 3.1 y 4 g A 2 '\ -4 0 X 3.1.1 1 ✓ gradient g(x)=-x+2 ✓ y intercept 2 (2) 3.1.2 q=4 ✓ q=4 y=ax 2 +4 sub (-4;0) ➔ 0=a(-4) 2 +4 ✓ substitution -4 I a=-=-- 1 16 4 ✓ a=-- (3) 4 3.1.3 B(4;0) ✓ answer (1) 3.1.4 y�4 ✓ values or ✓ notation (2) yE(--00; 4] 3.1.5 1 l k(x)=-x 2 -4 ✓ - ✓ -4 4 4 (2) 3.1.6 x=4 or x=-4 ✓ either value (l) Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline 3_1.7 -4<x<2 ✓ values or ✓ notation XE (-4;2) (2) 3_2 y t h I ✓ shape: increasing ✓ intercepts -7 ✓ asymptote -8 y=...... _/ ...................................... (3) r161 QUESTION 4 4.1 x=O ✓ answer ✓ answer (2) y=3 4_2 y=x+c ✓ gradient = + 1 sub (0;3) ➔ y =x+3 ✓ y-intercept: (0;3) (2) 4_3 I f y r ✓ shape: increasing g ✓ x-intercept ✓ asymptote y = 3 v=3 �- - - - -· · · · · · ·-· · · · · · ·�"' '7 3 ................................................... 0 ✓ g: x- and y- intercepts (4) [8] Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline QUESTIONS E EG 2 tan0=-=- EF 3 5.1.l ✓ answer (1) cosec0=-=- 5.1.2 .JI3 ✓ answer 2 GF (1) 2 GE cos(9O° -0) = cosG = .Ju 13 5.1.3 ✓ answer (1) 5.2 y 0( \ 15 X � 7 � P(l5 ; - 8) 15 17 5.2.1 cos0=- ✓ answer (1) =172 -15 2 :.OP=17units 5.2.2 y2 /=64 ✓ y=-8 :. y =-8 (4th quadrant) . y -8 :. tan0=-=- 15 ✓ answer (2) LHS: cos 2 0+sin2 0 X 5.2.3 = + 2 C�) (��J ✓ substitution 289 289 l =- = 289 289 ✓ =1 =RHS :. cos 0+sin 0 = 1 (2) 2 2 [8] Copyright Reserved Please turn over

Downloaded from hlayiso.com Mathe Common Test June 2022 QUESTION6 6.1.1 cos3 x-sin2 x ✓ substitution = cos3 l5 °-sin2 l5 ° =0,83 Answer only: full marks I ✓ answer (2) 6.1.2 5 sec2y cotx 5 sec2(22,5 °) = ✓ substitution cot l5 ° 5 tan1 5 ° = cos2(22, 5 °) =1,89 I Answer only: full marks I ✓ answer (2) 6.1.3 ✓cosec(y-x) 1 ✓ substitution = \ sin(22,5 -l5 ) = 2,77 ° ° I Answer only: full marks I ✓ answer (2) 6.2.1 tan2x =1,01 2x= 45,28505.... Penalty for incorrect ✓ 45,28505... rounding in this question ✓ answer x= 22,6 ° only. (2) 6.2.2 sin( 2x-20 ° ) = 0,099 3 ' ✓ 0,297 sin(2x-20 °) = 0,297 � ✓ 7,2775... 2x-20°=17,2775... ✓ answer x=18,6 ° (3) 6.3 sin 60 °. sec30 ° + tan2 6� 2 tal!5 ° 11 ✓3 .2-+ ( ./3 2 2 ✓3 ) ✓✓3 ✓ 2_ ✓ ✓3 = 2 ✓3 2.1 1+3 ✓ tan45 =1 ° I I =- 2 Answer only: max. 1/5 ✓ answer (5) =2 [16) Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline QUESTION7 y 2 100° 150" 250° -1 -2 -3 g 7.1 a=2 ✓ answer p=-1 ✓ answer (2) 7.2.1 amplitude ofg = 2 ✓ answer (1) 7.2.2 yER ✓ answer (I) or ✓ answer y E (-oo;oo) (I) 7.2.3 period off= 180° ✓ answer (I) 7.3 2 solutions ✓ answer (I) [6] GEOMETRY A mark for a correct statement (A statement mark is independent of a reason.) R A mark for a correct reason (A reason mark may only be awarded if the statement is correct.) Award a mark if the statement AND reason are both correct. S/R Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline QUESTIONS D 8.1.1 03 -90° (diags of rhombus) ✓ s (1) 8.1.2 In MOC: 03 =90° (diags of rhombus) B1 =B2 =x (diags of rhombus) ✓ S/R ✓ S/R C1 =3x-16 ° (given) : . 3x-16 ° +x+90° = 180° (sum Ls�) ✓ s (3) ° :. x=26,5 8.1.3 AI --c - 3x -16 I - ° (alt Ls; AD II BC) ✓ S/R .". A_ ! =3(26,5° ) - 16 ° = 63,5° ✓ s (2) 8.2 Q s 8.2.1 InMQT: ✓ S/R QTP=90° (diags of kite) :.Pi =180° -58° -90 ° (sum Ls�) ✓ S/R :. Pi= 320 (2) 8.2.2 In MRS: PSR = 110° (given) ✓ s ✓ R :. Pi = Pi = 320 (diags of kite) :. R 2 = 180° -110° -32° (sum Ls�) ✓ S/R :. R 2 = 3g o (3) [11] Copyright Reserved Please turn over

Downloaded from hlayiso.com Common Test June 2022 NSC - Grade fO iv!arking Guideline QUESTION9 9.1 In�EDF: FEP= 39° (alt Ls; EF II HG) EFH = 45 ° (diags of square) :. EDF= 180 ° - 45 ° -39° (sum Ls�) ✓ SIR :. EDF= 96° (4) s N Q 9.2 n parallelogram LMNP ✓ S/R MN=LP (opp sides of parm butPQ= MN ✓ :.LP= PQ :.QLP = X ✓ S/R :.LPS = 2x (ext LMPQ) ✓ s ✓ R LPS = QSN =2x (altLs; LP II Mn A A and LQS =x (given) :.QSN = 2LQS A A (5) (9) TOTAL: 100 Copyright Reserved