KZN GR 12 MATHS P2 SEPT 2024_hlayiso.com_.pdf

» | KWAZULU-NATAL PROVINCE EDUCATION REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 gee RRR R RRs E ERS EER ERR R REE RA EEES MATHEMATICS P2 seeeee ewe ee, PREPARATORY EXAMINATION SEPTEMBER 2024 MARKS: = 150 TIME: Jhours ‘This question paper consists of 13 pages and | information sheet. opytight Reserved Please Tum Over Powered by CamScanner

wanna DOWNIOAadEe FrQm hlayiSQuGQMaion september 2024 NSC QUESTION 2 The cumulative frequency graph (ogive) drawn below shows the ages of the people who voted in the Local Government olections at one voting station, Use the graph to answer the questions that follow. Ogive for ages of voters in one voting station 6000 | 5500 5000 4500 > Ss Ss Ss w a S s w Ss Ss Ss nN a S S Cumulative Frequency 0 5 10 15S 20 25 30 35 40 45 50 55 60 Ages of Voters in years 2.1 How many people voted at this voting station? (1) 2:2 Determine the interquartile range of the ages of the voters, (3) 2.3 What percentage of the voters was 25 years or younger? (2) (6] Copyright Reserved Please Turn Over Powered by ics] CamScanner

Matenaioor? DOWnIOaded frqm hlayiSQ.GQWAkion seprenber2024 NSC QUESTION 3 ABCD is a parallelogram with A(0;4), B(-6;-2), C(2;y) and D(10;-1) as shown below. AG is drawn. F, G and H are the x-intercepts of AB, AC and AD respectively. Eis a point on the x-axis to the left of F and Ja point on the x-axis to the right of H. y A(0;4) E FE H J [e) D(10;-1) B(-6;-2) Cay) 3.1 Determine the gradient of AB. (2) 3.2 Determine the equation of CD. GB) 3.3 Determine the coordinates of M, the midpoint of AC. @) 3.4 Determine the coordinates of C. Q) 3.5 Determine the size of BCD. tO) [16] Please Tum Over Copyright Reserved Powered by Ics) CamScanner

° ie... | Maematiove2 DOWNIOAdEM Fron hlayiSe@rOrVAination seprember 203, NSC QUESTION 4 4.1 The diagram below shows two circles touching at J (23-1) : The smaller circle has its centre at the origin and the bigger circle has centre K (a; -3), The length of the radius of the bigger circle is TWICE the length of the radius of the smaller circle, SR is a tangent to both circles, touching the bigger circle at S and the smaller circle at R, KO and SR are both produced to intersect in point P. y P ) J(2;-1) x R K(a;-3) Ss 4.1.1 Calculate the length of the radius of the smaller circle, (2) 4.1.2 Show that a=6. (3) 4.1.3 Determine the equation of the bigger circle. (2) 4.1.4 Does the point (10;—4) lie outside, inside or on the bigger circle? (3) 4.1.5 Calculate the length of PS. (5) 4.2 The length of the diameter of the circle with equation x? —4x4 y? +5y=-d is 24 Determine: : 4.2.1 the coordinates of the centre of the circle. (4) 4.2.2 the value of d. (3) [22] Copyright Reserved Please Turn Over Powered by ics] CamScanner

Matemaicve2 DOWnloaded from hlayiso.com Preparatory Examination September 2024 mie QUESTION 5 5.1 If ja . tan 58° =n, determine the following in terms of n without using a calculator. 3.1.1 sin 58° @) 5.1.2 sin 296° (4) 5.1.3 cos 2° G) 5.2 Given the following identity: 1-cos2x _ sin2x 5.2.1 Prove the identity. re)) 5.2.2 Use the identity to determine the value of tan15° in its simplest form. (3) No calculator may be used. 53 Simplify to a single trigonometric ratio: in (360°+ x).cos(90°+x) — a ae sin (860° 3) 009042) + en GSP) ©) 5.4 Determine the general solution of: cos2x -5 = sin (9) 5.5 For which values of & will sin(2x+30°)+ =3 have no solution? (5) [33] Copyright Reserved Please Turn Over Powered by Ics) CamScanner

Downloaded from hlayiga.G@Maion septerter20 NSC Mathematics/P2 QUESTION 6 eo th In the diagram below, the graphs of f(x)=tanbx and g(x)=cos(x—30°) are ssa aisha system of axes for —180°< x <180°. The point P(90°; 1) lies on f. P(90°;1) a 8 ze TBO 180° | Use the diagram to answer the following questions: 6.1 Determine the value of 5. (1) 6.2 Write down the period of g. () 6.3 Write down the coordinates of A, a turning point of g. (2) 6.4 Write down the equation(s) of the asymptote(s) of y= tan b( Se te 20°) for x e[-180°;180°]. () (2) [7] 6.5 Determine the range of A if h(x)= 2g(x)-1. Copyright Reserved me Please Tum Over Powered by Ics) CamScanner

Downloaded from hlayiso.com 9 Preparatory Examination September 2024 NSC Mathematics/P2 QUESTION 7 In the diagram, B, C and D lie in the sai BD =2CD. CBD=x, BCD=y» andCD =p meters. AB isa vertical tower. The angle of elevation of A from C is 0. me horizontal plane. A 6 a Cc B z 5 b D VA Show that sin y =2sinx. (2) 72 Prove that AB=btan@, [5+4cos(x+ y) (7) Hence, determine the height of the tower, rounded off to two decimal places, if: - b=54,8 metres , x=31°, 0=42,6° and y=75,84°, (2) [11] Powered by Ics) CamScanner

Downloaded from hlayiso.com. . NSC Seprember 2124 Mathematics/P2 my QUESTION 6 In the diagram below, the graphs of f(x)=tanbx and g(x)=cos(x—30°) are drawn on the system of axes for —180°< x <180°. The point P(90° ; 1) lies on f- P(90°;1) # ry See -) Cae Use the diagram to answer the following questions: 6.1 Determine the value of 5. a) 6.2 Write down the period of g. w 6.3 Write down the coordinates of A, a tuning point of g. @) 6.4 Write down the equation(s) of the asymptote(s) of y= tan b( x+20°) for x © [-180°;180°]. a) 6.5 Determine the range of h if h(x)=2g(x)-1. @) | Powered by Ics) CamScanner

sistas D own | oad ed from h | ay i PAmGyDAMnation September 2024 NSC GIVE REASONS FOR YOUR STATEMENTS IN QUESTIONS 8, 9 AND 10. QUESTION 8 In the diagram, A , C, E and G are points on the circumference of the circle with centre O. COE =136°. DEF is a tangent to the circle at E, with DF|CA. BOE is a straight line, with B a point on AC, AE is drawn. AC=14units. 8.1 Calculate, with reasons, the size of each of the following: 8.1.1 A, Q) 8.1.2 B, Q) 8.1.3 BCE Q) 8.1.4 6 @) 8.2 Calculate, with reasons, the length of AB. (5) [13] Copyright Reserved Please Turn Over Powered by Ics) CamScanner

maemiese2 DOwnloaded from hlayiso NSC reparatory com... September 2024 QUESTION 9 In the diagram, AQRS isa triangle with RS produced to P. U and V are points on QR such that PU|SV. PU intersects QS in T. _ nw and —— aT 2 R 3 TS 2° P S R T hy U Q ss PS Calculate, giving reasons, the value of SR (6) [6] QUESTION 10 10.1 In the diagram below, AABC and ADEF are drawn with A=D, B=E and C=F A D A B Cc Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, i.e. AB_ AC DE DF’ (6) Copyright Reserved Please Tum Over Powered by Ics) CamScanner

Mathenaicse2 DOWNIOAE From NlaydgaeQsMon september 2024 NSC 10.2 In the diagram below, AKLM is given. PN is drawn with P on KM and N on LM. ¢ LM=30 units, KL =25 units, KM = 40 units, MN = 16 units, PM = 12 units and PN = 10 units. L N M Prove that: 10.2.1 AMKL|||AMNP (4) 10.2.2 KLNP is a cyclic quadrilateral. @) Copyright Reserved Please Tum over Powered by ics] CamScanner

wieeaise2 Downloaded from hlayiso.com aratory Examination September 2024 NSC 10.3 In the diagram A, B, C and D are points ona circle. BD bisects ADC and BD and AC intersect in E. Let D, = Dd, =x. A Prove that: AD.CE 103.1 DE TBC (5) 10.3.2 AD.CD=DE?+DE.BE (5) [23] TOTAL: 150 Copyright Reserved Powered by Ics) CamScanner

Mathematics/P2 D ow n | 0 a d e d fro m h | ay 1SQeG QiMaion September 2024 NSC INFORMATION SHEET: MATHEMATICS ee —bvb2 —4ac 2a A= P(+ni) A= P(l=ni) A=P(l-i)" A=P(1+i)" T, =a+(n-l)d S, =F hea+ nd] T, =ar™ s,=MCo) sre S,=—-4;-1ler<l | l-r p= 2+ 1] paxla+9) i an ae tee) — yon Se +h) - f(x) G)* fine h d= (x, x1)? +0, -y,)? m(Ais21422) yamete yr-y =m(x-x,) m=22-"1 m=tan@ x2 —*x (x—a) +(y-b) =r? InAABC: ab ec sinA sinB_ sinc a? =b? +02 —2be.cosA areaAABC = 5ab.sinC sin(a + f)=sina.cos 2 +cosa.sin B cosa +A)= cosa@.cos#~sina,sin 8 cos’ a —sin? a cos2a =41—2sin? a sin(a -p)= sina.cosf - cosa.sin B cos(a — B)=cosa.cos B+sina.sinB sin2a@ = 2sina.cosa 2cos? a-1 o? = 4 n (A 6 P(A or B) = P(A) + P(B) — P(A and B) pratbx b= (x-¥ky-j) DCr- x) Copyright Reserved Powered by Ics) CamScanner