Mathematics P2 (SG) (English) June 2012 QP hlayiso.com

Downloaded from hlayiso.com basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION MATHEMATICS P2 STANDARD GRADE 2012 MARKS: 150 TIME: 3 hours MATHEMATICS SG: Paper 2 3012E 30122E This question paper consists of 12 pages, 1 formula sheet and a diagram sheet consisting of 4 pages. X@5 MMI 2 Copyright reserved tM HN IN INIA Il Please turn over

Downloaded from hlayiso.com Mathematics/P2/SG 2 DBE/2012 SCE INSTRUCTIONS AND INFORMATION r This question paper consists of NINE questions. 2. Use the formula sheet to answer this question paper. 3. Detach the diagram sheets from the question paper and place them inside your ANSWER BOOK. 4. The diagrams are not drawn to scale. Sh Answer ALL the questions. 6. Number ALL the answers correctly and clearly. 1h ALL the necessary calculations must be shown. 8. Non-programmable calculators may be used, unless otherwise stated. _ The number of decimal digits to which answers must be rounded off will be stated in the question, where necessary. 10. Write neatly and legibly. Comrise A Pease ove

Downloaded from hlayiso.com Mathematics/P2/SG a DBE/2012 SCE ANALYTICAL GEOMETRY NOTE: — USE ANALYTICAL METHODS IN THIS SECTION. — CONSTRUCTION AND MEASUREMENT METHODS MAY NOT BE USED. QUESTION 1 a In the diagram alongside, PC-137) y P(-1;7) and Q(3;-1) are two points in a Cartesian plane. 4 8 » < 0 x? QB 5-1) v Determine: Ta The length of PQ (Leave the answer in simplified surd form.) (3) 1.2 The coordinates of M, the midpoint of PQ (2) 1.3 The equation of straight line PQ in the form y=... (4) 1.4 The size of 0, the angle between PQ and the positive x-axis (rounded off to ONE decimal digit) (3) 1.5 The equation of the straight line which is perpendicular to PQ and which passes through the point (—5 ; 1). Write the equation in the form y=... (4) 1.6 Whether point T( 4 ; —2) lies on straight line PQ (3) [19] Cori rsd NL Please turn over

Downloaded from hlayiso.com Mathematics/P2/SG 4 DBE/2012 SCE QUESTION 2 In the diagram alongside, ty straight line y=x-—1 cuts the circle + year at points A(—3 ; -4) B(433) and B(4; 3). ~ Straight line CB is parallel to * the x-axis with C on the circle. AB cuts the x-axis at S. The circle intersects the x-axis at T. v Determine: 2.1 The equation of the circle with centre O and radius OB (3) 2.2 The equation of the circle with centre O and radius OS (3) 2.3 The coordinates of C qd) 2.4 The equation of the tangent to the circle at T (3) 2.5 The equation of the perpendicular bisector of AB (5) 2.6 Determine in each of the following the equation of the locus of point P(x ; y) which is: 2.6.1 3 units above the x-axis (2) 2.6.2 Equidistant from the origin and the point (—6 ; 0) 13) Comrise NT 08 1 Please turn over

Downloaded from hlayiso.com Mathematics/P2/SG a) DBE/2012 SCE TRIGONOMETRY QUESTION 3 Sal Use a calculator to determine the value of the following (rounded off to TWO decimal digits): 3.1.1 cot ( 233,3° + 96°) qd) 312 sin 230° — cos 115° (1) 3.2 In the diagram alongside, P(12 ; —5) is a point in a Cartesian plane. @ is the reflex angle that OP makes | \ > with the positive x-axis. \ o P(12 ; -5) Determine the numerical value(s) of the following, without the use of a calculator: 3.2.1 tan 6 dd) 3.2.2 cosec 0 (2) 3.2.3 5 cot @ — 13 cos 6 (3) 33 Simplify the following, without the use of a calculator: 3.3.1 cot 240°. cos 330°. sin 210° tan315° (9) B32 cos (180°— A).tan(360°— A). sec(90°— A) sin(90°—A).tan(180°+ A) (6) 3.4 Use trigonometric identities to prove that: cosx + tanx.sinx = secx (4) 35 Solve for xe [0°;180°]: 1-sin? x = ; (4) [31] Core cg ail GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 6 DBE/2012 SCE QUESTION 4 The diagram below shows sketch graphs of f and g which intersect at points P and Q for x € [0°;180°]. The coordinates of the turning point of g are A(90°; 2) and B(180°; 1) is the endpoint of f Given: f(x)=acosbx and g(x)=csindx A(90° ; 2) B(180° ; 1) Use the graph to answer the following questions: 41 Determine the numerical values of a, b, c and d. (4) 4.2 Complete: (90°) =... (¢3) 43 If (158,5° ; 0,73) are the coordinates of Q, write down the coordinates of P. (2) 44 Give: 4.4.1 The period of f qd) 4.4.2 The amplitude of g Q) 4.5 Determine the value(s) of x € [0°;180°] for which: 4.5.1 g(x) =2 r¢9) 4.5.2 fix) <0 (2) 4.5.3 g(x) —flx) =3 qa) [13] Consign reserved NAA Pease urn over GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 7 DBE/2012 SCE QUESTION 5 a1 In the diagram below, APQR is acute angled. P 5.1.1 Use the diagram on DIAGRAM SHEET 2, or redraw the diagram in your ANSWER BOOK, to prove that: snQ_ sinR q or (4) 5.1.2 Complete for APQR: cos Q= =~ 2 pr (1) 5.2 In the diagram below, ABCD is a quadrilateral with BC=CD= 12m, AD= 10m, i B = 43,2° and C; = 778° N 2 12 Calculate (rounded off to ONE decimal digit): 5.2.1 The length of AC (4) 522 Thesizeof D (3) Conprigin reserved NCU A Pease nurn over GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 8 DBE/2012 SCE 5.3 In the diagram below, the area of AKLM is 250 mm”. LM = 35 mm and KM = 25 mm 35 Calculate the size of M (rounded off to ONE decimal digit). (4) [16] Copyrighn reserved MOAN MA eile GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 9 DBE/2012 SCE EUCLIDEAN GEOMETRY NOTE: — DIAGRAMS FOR PROVING THEORY MAY BE USED ON THE DIAGRAM SHEETS OR REDRAWN IN THE ANSWER BOOK. — DETACH THE DIAGRAM SHEETS FROM THE QUESTION PAPER AND PLACE THEM IN YOUR ANSWER BOOK. — GIVE A REASON FOR EACH STATEMENT, UNLESS OTHERWISE STATED. QUESTION 6 In the diagram alongside, AB is a chord of a circle with centre O. ON 1 AB andcuts AB at M and meets the circle at N. If MN = 30 mm and the radius of the circle is 150 mm, calculate, with reasons, the length of AB. QUESTION 7 71 In the diagram alongside, O is the centre of the circle. A, B and C are points on the circle. Use the diagram on DIAGRAM SHEET 3, or redraw the diagram in your ANSWER BOOK, to prove the theorem which states that: AOB =2ACB Comrise TQ 5] (6) Please turn over

Downloaded from hlayiso.com Mathematics/P2/SG 10 DBE/2012 7.2 SCE In the diagram alongside, A O is the centre of circle ABDC. O isjoinedto B and C. OBC =x Determine, with reasons, the following angles in terms of x: 721 Oo Q) 722 A Q) 723. D 1] Cori ree Ng laa GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 11 DBE/2012 SCE QUESTION 8 8.1 In the diagram alongside, ST || QW and PQR isa tangent to circle QSTVW at point Q. Wi =70° and Q, = 30° Calculate, with reasons, the magnitudes of the following: 8.1.1 Vv (3) 812 Q, Q) 813 Q )) 8140 oT (2) A 8.2 In the diagram alongside, AABC is given. HG// AC, ED // BC, ED and HG intersect at K. AD _3 jg BG_2 E. K D v ——== an = DC 2 Ge) B > ral Cc If AB=15 units, determine with reasons: 8.2.1 The length of AE (3) 8.2.2 The length of AH G3) GK @) 8.2.3 The value of KH 17] dialled NAA Please turn over GAUTENG

Downloaded from hlayiso.com Mathematics/P2/SG 12 DBE/2012 SCE QUESTION 9 91 In the diagram alongside, AABC and APQR 9.2 are given. Use the diagram on DIAGRAM SHEET 3, or redraw the diagram in your ANSWER BOOK, to prove the theorem which states that: In the diagram alongside, ABCD isa cyclic quadrilateral. AC and BD intersect at P. E isa point on BD so that AE // DC. Prove that: 9.2.1 AAPE /// ABPA (5) 9.2.2 AP? = BP. PE (2) 923 AP PE (4) PC PD [18] TOTAL: 150 Copsright reserved ANNU A

Downloaded from hlayiso.com Mathematics/P2/SG DBE/2012 SCE Mathematics Formula Sheet (HG and SG) = —b+-b’ —4ac 2a T,=a+(n-1)d S,=~(a+T,) S,=F@+4) S, =F Ba+(n -1)d] T,=ar"7! S, = ( (r #1) s, =20 -1) 1-r r-1 (r #1) S. Sar (Ir|< 1) A=P(1+ 2) OR A=P(1-35) 100 f(x) = lim f(x+h)-f(x) h0 h d= (x, -x,) +(y, -y,) y=mx+e y-y, =m(x—x,) y27)1 X,—X, m= m =tan@d G33 y3) = (“4 25 = s} (x-p)’ + (y-g)’ =r? In A ABC: a snA sinB_ sinC a’ =b’ +c? —2be.cosA area AABC = 4ab.sinC Copsright reserved NINA AT GAUTENG

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Downloaded from hlayiso.com basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION/SENIORSERTIFIKAAT-EKSAMEN MATHEMATICS SG/WISKUNDE SG PAPER 2/VRAESTEL 2 2012 DIAGRAM SHEET/DIAGRAMVEL INSTRUCTION This diagram sheet must be handed in with your ANSWER BOOK. Ensure that your details are complete. INSTRUKSIE Hierdie diagramvel moet saam met jou ANTWOORDEBOEK ingelewer word. Maak seker dat jou besonderhede volledig ingevul is. EXAMINATION NUMBER EKSAMENNOMMER CENTRE NUMBER SENTRUMNOMMER Copyright reserved/Kopiereg voorbehou ATT || HAN WAMU Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/SG/P2/Wiskunde/SG/V2 2 DBE/2012 SCE/SSE — Diagram Sheet/Diagramvel DIAGRAM SHEET 1/DIAGRAMVEL 1 QUESTION 1/VRAAG 1 QUESTION 2.1/VRAAG 2.1 4 Ay P(-137) y B(4 33) : x 4 Q _— Q(3;-1) v v QUESTION 3.2/VRAAG 3.2 QUESTION 4/VRAAG 4 A y A(90° ; 2) YIN B(180° ; 1) LB a P(12 ;-5) v 1go° * Copyright reserved/Kopiereg voorbehou {IIL WI ill lll Ul Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/SG/P2/Wiskunde/SG/V2 DBE/2012 SCE/SSE — Diagram Sheet/Diagramvel DIAGRAM SHEET 2/DIAGRAMVEL 2 QUESTION 5.1/VRAAG 5.1 P r q Q p R QUESTION 5.3/VRAAG 5.3 L 35 K M 25 QUESTION 5.2/VRAAG 5.2 QUESTION 6/VRAAG 6 0) M A In. B N Copyright reserved/Kopiereg voorbehou {HAA ill INN | WIA Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/SG/P2/Wiskunde/SG/V2 4 DBE/2012 SCE/SSE — Diagram Sheet/Diagramvel DIAGRAM SHEET 3/DIAGRAMVEL 3 QUESTION 7.1/VRAAG 7.1 QUESTION 7.2/VRAAG 7.2 A | 1 , a D QUESTION 8.1/VRAAG 8.1 QUESTION 8.2/VRAAG 8.2 Copyright reserved/Kopiereg voorbehou HU Ill ITI A Ill Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/SG/P 2/Wiskunde/SG/V2 5 DBE/2012 SCE/SSE — Diagram Sheet/Diagramvel DIAGRAM SHEET 4/DIAGRAMVEL 4 QUESTION 9.1/VRAAG 9.1 QUESTION 9.2/VRAAG 9.2 Copprigh reseed Kopiereg voorbehow IIA MTA GAUTENG