Mathematics Grade 12 P2 NSC June 2012 Question Paper (Eng)

Downloaded from hlayiso.com basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION MATHEMATICS P2 HIGHER GRADE 2012 MARKS: 200 MATHEMATICS HG: Paper 2 TIME: 3 hours 3011E 30112E This question paper consists of 13 pages, 1 formula sheet and a diagram sheet | consisting of 4 pages. X05 | capri reson NA 00 0 ae

Downloaded from hlayiso.com Mathematics/HG/P2 2 DBE/2012 SCE INSTRUCTIONS AND INFORMATION L 2. 10. This question paper consists of TEN questions. Use the formula sheet to answer this question paper. Detach the diagram sheets from the question paper and place them inside your ANSWER BOOK. The diagrams are NOT drawn to scale. Answer ALL the questions. Number the answers correctly according to the numbering system used in this question paper. Show ALL the necessary calculations. You may use non-programmable calculators, unless stated otherwise. The number of decimal digits to which answers must be rounded off will be stated in the question, where necessary. Write neatly and legibly. a FAIA ON IAEA A diab

Downloaded from hlayiso.com Mathematics/HG/P2 3 DBE/2012 SCE ANALYTICAL GEOMETRY NOTE: - USE ANALYTICAL METHODS IN THIS SECTION. = CONSTRUCTION AND MEASUREMENT METHODS MAY NOT BE USED QUESTION 1 In the diagram alongside, Ay AG; 9) and D(5 ; 1) are points in a AG; 9) Cartesian plane. B and C are points on the y-axis such that ABD=ACD = 90° DS; 1) Ve 11 Show that (0 ; 6) are the coordinates of B. (6) 1.2 Hence, or otherwise, find the coordinates of C. (2) 1.3 Calculate the coordinates of M, the midpoint of AD. (2) 1.4 Calculate the length of the radius of circle ABD. Leave your answer in simplified surd form. (3) 15 Will circle ABD pass through point C? Give a reason for your answer. (2) 1.6 Calculate the size of BDA , rounded off to the nearest degree. (7) 1.7 Find the equation of the straight line which is parallel to BD and which passes through point A. (4) [26] aia FIO TAO A Please turn over GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 4 DBE/2012 SCE QUESTION 2 2.1.‘ In the diagram alongside, diameter AB of circle ACB in a Cartesian plane is given. ACI; 8) The coordinates of A and B are (— 1; 8) and (5; 0) respectively. C isa point on the x-axis. v c B(5 ; 0) 2.1.1 Determine: (a) The equation of circle ACB (5) (b) The coordinates of C (3) (c) The equation of the tangent to the circle at C (5) 2.1.2 Another circle with equation x? —4x+ y? —-8y—29=0 is given. (a) Determine whether the two circles are concentric. (4) (b) Determine, with reasons, whether this circle lies inside or outside circle ACB (2) 2.2 Determine the equation of the locus of P(x ; y) which is equidistant from point A(1 ; 3) and the line x = —2 (6) [25] Coil re 1 Pes moe GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 5 DBE/2012 SCE TRIGONOMETRY QUESTION 3 YOU MAY NOT USE A CALCULATOR IN THIS QUESTION. SHOW ALL CALCULATIONS. 3.1 In the diagram alongside, P is a point in the : first quadrant of a Cartesian plane such that y 13cos6 —-5=0 R(k ; 10) is a point in the second quadrant Pp such that POR =90° R(k; 10) 8 x > oO Determine the numerical value of: 3.1.1 tan 0 (3) 3.1.2 k (4) 32 Simplify the following expression: cos (90°—x) . cosec(90° + x) . cos(—22°) tan(360°—x) . sin 34°. cos 34°. sec(540°— x) (9) 3.3 If sin 24°= m, find the following in terms of m: 3.3.1 cos 114° (2) 3.3.2 cos (—156°) (2) [20] Cori rsa AN Pose tmove GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 6 DBE/2012 SCE QUESTION 4 Given: f(x) = tanx and g(x) = sin2x for x © [-90° ; 180°] 4.1 Determine the solution of the equation tanx = sin2x for x © [-90° ; 180°] 4.2 Use the set of axes provided on DIAGRAM SHEET 1 to draw sketch graphs of the curves of f and g forx € [-90° ; 180°]. Clearly show all intercepts with the axes, coordinates of all turning points and end points of both curves. Indicate the asymptotes by using dotted lines. 4.3 Use the graphs drawn in QUESTION 4.2 and the solution obtained in QUESTION 4.1 to determine the value(s) of x © [—90° ; 180°] for which: 43.1 f@)<g) 4.3.2 g (x) increases as x increases 43.3 f(x).g(x) = 1 QUESTION 5 5.1 Solve for x €(0°; 90°): i+ tan? x = cosec(2x + 30°) 5.2 Given the identity: cos*A — sin’ A =cos 2A 5.2.1 Prove the identity. 5.2.2 Hence, or otherwise, find the general solution of the equation 3cos'A —3sin'A =1 Give your answer rounded off to ONE decimal digit. Copied NN Pes oe GAUTENG (8) (10) (4) (3) (3) [28] (6) Q) i) [16]

Downloaded from hlayiso.com Mathematics/HG/P2 Zz DBE/2012 SCE QUESTION 6 6.1.1 Use the diagram on DIAGRAM SHEET 2 or redraw the diagram in A your ANSWER BOOK to prove that in AABC: B= a@ + c-2a\(c)cos B c a (6) 6.1.2 In the diagram alongside, ASRT is given. 8 SR = s units RT = 2s units and ST = runits Show that ” = ./5—4cosR (4) s a FADIA AU A Please tan over GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 8 DBE/2012 SCE 6.2 In the diagram alongside, LM represents a vertical mast. It is constructed so that M isat the same level as the foundations L of a thatched house. Points B and C are on the perimeter of the property. B, C and M lie in the same horizontal plane. BM = MC LM = A units MBC=30° a is the angle of elevation from C to L. 6.2.1 Express MC in terms of hand a. (2) 6.2.2 Show that distance BC = 3 h .cota@ (4) _ If it is further given that 4 =25 metres and a = 43°, calculate rounded off to ONE decimal digit: (a) The distance BC (2) (b) The areaof ABCM (5) [23] Coigh roeme NH Pes am oe GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 9 DBE/2012 SCE EUCLIDEAN GEOMETRY NOTE: — DIAGRAMS FOR PROVING THEORY MAY BE USED ON THE DIAGRAM SHEETS OR REDRAWN IN YOUR 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 7 In the diagram below, two concentric circles with centre at M and with radii 5 cm and 8,5 cm are given. PQRS isa chord of the larger circle cutting the smaller circle at Q and R. MYS isa straight line with Y on the smaller circle. QR=6cm Calculate, with reasons, the length of PS. (7] Co rceme A A Pe oe GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 10 DBE/2012 SCE QUESTION 8 8.1 In the diagram alongside, O is the centre of circle EBD. Use the diagram on DIAGRAM SHEET 2 or redraw the diagram in your ANSWER BOOK to prove the theorem which states that: D If BC is a tangent to the circle, then DBC=E (7) STA) AUN QUO AD AU Please arn over

Downloaded from hlayiso.com Mathematics/HG/P2 Ti DBE/2012 SCE 8.2 In the diagram below, PA and PBS are tangents to circle BCA with centre M. AC is produced to R. AB and PR are produced to Q. BC=AC and PQ || CB. PC is drawn. Let Ai=x Prove that: 8.2.1 Po =Ps (6) 8.2.2 ABRP is a cyclic quadrilateral (4) 8.2.3 PRQ is a tangent to circle BCR at R (4) [21] Copyrigt reserved HUNAN. AE Please urn over GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 12 DBE/2012 SCE QUESTION 9 9.1 In the diagram alongside, A ABC is A given. M and N are points on AB and AC such that MN || BC. Use the diagram on DIAGRAM SHEET 4 or redraw the diagram in your ANSWER BOOK to prove the theorem which states that: M > N If MN || BC then AM _ AN AB NC B > Cc 7) 9.2 In the diagram alongside, ABCD is a parallelogram with diagonals AC and BD intersecting at F. S P isapointon AD. CP and BA are produced to S. FK || BC with K on CP. BA _2 BS 5 A P D B id c 9.2.1 Show that K is the midpoint of PC. (4) 9.2.2 Determine, with reasons, the value of the ratio KC: SP. (4) 9.2.3 If BC = 15 cm, calculate the length of AP. (4) [19] Cog cen A Pe oe GAUTENG

Downloaded from hlayiso.com Mathematics/HG/P2 iE DBE/2012 SCE QUESTION 10 In the diagram below, AOB is the diameter of semi-circle ADB with O the centre. ADC is a straight line. CB isa tangent at B. OK 1 DB with K on DB. OK produced cuts CB at S. 10.1 Prove that: 10.1.1 OS |] AC 3) 10.1.2 A ABC ||| AADB ||| A BDC 6) 10.1.3 AB? = AD.DC (2) 10.2 Calculate the numerical value of Area of A ABC 5 Area of AOSB 118 TOTAL: 200 Comrise NAN

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Downloaded from hlayiso.com Mathematics/HG/P2 DBE/2012 SCE Mathematics Formula Sheet (HG and SG) _ ~b+vb? —4ac 2a n n n T,=a+(n-1)d 8,-7@+,) S.-7@+9 S, = 5 Bat (a —1)d] T, =ar"-! S, = a(1-r") (r #1) s, <8 =D xy 1-r r-1 S, = a (\r| <1) A=P(1+i5) OR A=p(1-i5] 100 100 f"(x)=lim f(x+h)-f@) h>0 h d=, -x,) +(y, -y,) y=mx+e y-y, =m(x-x,) m=227%1 x, —Xy m = tanO xX,+x + ty =r (x-p)’ + (y-q)* =r? - £F snA sinB_ sinC In A ABC: a’ =b’ +c? —2be.cosA area A ABC = tab.sinC Cori reed 0 A A

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Downloaded from hlayiso.com basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATION/SENIORSERTIFIKAAT-EKSAMEN MATHEMATICS HG/WISKUNDE HG 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 LIMA WU HALAL Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/HG/P2/Wiskunde/HG/V2 2, DBE/2012 SCE — Diagram sheet/Diagramvel DIAGRAM SHEET 1/DIAGRAMVEL 1 QUESTION 1/VRAAG 1 QUESTION 2.1/VRAAG 2.1 ty y poe A(-1 ; 8) Cc [e} xy CN B(S ; 0) D6; 1) a oO QUESTION 4.2/VRAAG 4.2 ~~ y 34 24 1+ — -90 45 45 90 135 180 -l- 24+ 3-4 Copyright reserved/Kopiereg voorbehou | HTT WII nT | lll Ill Please turn over/Blaai om asseblief GAUTENG

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Downloaded from hlayiso.com Mathematics/HG/P2/Wiskunde/HG/V2 3 DBE/2012 SCE — Diagram sheet/Diagramvel DIAGRAM SHEET 2/DIAGRAMVEL 2 QUESTION 6.1.1/VRAAG 6.1.1 A QUESTION 6.2/VRAAG 6.2 L 50°) B 3 Cc B QUESTION 6.1.2/VRAAG 6.1.2 S r Ss T R 2s QUESTION 7/VRAAG 7 M y P Q T R . Copyright reserved/Kopieregvoorbehiou IIIT IIIT GAUTENG QUESTION 8.1/VRAAG 8.1 Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/HG/P2/Wiskunde/HG/V2 4 DBE/2012 SCE — Diagram sheet/Diagramvel DIAGRAM SHEET 3/DIAGRAMVEL 3 QUESTION 8.2/VRAAG 8.2 Copyright reserved/Kopiereg voorbehou (Wi TUMHAUAULUL AH Il lll ll Hil Please turn over/Blaai om asseblief

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Downloaded from hlayiso.com Mathematics/HG/P2/Wiskunde/HG/V2 3 DBE/2012 SCE — Diagram sheet/Diagramvel DIAGRAM SHEET 4/DIAGRAMVEL 4 QUESTION 9.1/VRAAG 9.1 QUESTION 9.2/VRAAG 9.2 S A QUESTION 10/VRAAG 10 Cc Déy : s K 2 14 1 A Oo B Coprigu reek voicow {MN