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MAT HEMAT I CS /WI SKUNDE / P2 / SG DOE/M2/l/01

MATHEMATICS PAPER 2WISKUNDE VRAESTEL 2

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STANDARD GRADESTANDAARDGRAAD

NOVEMBER 2001

09HOO

MARKS: 150PUNTE: 1503 HOURS3 URE

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DEPARTMENT OF EDUCATIONDEPARTEMENT VAN ONDERWYS

Hierdie vraestel bestaan uit 'n voorblad, 18 bladsye,3 diagramvelle en 'n formulebladThis question paper consists of a cover page, 18 pages, 3 diagramsheets and a formula sheet.

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WISKUNDEIP2/SG DOEIM,/IIOI

SENIOR SERTIFIKAAT EKSAMEN - 2001-1-

INSTRUKSIES

1. Beantwoord AL die vrae.

2. 'n Formuleblad is by die vraestel ingesluit.

3. Toon AL die bewerkings.

4. Nommer AL die antwoorde duidelik en korrek.

5. Die diagramme is Dievolgens skaal geteken Die.

6. Drie diagramvelle is ingesluit. Plass dit in die ANTWOORDEBOEK.

7. Nie-programmeerbare sakrekenaars mag gebruik word, tensy die vraag aDders aandui.

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MA THEMA TICSiP2/SG DOEIM,/IiOl

SENIOR CERTIFICATE EXAMINATION -2001-2-

INSTRUCTIONS

1. Answer ALL the questions.

2. A formula sheet is included in the question paper.

3. Show ALL the calculations.

4. Number ALL the answers clearly and correctly.

5. The diagrams are not drawn to scale.

6. Three diagram sheets are included. Place it in the ANSWER BOOK.

7. Non-programmable calculators may be used, unless the question states otherwise.

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WISKUNDElP2/SG DOEIM,/IIOI

SENIOR SERTIFlKAAT EKSAMEN - 2001

-3-

ANALITIESE MEETKUNDE

LET WEL: - IN HIERDIE AFDELING MAG SLEGS ANALITIESE METODESGEBRUIK WORD.

- AKKURATE KONSTRUKSIES EN METING MAG NIE GEBRUIK WORDNIE.

VRAAG 1

1.1 Bereken die lengte van AB met A(13; 2) en B(I; )3) (3)

1.2 Bepaal die vergelyking van die reguit Iyn deur:

1.2.1 Die oorsprong met gradient 3 (2)

(4)1.2.2 Die punt (5; )3) en ewewydig aan die Iyn 2x + y = 5

1.3 Die diagram hieronder toon APQR met hoekpunte P(2; 3), Q(-4; )1)en R(6; )3)

y P (2; 3)

x

R (6; )3)

1.3.1 Bepaal die koordinate van M, die middelpunt van QR. (2)

(2)1.3.2 Bereken die gradient van PQ.

1.3.4 Bewys dat APQR reghoekig is by P.

(2)

(2)

1.3.3 Bereken die gradient van PRo

1.3.5 Watter meetkundige verwantskap kan tosseD QR en die sirkeldeur P, Q en R afgelei word? (1)

(3)[21]

1.3.6 Bepaal die leugte van die radius van die sirkel deur P, Q eu R.


MATHEMATICSIP2/SG


ANALYTICAL GEOMETRY

NOTE: - IN TIllS SECTION ONLY ANALYTICAL METHODS MAY BE USED.- ACCURATE CONSTRUCTIONS AND MEASUREMENTS MAY NOT BE

USED.

QUESTION 1

1.1 Calculate the length of AB with A(13; 2) and B(l; )3)

1.2 Determine the equation of the straight line through:

1.2.1 The origin with gradient 3

1.2.2 The point (5; )3) and parallel to the line 2x + Y = 5

1.3 The diagram below shows APQR with vertices P(2; 3), Q(-4; )1)and R(6; )3)

y P (2; 3)

x

R (6; )3)

1.3.1 Determine the coordinates oeM, the midpoint ofQR.

1.3.2 Calculate the gradient ofPQ.

1.3.3 Calculate the gradient ofPR.

1.3.4 Prove that APQR is right-angled at P.

1.3.5 What can be deduced geometrically about QR in relation to thecircle through P, Q and R?

1.3.6 Determine the length of the radius of the circle through P, Q, and R.

DOEiM,/l/Ol

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VRAAG2

2.1

B

x

In die diagram hierbo soy die sirkel, met die oorsprong as middelpunt, die IynACby A(O;5) en C(x;y). M(2;4)isdiemiddelpuntvanAc.BC is 'n middellyn van die sirkel.

2.1.1 Bepaal die vergelyking van die sirkel.

2.1.2 Bereken die koiirdinate van C.

2.1.3 Bewys dat OM 1- AC.

2.1.4 Skryf die grootte van BAC Deer.

2.1.5 Bepaal die vergelyking van AB.

2.2 2.2.1 P(x; y) is 3 eenhede vanaf die x-as. Skets die lokus van die punt P.

2.2.2 Skryf die vergelyking(s) van die lokus van P Deer.

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MA TIJEMA TICSiP2/SG


QUESTION 2

2.1

B

In the diagram above, the circle with the origin as centre cuts the line ACat A(O; 5) and C(x; y). M(2; 4) is the midpoint of Ac.BC is a diameter of the circle.

2.1.1 Determine the equation of the circle.

2.1.2 Calculate the coordinates ofC.

2.1.3 Prove that OM 1-AC.

2.1.4 Write down the size of BAC.

2.1.5 Determine the equation of AD.

2.2 2.2.1 P(x;y) is 3 units from the x-axis. Sketch the locus of the pointP.

2.2.2 Write down the equation(s) of the locus ofP.

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WISKUNDEJP2/SG


TRIGONOMETRIE

VRAAG 3

3.1

3.2

As 2 see a. = 3, en a. 'n skerphoek, gebruik 'n skets en bereken die waardevan die volgende SONDER om 'n sakrekenaar te !!ebruik:(Laat antwoorde in wortelvorm indien nodig.)

3.1.1 tan a.

3.1.2 tsin2 a.

As A = 287° en II =49,8°, bereken die waarde van die volgende, benaderd tottwee desimale syfers:

3.2.1 tcos2B

3.2.2 cosec A

3.3 Bepaal die waarde van die volgende SONDER om'n sakrekenaar te !!ebruik:

CDS3000 + sin2 2250

3.4 Vereenvondig die volgende uitdrukking tot EEN trigonometriese verhoudingvan 0:

sec(3600- 0). sin (900- 0) - sin2 (1800+ 0)cot(1800+ 0).cos(1800- 0)

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MATHEMATICS/P2ISGSENIOR CERTIFICATE EXAMINATION - 2001

-8-

TRIGONOMETRY

QUESTION 3

3.1

3.2

If 2 sec a = 3, and a is acute, use a sketch to calculate the value of the followingWITHOUT the use of a calculator:(Leave answers in surd form if necessary.)

3.1.1 tan a

3.1.2 tsin2 a

If A = 287"and B = 49,8", calculate the value of the following,rounded otTtotwo decimal digits:

3.2.1 tcos2B

3.2.2 cosec A

3.3 Determine the value of the following WITHOUT usin!! a calculator:

cos 300° + sin2 225°

3.4 Simplify the following expressiou to ONE trigonometric ratio of 8:

sec(3600- 9). sin (90°- 9) - sin2 (180°+ 9)cot(1800+ 9).cos(1800- 9)

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WISKUNDE/P2JSG

SENIOR SERTIFlKAAT EKSAMEN -2001-9-

VRAAG 4

Diediagramhierondertoondiegrafiekevan f(x)= sin2x en g(x)= acosx

virx E [0°; 270°]. fen g sny in A(300; 0,87) en in B.

4.1

4.2

4.3

4.4

y1

x270°0

Skryf die waarde van a neer.

Wat is die periode van f?

Skryf die koiirdinate van B Deer.

Vir watter waarde(s) van x E [0°; 270°] is:

4.4.1 f(x) =g (x)

4.4.2 f(x) ~ g (x)

4.4.3 g(x) dalend soos wat x toeneem

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MATHEMATICSiP2ISG

SENIOR CERTIFICATE EXAMINATION - 2001

0-0-

QUESTION 4

The diagram below shows the graphs of f(x)= sin2x and g(x)= acosx

for x E [0° ; 270°]. f and g intersect at A(300; 0,87) and at B.

4.1

4.2

4.3

y

1

270°0

Write down the value of a.

What is the period of f?

Write down the coordinates ofB.

4.4 For which value(s) of x E [0°; 270°] is:

4.4.1 f(x) = g (x)

4.4.2 f(x) ~ g (x)

4.4.3 g (x) decreasing as x increases

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WISKUNDE/P2ISG

SENIOR SERTIFIKAAT EKSAMEN - 20011111

VRAAG5

5.1

5.2

5.1.1 Gebruik fundamentele identiteite en bewys dat:cosec e (see e - cos e ) = tan e

5.1.2 Bereken vervolgens die waarde van e benaderd tot een desimale syferas cosece (sece - cose) = -2 waar e E [0°; 180°]

Los op vir x, benaderd tot een desimale syfer, as 2tan(2x - 30°) = 1,58

en x E [0° ; 360°]

VRAAG6

6.1

c

In die diagram langsaan, is AABC'n skerphoekige driehoek.Gebruik die diagram om te bewys dat:

sin A sinB--

a b cA

p

6.2 In die diagram langsaan is PN'n vertikale toring. Die punicL en M Ie op dieselfde horisontalevlak as N, die voet van die toring.L, M en N Ie in 'n reguit Iyn.LM=43,5m, LPM=13,9° en

PMN = 50°

M 43,5

6.2.1 Toon deur berekening dat PM'"

106,7 m.

6.2.2 Bereken die boogie van die toring tot die naaste meter.

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MAlliEMATICSIPVSG

SENIOR CERTIFICATE EXAMINATION - 2001

2-2-QUESTION 5

5.1

5.2

5.1.1 Use fundamental identities to prove that:cosec a ( sec a - cos a ) = tan a

5.1.2 Hence, if cosec a ( sec a - cos a ) = -2 calculate the value ofarounded off to one decimal digit where a E [0° ; 180°]

Solve for x, rounded off to one decimal digit, if 2tan(2x - 30°) = 1,58

and x E [0° ; 360°]

QUESTION 6 c

6.1 In the diagram alongside, L\.ABC isan acute-angled triangle.Use the diagram to prove that:

sin A sinB---

a h cA

p

6.2 In the diagram alongside, PN is avertical towel". The points Land Mlie on the same horizontal plane as N,the foot ofthe tower.L, M and N lie on a straight line.

LM = 43,5 m, LPM = 13,9° and

PMN=500

M 43,5

6.2.1 Show by calculation that PM ,., 106,7 m.

6.2.2 Calculate the height of the tower to the nearest metre.

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WISKUNDE/P2JSG DOEiM,/1/01

SENIOR SERTIFIKAAT EKSAMEN -20013-3-

MEETKUNDE

LET WEL: -DIAGRAMME VIR DIE BEWYS VAN TEORIE W AT OP DIE DIAGRAM-VEL VERSKYN, MAG GEBRUIK WORD OF IN DIE ANTWOORDEBOEKOORGETEKEN WORD.

-MAAK DIE DIAGRAMVEL VAN DIE VRAESTEL LOS EN PLAAS DIT INDIE ANTWOORDEBOEK.

-GEE 'n REDE VIR ELKE BEWERING.

VRAAG 7

7.1 In die diagram langsaan is AB 'n koord vansirkelO. OC soy koord AB in D.AB 1- OC. As OA = 15 em enDC = 6 em, bereken die lengte van AB.

(6)

7.2 In die diagram langsaanis PQRS 'n koordevierhoeksodat QS vir PQR halveer.RS word verleng na T sodatST = SP. T = 420 en 83 =370

T

Bereken met redes:

7.2.1 S2 (4)

7.2.2 Q1 (3)

7.2.3 iT 1(3)

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MATIIEMATICSiP2ISG

SENIOR CERTIFICATE EXAMINATION - 20014-4-

DOE/M,IIIOt

GEOMETRY

NOTE: -DIAGRAMS FOR PROVING THEORY MAYBE USED ON THE DIAGRAMSHEET OR REDRAWN IN THE ANSWER BOOK.

-DETACH THE DIAGRAM SHEET FROM THE QUESTION PAPER ANDPLACE IT IN THE ANSWER BOOK.

-GIVE A REASON FOR EACH STATEMENT.

QUESTION 7

7.2 In the diagram alongside,PQRS is a cyclic quadrilateral

such that QS bisects PQR.RS is produced to T such thatST= SP. 1'=420 and 83 =370

T

A7.1 In the diagram alongside, AB is a chord

of circle O. OC intersects chord AB at D.AB 1- OC. IfOA = 15 cm andDC = 6 cm, calculate the length of AB.

0

(6)

Calculate, with reasons:

7.2.1 82 (4)

7.2.2 Q1 (3)

7.2.3 U1 (3)

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WISKUNDE/P2JSG


VRAAG8

8.1

8.2

In die diagram langsaan is PRST'n koordevierhoek van sirkel O.Gebruik die diagram om diestelling te bewys waf meld dat:

P + S = 1800

In die diagram langsaan is EF en EGraaklyne aan sirkel O. FH II EK.EK sny FG in J en ontmoet GH by K.

H=x

Bewys, met redes, dat:

8.2.1 FOGE'n koordevierhoek is

8.2.2 EG'n raaklyn aan sirkel GJK is

8.2.3 FEG = 1800- 2x

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MA TIffiMA TICS>?2ISG


QUESTION 8

8.1

8.2

In the diagram alongside, PRSTis a cyclic quadrilateral ofcircle O. Use the diagram toprove the theorem whichstates that:

P + § = 1800

In the diagram alongside, EF andEG are tangents to circle O. FH II EK.EK intersects FG at J and meets GH at K.

H=x

Prove, with reasons, that:

8.2.1 FOGE is a cyclic quadrilateral

8.2.2 EG is a tangent to circle GJK

8.2.3 FEG = 1800- 2x

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WISKUNDEIP2ISG

SENIOR SERTIFlKAAT EKSAMEN - 20011717

VRAAG9

9.1

9.2

Die diagram langsaan fOODMBC.DE II BC metD opAB en E opAC.Gebruik die diagram om diestelling te bewys waf meld dat:

AD AE- -DB EC

In die diagram langsaan isM, N en P op die sirkel.NM word na L verleng sodatLM = MN. PL soy diesirkel by Q. NQ wordverleng na K sodatKL II PM.PM soy KN by R.

Bewys, met redes, dat:

9.2.1NR 1NK 2

9.2.2 ANQLIII APML

9.2.3 QL.PL=2NM2

Kopiereg voorbehou

A

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p

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TOTAAL: 150

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MATHEMATICS/P2/SG


DOEIMJlIOI

QUESTION 9

9.1 The diagram alongside shows AABC.DE II BC with D onAB and E onAC.Use the diagram to prove the theoremwhich states that:

AD AE- -DB EC

9.2 In the diagram alongsideM, Nand P are on the circle.NM is produced to Lsuch that LM =MN.PL cuts the circle at Q.NQ is produced to Ksuch that KL II PM.PM intersects KN at R.

Prove, with reasons, that:

9.2.1NR 1

NK 2

9.2.2 ~NQL III ~PML

9.2.3 QL.PL=2NM2

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MA THEMA TICSIWISKUNDEIP2/SG

SENIOR CERTIFICATE EXAMINATION/SENIOR SERTIFlKAAT EKSAMEN - 2001DOE/MillOl

Mathematics Formula Sheet (KG and SG)Wiskunde Formuleblad (KG en SG)

:t:t~b2 acacx=

2a

Tn =a + ( n - 1 )d Sn = !!. (a + I)2

Sn =!!. [2a + (n )d)d]2

Tn = a.r n-l Sn =a(l-rn)

1-rS =

a(rn )1)n

r-1Soo = ~

1-r

A = P( I + ~)

n

100 A=P(I-I~O)"

f' (x) = limf(x +h) - f(x)

h O h

d =~(X2 lili +(Y2 - Yli

y=mx+c

Y - Yl =m(x-xl)

m=Y2-YIX2 lXl

m = tan()

(Xl+X2.Yl+Y2

)2 ' 2

X2 + y2 =r2 (X - pi + (y - qi = r2

1n MBC:a b c

- ---- - -sin A sinn sinC

a2 =b2 +C2 ososAcosA

areaMBC =~ab.sinC

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NATIONALDEPARTMENT OF EDUCATION

NASIONALEDEPARTEMENT VAN ONDERWYS

DEPARTMENT OF EDUCATION

SENIOR CERTIFICATE EXAMINATION I SENIOR SERTIFIKAA T EKSAMENMATHEMA TICS SG I WISKUNDE SG

PAPER n I VRAESTEL nNOVEMBER 2001

DIAGRAM SHEET I DIAGRAMVEL

INSTRUCTION

This diagram sheet must be handed in with the answer book. Please ensure that your details arecomplete.

INSTRUKSlE

Hierdie diagramvel moet saam met die antwoordeboek ingelewer word. Maak asseblief sekerdat u besonderhede volledig ingevul is.

EXAMINATION NUMBEREKSAMENNOMMER

CENTRE NUMBERSENTRUMNOMMER

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MA THEMA TlCSIWIS KUND EIP2ISG

SENIOR CERTIFICATE EXAMINATION/SENIOR SERTIFIKAAT EKSAMEN - 2001-2-

DOEIM,II/OI

DIAGRAM SHEET 1DIAGRAMVEL

OUESTION 41 VRAAG 4

y1

0 270. x

OUESTION 6.11 VRAAG 6.1

cOUESTION 6.2 1VRAAG 6.2

P

A c B M L

OUESTION 7.1/VRAAG 7.1 OUESTION 7.2/ VRAAG 7.2

T

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MA THEMA 11 CSIWI S KUND EIP2ISG

SENIOR CERTIFICATE EXAMINATION/SENIOR SERTIFIKAAT EKSAMEN -2001-3-

DOEiM,I11O1

OUESTION 8.1 / VRAAG 8.1 OUESTION 8.2 / VRAAG 8.2

E

T

H

OUESTION 9.1 / VRAAG 9.1 OUESTION 9.2 / VRAAG 9.2

p

L

A

B c

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