mathematics grade 12 - western · pdf filewestern cape education department examination...

Western Cape Education Department

Examination Preparation Learning Resource 2016

GEOMETRY

MATHEMATICS Grade 12

Razzia Ebrahim

Senior Curriculum Planner for Mathematics

E-mail: [email protected]

Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835-

grade-12?Itemid=-1

Website: http://wcedeportal.co.za

Tel: 021 467 2617

Cell: 083 708 0448

mailto:[email protected]

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2

Index Page

1. 2016 Feb-March Paper 2 3 – 5

2. 2015 November Paper 2 6 – 7

3. 2015 June Paper 2 8 – 9

4. 2015 Feb-March Paper 2 10 – 11


6. 2014 Exemplar Paper 2 16 – 17




10. 2010 November Paper 3 24 – 25

11. 2009 November Paper 3 26 – 28

12. 2008 November Paper 3 29 – 30

Mathematics/P2 DBE/November 2015 NSC

Copyright reserved Please turn over

QUESTION 10 In the diagram below, BC = 17 units, where BC is a diameter of the circle. The length of chord BD is 8 units. The tangent at B meets CD produced at A.

10.1 Calculate, with reasons, the length of DC.

(3) 10.2 E is a point on BC such that BE : EC = 3 : 1. EF is parallel to BD with

F on DC.

10.2.1 Calculate, with reasons, the length of CF.

(3)

10.2.2 Prove that ∆BAC | | | ∆FEC.

(5) 10.2.3 Calculate the length of AC.

(4)

10.2.4 Write down, giving reasons, the radius of the circle passing through points

A, B and C.

(2) [17]

17

8

B

C

E

A

D

F

6


Copyright reserved

QUESTION 11 11.1 Complete the following statement: If the sides of two triangles are in the same proportion, then the triangles are ... (1) 11.2 In the diagram below, K, M and N respectively are points on sides PQ, PR and

QR of ∆PQR. KP = 1,5; PM = 2; KM = 2,5; MN = 1; MR = 1,25 and NR = 0,75.

11.2.1 Prove that ∆KPM | | | ∆RNM. (3) 11.2.2 Determine the length of NQ. (6)

[10]

P

Q

N

R

M

K

1,5 2

2,5

1 1,25

0,75

7

Mathematics/P2 DBE/November 2014

NSC


QUESTION 9

9.1 In the diagram, points D and E lie on sides AB and AC of ABC respectively

such that DE | | BC. DC and BE are joined.

9.1.1 Explain why the areas of DEB and DEC are equal. (1)

9.1.2 Given below is the partially completed proof of the theorem that states

that if in any ABC the line DE | | BC then EC

AE

DB

AD .

Using the above diagram, complete the proof of the theorem on

DIAGRAM SHEET 4.

Construction: Construct the altitudes (heights) h and k in ADE .

........

BD2

1

AD2

1

DEBarea

ΔADEarea

h

h

EC

AE......................

DECarea

ΔADEarea

But areaDEB = .............................. (reason: .................................)

DEBarea

ΔADEarea ...............................

EC

AE

DB

AD

(5)

A

B

C

D

E

h

1

k

1

12


NSC


9.2 In the diagram, ABCD is a parallelogram. The diagonals of ABCD intersect in M.

F is a point on AD such that AF : FD = 4 : 3. E is a point on AM such that

EF | | BD. FC and MD intersect in G.

Calculate, giving reasons, the ratio of:

9.2.1

AM

EM

(3)

9.2.2

ME

CM

(3)

9.2.3

BDCarea

FDCarea

(4)

[16]

A

B C

D

M

E

F

G

13


NSC


QUESTION 10

The two circles in the diagram have a common tangent XRY at R. W is any point on the

small circle. The straight line RWS meets the large circle at S. The chord STQ is a tangent

to the small circle, where T is the point of contact. Chord RTP is drawn.

yx 24 RandRLet

10.1 Give reasons for the statements below.

Complete the table on DIAGRAM SHEET 6.

yx 24 RandRLet

Statement Reason

10.1.1 3T = x

10.1.2 1P = x

10.1.3 WT | | SP

10.1.4 1S = y

10.1.5 2T = y

(5)

Y

X

R

W

S

P

Q

T

1

2 3

4

1

2 1

2 3

1 2

1

2

1 2

3 4 X

R

W

S

P

T

y

1

2 3

4

1

2 1

2 3

1 2

1

2

1 2

3 4

x

14


NSC

Copyright reserved

10.2 Prove that RS

WR.RPRT

(2)

10.3 Identify, with reasons, another TWO angles equal to y. (4)

10.4 Prove that 23 WQ . (3)

10.5 Prove that RTS | | | RQP. (3)

10.6 Hence, prove that 2

2

RP

RS

RQ

WR .

(3)

[20]

15

Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar


QUESTION 9 In the diagram, M is the centre of the circle and diameter AB is produced to C. ME is drawn perpendicular to AC such that CDE is a tangent to the circle at D. ME and chord AD intersect at F. MB = 2BC.

9.1 If 4D = x, write down, with reasons, TWO other angles each equal to x. (3) 9.2 Prove that CM is a tangent at M to the circle passing through M, E and D. (4) 9.3 Prove that FMBD is a cyclic quadrilateral. (3) 9.4 Prove that DC2 = 5BC2. (3) 9.5 Prove that ∆DBC | | | ∆DFM. (4) 9.6 Hence, determine the value of

FMDM .

(2) [19]

M A C

D

B

E

3 2

1 2 1

F 1

2 3

1 2 3 4

x

16

Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar

Copyright reserved

QUESTION 10 10.1 In the diagram, points D and E lie on sides AB and AC respectively of ∆ABC such

that DE | | BC. Use Euclidean Geometry methods to prove the theorem which states that

ECAE

DBAD

= .

(6)

10.2 In the diagram, ADE is a triangle having BC | | ED and AE | | GF. It is also given that AB : BE = 1 : 3, AC = 3 units, EF = 6 units, FD = 3 units and CG = x units.

Calculate, giving reasons: 10.2.1 The length of CD (3) 10.2.2 The value of x (4) 10.2.3 The length of BC (5)

10.2.4 The value of ΔGFDareaΔABCarea

(5) [23]

A

B

E

C

D

G

F

3

6 3

x

A

B C

D E

17



NOTE: Give reasons for all statements made in QUESTION 7, QUESTION 8,

QUESTION 9 and QUESTION 10. QUESTION 7

7.1 If in ∆ LMN and ∆ FGH it is given that FL = and GM = , prove the theorem that

states FHLN

FGLM

= .

(7) 7.2 In the diagram below, ∆VRK has P on VR and T on VK such that PT || RK.

VT = 4 units, PR = 9 units, TK = 6 units and VP = 2x – 10 units. Calculate the value of x.

(4)

[11]

9

6

9

K

T P

42x – 10

6

9

V

R

M

L

N

F

G H

20



QUESTION 9 O is the centre of the circle CAKB. AK produced intersects circle AOBT at T.

9.1 Prove that x2180T −°= . (3) 9.2 Prove AC || KB. (5) 9.3 Prove ∆BKT ||| ∆CAT (3)

9.4 If AK : KT = 5 : 2, determine the value of KBAC

(3) [14]

C x

A

O

B

K

T 1 2 3

4

1 2

1 2 3

x=BCA

21



QUESTION 9 AB is a diameter of the circle ABCD. OD is drawn parallel to BC and meets AC in E. If the radius is 10 cm and AC = 16 cm, calculate the length of ED. [5] QUESTION 10 CD is a tangent to circle ABDEF at D. Chord AB is produced to C. Chord BE cuts chord AD in H and chord FD in G. AC || FD and FE = AB. Let x=4D and y=1D .

10.1 Determine THREE other angles that are each equal to x. (6) 10.2 Prove that ΔBHD ||| ΔFED. (5) 10.3 Hence, or otherwise, prove that AB.BD = FD.BH. (2) [13]

D

A

B

C

E

F

4

1

1

1 1 G

H 2

2 2

2

3

3

3 1 2

3

A

C

B

D

E

O

22


Copyright reserved

QUESTION 11 ABCD is a parallelogram with diagonals intersecting at F. FE is drawn parallel to CD. AC is produced to P such that PC = 2AC and AD is produced to Q such that DQ = 2AD.

11.1 Show that E is the midpoint of AD. (2) 11.2 Prove PQ || FE. (3) 11.3 If PQ is 60 cm, calculate the length of FE. (5)

[10]

P Q

C D

B A

F E

23



8.2 ED is a diameter of the circle, with centre O. ED is extended to C. CA is a tangent to the circle at B. AO intersects BE at F. BD || AO. x=E .

8.2.1 Write down, with reasons, THREE other angles equal to x. (4)

8.2.2 Determine, with reasons, EBC in terms of x. (3) 8.2.3 Prove that F is the midpoint of BE. (4) 8.2.4 Prove that ∆CBD ||| ∆CEB. (2) 8.2.5 Prove that 2EF.CB = CE.BD. (3)

[21]

A

O

E

F

B

D

C

x

1 234

1

2

1 2

32 1

24

Mathematics/P3 7 DBE/November 2010 NSC

Copyright reserved

QUESTION 9 In the diagram below A, B, C and D are points on the circumference of the circle. BD and AC intersect at E. Also, EB = 8 cm, DC = 8 cm and AE : EC = 4 : 7.

If DE = x units and AB = y units, calculate x and y. [6] QUESTION 10 In the diagram below M is the centre of the circle. FEC is a tangent to the circle at E. D is the midpoint of AB.

10.1 Prove MDCE is a cyclic quadrilateral. (3) 10.2 Prove that MC2 = MB2 + DC2 – DB2. (3) 10.3 Calculate CE if AB = 60 mm, ME = 40 mm and BC = 20 mm. (4)

[10]

A

B

C

D

E

x 8

8y

A

M

D B C

E

F

|| ||

25

Mathematics/P3 DoE/November 2009 NSC


C

O B

M

A

QUESTION 9 O is the centre of the circle below. OM ⊥ AC. The radius of the circle is equal to 5 cm and BC = 8 cm.

9.1 Write down the size of A.CB (1) 9.2 Calculate: 9.2.1 The length of AM, with reasons (3) 9.2.2 (3) Area ΔAOM : Area ΔABC

26

Mathematics/P3 9 DoE/November 2009 NSC


QUESTION 10

In the figure below, GB || FC and BE || CD. AC = 6 cm and 2

BCAB

= .

D

A

C

B

E

G

F

H

10.1 Calculate with reasons: 10.1.1 AH : ED (4)

CDBE

10.1.2 (2)

10.2 If HE = 2 cm, calculate the value of AD × HE. (2)

[8]

27

Mathematics/P3 10 DoE/November 2009 NSC

Copyright reserved

QUESTION 11 In the figure below, AB is a tangent to the circle with centre O. AC = AO and BA || CE. DC produced, cuts tangent BA at B.

B

C

D

O

E A

F

1 2

3

4

12

3 4

1

1

1

2

2 2

3

11.1 Show . 12 DC = (3) 11.2 Prove that ΔACF ||| ΔADC. (3) 11.3 Prove that AD = 4AF. (4)

[10]

28



QUESTION 9 In the figure below, PQ is a diameter to circle PWRQ. SP is a tangent to the circle at P.

Let x=∧

1P

9.1 Why is QRP

∧

= 90°?

(1)

9.2 Prove that ∧∧

= SP1 .

(3) 9.3 Prove that SRWT is a cylic quadrilateral. (3) 9.4 Prove that ∆QWR /// ∆QST. (3) 9.5 If QW = 5 cm, TW = 3 cm, QR = 4 cm and WR = 2 cm, calculate the length of: 9.5.1 TS (3) 9.5.2 SR (3)

[16]

P Q

R

W

T

S

x 1

1

2

2 12

3

1 2

12

29


Copyright reserved

QUESTION 10 In the figure below, ∆ABC has D and E on BC. BD = 6 cm and DC = 9 cm. AT : TC = 2 : 1 and AD || TE.

10.1 Write down the numerical value of EDCE

(1) 10.2 Show that D is the midpoint of BE. (2) 10.3 If FD = 2 cm, calculate the length of TE. (2) 10.4 Calculate the numerical value of:

104.1 ABD of AreaADC of Area

∆∆

(1)

10.4.2 ∆ABC of Area∆TEC of Area

(3) [9]

A

B D

E C

T

F

30

mathematics grade 12 - western · pdf filewestern cape education department examination...

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