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Western Cape Education Department
Examination Preparation Learning Resource 2016
GEOMETRY
MATHEMATICS Grade 12
Razzia Ebrahim
Senior Curriculum Planner for Mathematics
E-mail: Razzia.Ebrahim@wced.info
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
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
5. 2014 November Paper 2 12 – 15
6. 2014 Exemplar Paper 2 16 – 17
7. 2013 November Paper 3 18 – 19
8. 2012 November Paper 3 20 – 21
9. 2011 November Paper 3 22 – 23
10. 2010 November Paper 3 24 – 25
11. 2009 November Paper 3 26 – 28
12. 2008 November Paper 3 29 – 30
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4
5
Mathematics/P2 DBE/November 2015 NSC
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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
Mathematics/P2 DBE/November 2015 NSC
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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
8
9
10
11
Mathematics/P2 DBE/November 2014
NSC
Copyright reserved Please turn over
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
Mathematics/P2 DBE/November 2014
NSC
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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
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Mathematics/P2 DBE/November 2014
NSC
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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
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Mathematics/P2 DBE/November 2014
NSC
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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]
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Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar
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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
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Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar
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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
18
19
Mathematics/P3 DBE/November 2012 NSC
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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
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Mathematics/P3 DBE/November 2012 NSC
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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
Mathematics/P3 DBE/November 2011 NSC
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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
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Mathematics/P3 DBE/November 2011 NSC
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
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Mathematics/P3 DBE/November 2010 NSC
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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
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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
|| ||
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Mathematics/P3 DoE/November 2009 NSC
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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
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Mathematics/P3 9 DoE/November 2009 NSC
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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
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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]
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Mathematics/P3 DoE/November 2008 NSC
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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
Mathematics/P3 DoE/November 2008 NSC
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
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