level 1 mathematics and statistics (91031) 2011
TRANSCRIPT
910310
1SUPERVISOR’S USE ONLY
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© New Zealand Qualifications Authority, 2011. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
ASSESSOR’S USE ONLY
TOTAL
Level 1 Mathematics and Statistics, 201191031 Apply geometric reasoning in solving problems
9.30 am �onday 1� November 2011am �onday 1� November 2011 �onday 1� November 20111� November 2011 November 2011 Credits: Four
Achievement Achievement with Merit Achievement with ExcellenceApply geometric reasoning in solving problems.
Apply geometric reasoning, using relational thinking, in solving problems.
Apply geometric reasoning, using extended abstract thinking, in solving problems.
Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.
You should attempt ALL the questions in this booklet.
Show ALL working.
If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.
Check that this booklet has pages 2 – 11 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
You are advised to spend 60 minutes answering the questions in this booklet.
QUESTION ONE
(a) Metalrailingsarefittedtotheedgeofadeck.
XZisparalleltoAD.
BYisparalleltoCZ.
Onesectionofrailingisshowninthediagrambelow:
62°
64°
X
Y
Z
C
BA
D
Diagram is NOT to scale
(i) FindthesizeofangleYBC.
Give geometric reasons for each step in your solution.
CALCULATION REASON
(ii) XYBisanisoscelestriangle.
UsegeometricreasoningforeachsteptoshowthatXBandYCcannotbeparallel.
CALCULATION REASON
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D
A
BC
O
72°
38°
y
x
Diagram is NOT to scale
(b) Forthediagramabove:
(i) FindthesizeofanglereflexCOA,x.
Explain your reasoning.
(ii) FindthesizeofangleDCO,y.
Give geometric reasons for each step in your solution.
CALCULATION REASON
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(iii) Forthediagrambelow,provethatangleC+angleA=180°.
D
A
BC
O
Diagram is NOT to scale
Give geometric reasons for each step in your solution.
CALCULATION REASON
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QUESTION TWO
Achild’spracticegoalposthasonepoleandtwosupports, asshownontheleft.
Thetwosupportsareeach90cmlong.
Thepoleisalwaysperpendiculartotheground.
(a) Thediagramontherightshowstheviewfromtheside.
OTis90cmlong.
OPis70cmlong.
(i) FindthelengthofPT,x,thedistancebetweenthepoleand asupportalongtheground.
(ii) CalculatethesizeofanglePTO.
(iii) Asupportbar,QR,isaddedatQ,whereOQ=30cm.
CalculatethedistanceofthepointRbelowO.
Show your working and explain your reasoning.
O
PxT
Diagram is NOT to scale
O
Q
R
PxT
Diagram is NOT to scale
O
T
P
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(b) AnothergoalposthasadifferentlengthforOP.
A3-Dsketchofthegoalpostisgivenbelow.
OTandONareboth90cmlong.
PT,PNandNTareall40cmlong.
PointAishalfwayalongNT.
(i) CalculatethesizeofangleTAP.
Explain your reasoning.
(ii) CalculatethelengthofAP.
(iii) CalculatetheangleOAP.
Show your working clearly.
O
P
T
N
A
Diagram is NOT to scale
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QUESTION THREE
(a) PRandQRarethesamelength.
AngleRQSisx.
RTisparalleltoPS.
x
RT
Q
P
S
Diagram is NOT to scale
(i) Ifxis110°,findthesizeofanglePRQ.
Give geometric reasons.
CALCULATION REASON
(ii) ProvethatanglePRTandangleRQSareequalforallvaluesofx.
Explain your geometric reasoning clearly and logically.
CALCULATION REASON
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E
D
F
O
82°
G
H J
Diagram is NOT to scale
(b) EFGDisacyclicquadrilateralwithangleEDG=82°.
Oisthecentreofthecircle. FindthesizeofangleHFJ.
Explain your geometric reasoning clearly and logically.
CALCULATION REASON
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A
N
O
x°
Z
QDiagram is
NOT to scale
(c) (i) ThepointsA,Q,Z,NlieonthecircumferenceofacirclecentreO.
AQisparalleltoNZ.
FindthesizeofangleNZQ,intermsofx.
Explain your geometric reasoning clearly and logically.
CALCULATION REASON
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O
(ii) Whatanglepropertiesdoesacyclic parallelogramhave?
Explain your answer with geometric reasoning.
Use the blank diagram above (where O is the centre of the circle) if you wish.
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QUESTION NUMBER
Extra paper if required.Write the question number(s) if applicable.
91
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