tuanaki, kaupae 3, 2006 - nzqa...tuanaki 90639, 2006 pĀtai tuaono ko te motuhanga o tētahi...

Tuanaki, Kaupae 3, 200690639 Te tuhi kauwhata o ngā motuhanga koeko

me te tuhi whārite e pā ana ki ngā motuhanga koeko

9 0 6 3 9 M

© Mana Tohu Mātauranga o Aotearoa, 2006Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.

For Supervisor’s use only 3

Mā te kaimāka anahe Paearu Paetae

Paetae PaetaeKaiaka

PaetaeKairangi

Te tuhi kauwhata o ngā motuhanga koeko me te tuhi whārite e pā ana ki ngā motuhanga koeko.

Te whakaoti rapanga e whai wāhi mai ana ngā motuhenga koeko.

Te whakaoti rapanga uaua e whai wāhi mai ana ngā motuhenga koeko.

Whakakaotanga o te tairanga mahinga

Whiwhinga: Toru9.30 i te ata Rāapa 29 Whiringa-ā-rangi 2006

Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu kei tō pepa whakauru ki te tau kei runga ake nei.

E tika ana kia riro i a koe tētahi Pukapuka o ngā Tikanga Tātai me ngā Tūtohi L3-CALMF hei whakamahi māu i roto i tēnei whakamātautau.

Me whakautu e koe ngā pātai KATOA kei roto i te pukapuka nei.

Whakaaturia ngā mahinga KATOA mō ngā pātai KATOA.

Ki te hiahia koe ki ētahi atu wāhi hei tuhituhi whakautu, whakamahia ngā whārangi kei muri i te pukapuka nei, ka āta tohu ai i ngā tau pātai.

Tirohia mehemea kei roto nei ngā whārangi 2–31 e raupapa tika ana, ā, kāore hoki he whārangi wātea.

HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE I TE MUTUNGA O TE WHAKAMĀTAUTAU.

See back cover for an English translation of this cover

Tuanaki 90639, 2006

Kia 40 meneti hei whakautu i ngā pātai o tēnei pukapuka.

PĀTAI TUATAHI

Tuhia te kauwhata o y2 = – 8x.

Whakaingoatia ngā āhuatanga pēnei i ngā haukotinga, ngā rārangi pātata me ngā arotahi.

y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104

Ki te hiahia koe ki te tuhi anō i tēnei kauwhata, whakamahia te whārangi 22, whārangi 24,

whārangi 26 rānei.

2

Mā te kaimāka anahe

Calculus 90639, 2006

You are advised to spend 40 minutes answering the questions in this booklet.

QUESTION ONE

Sketch the graph of y2 = – 8x.

Label features such as intercepts, asymptotes and foci.

y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104

If you need to redraw this graph,

use page 23, 25 or 27.

3

Assessor’suse only

Tuanaki 90639, 2006

PĀTAI TUARUA

Tuhia te kauwhata o x2 + y2 + 8x – 10y + 16 = 0.


y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104



�



QUESTION TWO

Sketch the graph of x2 + y2 + 8x – 10y + 16 = 0.


y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104



�


Tuanaki 90639, 2006

PĀTAI TUATORU

Tuhia te kauwhata o te ānau e tohua ana ki te x = 4cos t, y = 3sin t.


y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104



6



QUESTION THREE

Sketch the graph of the curve defined by x = 4cos t, y = 3sin t.


y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104



�


Tuanaki 90639, 2006

PĀTAI TUAWHĀ

(a) Whiriwhiria te whārite o te motuhanga koeko e whakaaturia nei.

Tuhia te whārite mā te ritenga tuaka (cartesian).

y

x–2–4–6 –1–3–5

2

4

1

3

5

–2

–4

–1

–3

–5

2 61 3 54

y =32

xy =-32

x

�



QUESTION FOUR

(a) Find the equation of the conic section shown.

Write the equation in cartesian form.

y

x–2–4–6 –1–3–5

2

4

1

3

5

–2

–4

–1

–3

–5

2 61 3 54

y =32

xy =-32

x

�


Tuanaki 90639, 2006

(b) Whiriwhiria te whārite o te motuhanga koeko e whakaaturia nei.

Tuhia te whārite mā te ritenga tuaka (cartesian).

y

x–2–4 –1–3–5

2

4

1

3

5

–2

–4

–1

–3

–5

21 3 54

10



(b) Find the equation of the conic section shown.

Write the equation in cartesian form.

y

x–2–4 –1–3–5

2

4

1

3

5

–2

–4

–1

–3

–5

21 3 54

11


Tuanaki 90639, 2006

PĀTAI TUARIMA

Whiriwhiria te whārite o te pātapa o te pūwerewere x y3 8 1

2 2

- = i te pūwāhi (3,4).

12



QUESTION FIVE

Find the equation of the tangent to the hyperbola x y3 8 1

2 2

- = at the point (3,4).

13


Tuanaki 90639, 2006

PĀTAI TUAONO

Ko te motuhanga o tētahi kauhanga raro mō ngā motukā he rite tōna āhua ki te taha whakarunga o tētahi pororapa.

10 m

4 m

10 mita te whānui o te kauhanga raro i te taumata o te huarahi.

E 4 mita te pūwāhi teitei i runga ake i te huarahi.

Kua whakamaua ngā rama ki te tuanui o te kauhanga raro i runga ake i ia arotahi o te pororapa.

Tātaihia te teitei o ngā rama i runga ake i te huarahi.

14



QUESTION SIX

The cross-section of a road tunnel for cars has the shape of the top half of an ellipse.

10 m

4 m

The tunnel is 10 metres wide at the level of the road.

Its highest point is 4 metres above the road.

Lights are located on the roof of the tunnel above each focus of the ellipse.

Calculate the height of the lights above the road.

15


Tuanaki 90639, 2006

PĀTAI TUAWHITU

Tokorua ngā kaiwhakarere i mahi i ngā mahi manawa kaitūtae i tētahi whakaaturanga wakarererangi.I whai haere tēnā wakarererangi o rāua i tēnā i tētahi ara porowhita i te papa poutū.

Ground

I rerekē te teitei o ngā wakarererangi i runga ake i te whenua mai i te itinga o te 100 mita ki te nuinga o te 1800 mita.

Ina 1500 mita tētahi wakarererangi i runga i te whenua, e 500 mita te teitei o tērā atu i runga ake i te whenua i tērā atu taha o te porowhita.

Tātaihia te tawhiti i waenganui i ngā wakarererangi e rua mena i tēnei pūwāhi ēnei wakarererangi.

Whenua

16



QUESTION SEVEN

Two pilots were performing stunts at an air show.Their planes were following each other in a circular path in the vertical plane.

Ground

The heights of the planes above the ground ranged from a minimum of 100 metres to a maximum of 1800 metres.

When one plane was 1500 metres above the ground, the other was 500 metres above the ground on the other side of the circle.

Calculate the distance between the two planes when they were in this position.

17


Tuanaki 90639, 2006

PĀTAI TUAWARU

Ka taea te tāutuhia he unahi hei huanui mō tētahi pūwāhi, ka neke kia tawhiti rite i tētahi pūwāhi pūmau, arā, te arotahi, me tētahi rārangi pūmau, arā, te rārangi whakarite (directrix).Ko te akitu o tētahi unahi i V(h,k) me te arotahi i F(a+h,k).Ko P(x,y) tētahi pūwāhi o te unahi.Ko L te rārangi whakarite o te unahi.Ko PN te rārangi hāngai mai i P(x,y) ki te rārangi whakarite, L.

y

x

V(h,k) F(a+h,k)

P(x,y)N

L

Tuhia he kīanga mō ngā roanga PN me PF.

Whakamahia ēnei kīanga hei whakaatu ko te whārite o te unahi ko (y – k)2 = 4a(x – h).

18



QUESTION EIGHT

Any parabola can be defined as the locus of a point, which moves so that it is equidistant from a fixed point, the focus, and a fixed line, the directrix.One such parabola has its vertex at V(h,k) and focus at F(a+h,k).P(x,y) is a point on the parabola.L is the directrix of the parabola.PN is the perpendicular from P(x,y) to L the directrix.

y

x

V(h,k) F(a+h,k)

P(x,y)N

L

Write expressions for the lengths PN and PF.

Use them to show that the equation of the parabola is (y – k)2 = 4a(x – h).

19


Tuanaki 90639, 2006

20



21


Tuanaki 90639, 2006

y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104

Mena kua hapa tō tuhi i tētahi o ngā kauwhata, me tuhi anō ki te puka kauwhata nei. Āta tohua te tau o te pātai.

22



y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104

If you have made a mistake and need to redraw a graph, use the appropriate copy printed here and clearly number the question.

23


Tuanaki 90639, 2006

y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104


24



y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104


25


Tuanaki 90639, 2006

y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104


26



y

x–2–4–6–8–10

2

4

6

8

10

–2

–4

–6

–8

–10

2 6 8 104


27


Tuanaki 90639, 2006

28


Taupātai

He pepa tāpiri hei whakaoti i ō whakautu mē e hiahiatia ana. Āta tohua te tau o te pātai.

Tuanaki 90639, 2006

29


Taupātai

He pepa tāpiri hei whakaoti i ō whakautu mē e hiahiatia ana. Āta tohua te tau o te pātai.


30


Question number

Extra paper for continuation of answers if required.Clearly number the question.


31


Question number

Extra paper for continuation of answers if required.Clearly number the question.

Level 3 Calculus, 200690639 Sketch graphs of conic sections

and write equations related to conic sections

© New Zealand Qualifications Authority, 2006All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

For Assessor’s use only Achievement Criteria

Achievement Achievement with Merit

Achievement with Excellence

Sketch graphs of conic sections and write equations related to conic sections.

Solve problems involving conic sections.

Solve more complex conic section problems.

Overall Level of Performance

Credits: Three9.30 am Wednesday 29 November 2006

Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.

Make sure you have a copy of the Formulae and Tables booklet L3-CALCF.

You should answer ALL the questions in this booklet.

Show ALL working for ALL questions.

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 2–31 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.

English translation of the wording on the front cover

tuanaki, kaupae 3, 2006 - nzqa...tuanaki 90639, 2006 pĀtai tuaono ko te motuhanga o tētahi...

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