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Cambridge University Press 978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions Manual Paul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward Excerpt More Information www.cambridge.org © in this web service Cambridge University Press

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Page 1: Cambridge University Press Paul Fannon , Vesna Kadelburg ...assets.cambridge.org/97811075/79378/excerpt/... · Paul Fannon , Vesna Kadelburg , Ben Woolley , ... b eg nn ng t h] ^

Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

www.cambridge.org© in this web service Cambridge University Press

✶� ✁✂✄☎✆✝☎✞ ✟✠✝☎✡✝✟☛☞✌

❊✍✎✏❝✑✒✎ ✓✔

✹ ✥✕✖✗✘✗✙

✺ ✚ ✛✖✻✖✸✘✾✜

❜ ✻✖✭✛✢✸✣✘✗✙

✤ ✛✖✻✢✛✖✸✢✻✖✸✘✻✸

✦ ✛✖✸✘✥✛

✼ ✚ ✛✙✖✻✙✖✻✥✖✻✛✘✥✛✻✸✧✾✻✜

❜ ★✩ ✪✫✬ ✮✪✯✰✱

✪✩★✯★✯✰ ✩✱✲✳ ✴

✽ ✵ ✷✿❀ ❁ ❂❂❃ ❄❂❀❂❄❀❅ ❆

❇ ❈❡r❡ ❉r❡ ✗ ❋♣t✐❋♥s ●❋r t❤❡ ❍❋✉r♥❡② t❋ t❤❡♠✐■■❧❡❏ t❤❡♥ t✇❋ ♣❋ss✐❑❧❡ ■✐r❡▲t✐❋♥s ✭❧❡▼❋r r✐◆❤t✣❏ ❉♥■ t❤❡♥ ❉ s✐♥◆❧❡ ▲❤❋✐▲❡ ❋● ♣❉t❤✉♣✇❉r■s❖ P❋ t❤❡ t❋t❉❧ ♥✉♠❑❡r ❋● ♣❋ss✐❑❧❡r❋✉t❡s ✐s ✗✖✕✖✥✘✙

◗❘ ✚ ❙ ❙ ❚❯❱ ❲ ❯❳ ❨❳❩

❜ ❈❡r❡ ❉r❡ t✇❋ ▲❉s❡s❬

❭ s❤✐rt ✐s ♥❋t ♣✐♥❦ ❪ ♥❋ ▲❋♥str❉✐♥ts❋♥ t✐❡ ❋r ✇❉✐st▲❋❉t❬ ❫ ❫ ❴❵❛ ❞ ❢ ❛❞❣

❭ s❤✐rt ✐s ♣✐♥❦ ❪ t✐❡ ♥❋t r❡■❏ ✇❉✐st▲❋❉t♥❋t r❡■❬ ❥ ❥ ♦q✈ ① ① q③④

⑤❋t❉❧ ♥✉♠❑❡r ❋● ♣❋ss✐❑❧❡ ❋✉t⑥ts ✐s⑦ ⑧⑨⑩❶ ❷❶❸ ❹⑩❸

◗◗ ✚ ✛✖✗✖✸✘✻✜

❜ ❺❻ ❼❽❻❾

◗❿ ✚ ➀➁ ➂➃➄

❜ ➅➆ ➇➈➆➉

❊✍✎✏❝✑✒✎ ✓➊

✹ ✚ ➋➌➍ ➎➏➐➏

❜ ➑✉♠❑❡r ❋● ❉rr❉♥◆❡♠❡♥ts ✇✐t❤ ❧❉r◆❡st❑❋❋❦ ❉t ❋♥❡ ❡♥■❬ ➒ ➓➔ →➣ ↔↕↕➙

P❋ ♥✉♠❑❡r ❋● ❉rr❉♥◆❡♠❡♥ts ✇✐t❤❧❉r◆❡st ♥❋t ❉t ❡✐t❤❡r ❡♥■❬

➛ ➜➝➞➟➞ ➠➟➟➞ ➡➢➞➞

✺ ✚ ➤➥➦ ➧➨➩

❜ ➫❡➭✉✐r❡ t❤❡ ⑥♥❉❧ ■✐◆✐t t❋ ❑❡ ✛❏ s❋ t❤❡r❡❉r❡ ➯➲➳ ➵➲ s✉▲❤ ♥✉♠❑❡rs❖

✦ ✚ ➸➺➻➼ ➽➾➾➚➪ ➻➶➹ ➪➘➴ ➚➘➘

❜ ➷➬➮➱ ✃❐ ❒✃✃ ❮❰❒ ❰❰❰ ❐❐❐

✼ Ï ÐÑÒ ÓÒ ÔÕÕÖ

✽ ✚ ×ØÙ ÚÛÜ

❜ ➫❡➭✉✐r❡ t❤❡ ⑥rst ■✐◆✐t t❋ ❑❡ ✥ ❋r ✕❏ s❋t❤❡r❡ ❉r❡ Ý Þßà áà ßâã s✉▲❤ ♥✉♠❑❡rs❖

❇ ä åæçè éêëì íç îæïðñòó

◗❘ ô❡▲❉✉s❡ t❤❡r❡ ❉r❡ ❉♥ ❋■■ ♥✉♠❑❡r ❋● r❡■t❋②s❏ t❤❡ ❋♥❡ ✐♥ t❤❡ ♠✐■■❧❡ ♠✉st ❑❡ r❡■õt❤❡♥❏ ❋♥ t❤❡ ❧❡▼ ❉rr❉♥◆❡ t✇❋ r❡■ ❉♥■ t✇❋❑❧✉❡❏ t❋ ❑❡ ♠❉t▲❤❡■ ❋♥ t❤❡ r✐◆❤t❖ ❈❡●❋❧❧❋✇✐♥◆ ▲❋❧❋✉r ♣❉tt❡r♥s ❉r❡ ♣❋ss✐❑❧❡❋♥ t❤❡ ❧❡▼❬➫➫ôô➫ô➫ô➫ôô➫ô➫➫ôô➫ô➫ôô➫➫❈❡r❡ ❉r❡

ö÷

øù

úù úùû ▲❋❧❋✉r ♣❉tt❡r♥s

❉ü❉✐❧❉❑❧❡❖

Page 2: Cambridge University Press Paul Fannon , Vesna Kadelburg ...assets.cambridge.org/97811075/79378/excerpt/... · Paul Fannon , Vesna Kadelburg , Ben Woolley , ... b eg nn ng t h] ^

Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

www.cambridge.org© in this web service Cambridge University Press

❚♦�✁✂ ✄☎ ✆♦✝✞✟✁✞✠ ✡☛☞☛✂✟✁♦✞✡✷

❲✐t❤✐✌ ✍❛❝❤ ✎❛tt✍r✌♥ t❤✍r✍ ❛r✍ ✤✺✏ ✹✏❛rr❛✌❣✍♠✍✌t✑ ✒❢ t❤✍ ✐✌❞✐✈✐❞✉❛❧ t✒②✑s

❙✒ t✒t❛❧ ✌✉♠❜✍r ✒❢ ✎✒✑✑✐❜❧✍ ❛rr❛✌❣✍♠✍✌t✑✐✑ ✓ ✓ ✣✔✕ ✖✕ ✻ ✗✼✘✙✚

❊✛✜✢✥✦✧✜ ✶★

✩✪ ✫

✪ ✫✬

✭ ✬ ✭✮

✮✮ ✮

✰ ✯✱ ✰✵

✲ ✳✲ ✳

✴ ✴ ✸

✴ ✽ ✸

✾ ✸

✮ ✮

✮ ✮

✮ ✮

✿❀ ❀

❁ ❂ ❀

❁ ❃❄❅ ❆❅ ❄ ❇❈❅❆❉❆❋❡ ❆●❉❡❍❡■❏

▲▼ ◆

▼ ◆▼ ◆▼ ◆▼ ◆

P◗ ❖ P ◗

❘ P P ◗

✮✮ ✮ ✮

✮ ✮

❯ ❱

❳ ❱❯ ❯ ❨❨❩

❨❨❩ ❩❬

❭r✒♠ ❪❫❴♥ ✮❵❥❦

♣ q ✇ q ✇① ① ③ ①✮ ✮ ✮④ ⑤ ④ ⑤⑥ ⑦ ④

❫✐✈✐❞✐✌❣ t❤r✒✉❣❤ ❜② ⑧ ⑨⑩✮ ❶ ❷❸

❺ ❻ ❺ ❻

❺ ❻❺ ❻

❼ ❼ ❼ ❽

❼ ❼ ❽

❼ ❾ ❽

❿ ❽ ➀

✮ ✮ ✮

✮ ✮

✮ ✮

✮ ✮

➁ ➁ ➁➂

➃ ➁➄ ➅

➄ ➆ ➅

➄ ➇➈➉➊ ➋

❊✛✜✢✥✦✧✜ ✶➍

✩ ❴❤✒✒✑✍ ➎ ❢r✒♠ ❥➏❸

➑➒➓

➔→➣

↔↕

➙↕➛➛↕

▲ ➜ ❴❤✒✒✑✍ ➝ ❢r✒♠ ➞❸

➠➡➢

➤➥➦

➨➨➩

➫ ❴❤✒✒✑✍ ➭ ❢r✒♠ ➯❸

➳➵➸

➺➻➼

➾➚➪

♣➶

➹➘➴

➷➬➮

➱✃

❐❒❮ ➱❰Ï ✃➱❐

Ð ❴❤✒✒✑✍ ➝ ❜✒②✑ ❢r✒♠ ❥➯ ❛✌❞ ➭ ❣✐r❧✑ ❢r✒♠ ❥➭❸

Ñ

ÒÓÔ

ÕÖ×Ñ

ÒÓÔ

ÕÖØ × Ø

ÙÚ

Û

ÙÜ

ÜÝÚÞ ÚÚ ÛÚßÚÞ

à ❴❤✒✒✑✍ ❥ ❣✒❛❧á✍✍✎✍r ❢r✒♠ ➝♥ â ❞✍❢✍✌❞✍r✑❢r✒♠ ã♥ â ♠✐❞ä✍❧❞✍r✑ ❢r✒♠ ➯♥ ❛✌❞ ➭❢✒rå❛r❞✑ ❢r✒♠ ➏❸

æ

çèé

êëìæ

çèé

êëìæ

çèé

êëìæ

çèé

êë

í ì ì ì í

î

ï

ð

ñ

ò

ñ

ó

ô

î õö ïó ïö îï óöö

÷ ❴❤✒✒✑✍ ❥➭ ❢r✒♠ ❥â❦♥ t❤✍✌ ❥❦ ❢r✒♠ ❥➭ã♥t❤✍✌ ❥❦ ❢r✒♠ t❤✍ r✍♠❛✐✌✐✌❣ ❥❥ã❸

ø

ùúû

üýþø

ùúû

üýþø

ùúû

üý

ÿ þ

✶�✁

✶✂

✶✂✄

✶✁

✶✶✄

✶✁

✶☎✻✶ ✶✁ ✭✆❙✝✮✹✞

✟✠ ➜ ❴❤✒✒✑✍ ➝ ❜✒②✑ ❢r✒♠ ❥â ❛✌❞ ➭ ❣✐r❧✑❢r✒♠ ❥➯❸

✣✢✜

✛✚✙✤

✣✢✜

✛✚✘ ✙ ✘

✡☛

✡☞

✷✸☞☛ ✡✷✌ ☛✸ ☞✍✌

➫ ❴❤✒✒✑✍ ➝ ❜✒②✑ ❢r✒♠ ❥â♥ ❛✌❞ ❢✒r t❤✍❣✐r❧✑ ❝❤✒✒✑✍ ➭ ❢r✒♠ ❥➏ ♠t❤✍ ➭ ✎❧❛❝✍✑ ✌✒tt❛á✍✌ ❜② ②✒❜✍rt❛ ✒r rr✐②❛✎ ❛✌❞ ❝❤✒✒✑✍❥ ❢r✒♠ ➭ ♠②✒❜✍rt❛ ✒r rr✐②❛✎❸

✑✒✓

✔✕✖✏

✑✒✓

✔✕✖ ✗ ✖ ✖ ✗

✥✦

✥★

✩✩ ✧✪✦ ✥✫★ ✩ ✬✪✦✦✫

✟✟ ➜ ❊①❛❝t❧② ➯✯ ✐✑ ✑✎✍✌t ✐❢ ②✒✉ ❝❤✒✒✑✍ ➝ ✒❢t❤✍ ➭✯ ✑å✍✍t✑ ❢r✒♠ ➞❸

✱✲✳

✴✵✺

✽✽✾

➫ ❊①❛❝t❧② ➞✯ ✐✑ ✑✎✍✌t ✐❢ ②✒✉ ❝❤✒✒✑✍ ❥ ✒❢t❤✍ ➭✯ ✑å✍✍t✑ ❢r✒♠ ➞ ❛✌❞ ❥ ✒❢ t❤✍ ➏✯✑å✍✍t✑ ❢r✒♠ ➏❸

❀❁❂

❃❄❅✿

❀❁❂

❃❄❆

❈❋❉

Page 3: Cambridge University Press Paul Fannon , Vesna Kadelburg ...assets.cambridge.org/97811075/79378/excerpt/... · Paul Fannon , Vesna Kadelburg , Ben Woolley , ... b eg nn ng t h] ^

Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✸✶ �✁✂✄☎✆✄✝ ✞✟✆✄✠✆✞✡☛☞

❝ ❊①✌✍t❧② ✎✏✑ ✒s s♣❡♥t ✒✓ ②♦✉ ✍❤♦♦s❡✺ ♦✓ t❤❡ ✔✑ s✇❡❡ts ✓r♦♠ ✕ ♦r ✔ ♦✓ t❤❡ ✺✑s✇❡❡ts ✓r♦♠ ✺✖

✣✢✜

✛✚✙✤

✣✢✜

✛✚✘ ✙ ✘

✷✷✥ ✥✦ ✧✥

❞ ❆t ♠♦st ✺✑ ✒s s♣❡♥t ✒✓ ②♦✉ ✍❤♦♦s❡✎ ♦r ✔ ♦✓ t❤❡ ✔✑ s✇❡❡ts ✓r♦♠ ✕ ♦r ✎ ♦✓t❤❡ ✺✑ s✇❡❡ts ✓r♦♠ ✺✖

✩✪✫

✬✭✮★

✩✪✫

✬✭✮★

✩✪✫

✬✭✯ ✮ ✮ ✯

✱✰ ✲✱ ✳ ✴✴

✵✹ ✻ ❈❤♦♦s❡ ✽ ✓r♦♠ ✾✖✿

❀❁❂

❃❄❅

❉❋●❍

❜ ❊①✍❧✉■❡ t❤❡ ♣♦ss✒❏✒❧✒t✒❡s ✇❤❡r❡ ✌❧❧q✉❡st✒♦♥s ✌r❡ ✓r♦♠ t❤❡ s✌♠❡ s❡✍t✒♦♥❑❆❧❧ q✉❡st✒♦♥s ✓r♦♠ ❆✖ ✍❤♦♦s❡ ✽ ✓r♦♠ ✺✖

▼◆❖

P◗❘

❚❙

❆❧❧ q✉❡st✒♦♥s ✓r♦♠ ❯✖ ✍❤♦♦s❡ ✽ ✓r♦♠ ✽✖

❲❳❨

❩❬❭

❪❫

❴ ♥✉♠❏❡r ♦✓ ✇✌②s ♦✓ ✍❤♦♦s✒♥❣✌t ❧❡✌st ♦♥❡ ✓r♦♠ ❡✌✍❤ s❡✍t✒♦♥ ✒s✎✔❵ ❛ ✺ ❛ ✎ ❢ ✎✔✏

✵✐ ❥♦ ■❡❧✒❏❡r✌t❡❧② ■♦✉❏❧❡❦✍♦✉♥t✖ ❡✌✍❤ ♣♦✒♥t✍♦♥♥❡✍ts t♦ ✎✽ ♦t❤❡r ♣♦✒♥ts✈ ③ ④⑤⑥ ⑦⑧ ⑦⑨⑩ ❶ ❷ ❶⑤ ❸❹ ❺ ❸❻❹ ❧✒♥❡s

✵❼

❽❾❿❿➀➁➂

➃➄ ➅➆ ➇➆➆➈➉➊➋ ➄➌➇➄ ➍➊➎➄➅➏➊➆ ➐➑ ➄➌➊ ➄➎➅➇➒➓➔➊➆→➣➈➇➋➎➅➔➇➄➊➎➇➔➆ ➏➇➒ ➐➒➔↔ ↕➊ ➇➄ ➄➌➊ ➐➎➅➓➅➒➇➔➄➊➒ ➙➐➅➒➄➆ ➇➒➋ ➒➐➄ ➇➄ ➇➒↔ ➅➒➄➊➎➆➊➏➄➅➐➒➆➏➎➊➇➄➊➋ ↕↔ ➔➅➒➊➆ ➛➐➅➒➅➒➓ ➄➌➊➆➊ ➄➊➒ ➙➐➅➒➄➆➜

✻ ❊✌✍❤ tr✒✌♥❣❧❡ ✒s ■❡➝♥❡■ ❏② t❤r❡❡ ♣♦✒♥ts❑➞✉♠❏❡r ♦✓ ■✒➟❡r❡♥t s❡ts ♦✓ ➠ ♣♦✒♥ts ❢

➢➤➥

➦➧➨

➩➫

➭➩➯➫

❜ ❊✌✍❤ q✉✌■r✒❧✌t❡r✌❧ ✒s ■❡➝♥❡■ ❏② ✓♦✉r♣♦✒♥ts❑➞✉♠❏❡r ♦✓ ■✒➟❡r❡♥t s❡ts ♦✓ ✽ ♣♦✒♥ts ❢

➳➵➸

➺➻➼

➽➾

➚➪➽➾

✵➶ ➹✒t❤ ⑤ ♣❡♦♣❧❡➘ ❡✌✍❤ s❤✌➴❡s ❤✌♥■s ✇✒t❤➷⑤ ➬ ♦t❤❡rs❑ ➮✒s ■♦✉❏❧❡❦✍♦✉♥ts t❤❡ t♦t✌❧

♥✉♠❏❡r ♦✓ ❤✌♥■s❤✌➴❡s❑

➱ ✃

➱ ✃➱ ✃

❐ ❒ ❮ ❰

❒ ❒ ❮

❒ Ï ❮

Ð ❮

⑤ ⑤

⑤ ⑤

⑤ ⑤

Ñ Ò ÒÓÔ

ÕÕÒ Ö

Ò× ÒØ Ö

Ò×

Ù

✵Ú Û♥✍❡ t❤❡ r♦✇s ✌r❡ ■❡t❡r♠✒♥❡■➘ t❤❡r❡ ✒s♦♥❧② ♦♥❡ ✌rr✌♥❣❡♠❡♥t ✓♦r ❡✌✍❤ r♦✇❑❈❤♦♦s❡ ✎✺ ✓r♦♠ ✽✺ ✓♦r t❤❡ ➝rst r♦✇ ✌♥■✎✺ ✓r♦♠ ➠✏ ✓♦r t❤❡ s❡✍♦♥■ r♦✇✖

Ü

ÝÞß

àáâÜ

ÝÞß

àáã â â â

ã â

äå

æå

çè

æåçéäå æè æéåå æè

åéçå æè êçëìí

îî ï

îð

ñòóôõö÷ó øñ

✵ ➞✉♠❏❡rs ■✒ù✒s✒❏❧❡ ❏② ✺ r✌♥❣❡ ✓r♦♠ ✔✎ ú ✺t♦ ✎❵✏ ú ✺✖ ✎✽✏ ♠✉❧t✒♣❧❡s ♦✓ ✺û♦ t❤❡r❡ ✌r❡ ü ýþÿÿ ✶�ÿ ✺✁ÿ ♥✉♠❏❡rs ♥♦t■✒ù✒s✒❏❧❡ ❏② ✺❑

❽❾❿❿➀➁➂

❆➔➄➊➎➒➇➄➅➍➊➔↔✱ ✂ ➐➈➄ ➐➑ ➊➍➊➎↔ ② ➐➑ ➄➌➊➒➈➉↕➊➎➆ ↕➊➄♥➊➊➒ ✄☎✄ ➇➒➋ ✆☎☎ ➇➎➊ ➒➐➄

➋➅➍➅➆➅↕➔➊ ↕↔ ②✱ ➆➐ ➄➌➊➎➊ ➇➎➊ ✤ ✝✹

✞✼✟✟ ✞✠✟

➐➑ ➄➌➊➉➜

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✹ ❚♦�✁✂ ✄☎ ☎✆✂✝✞✟✁♦✠ �✡✁✠✂✁�✝☛

✷ ✤❡r❡ ☞r❡ ✌ ✍✎✏❡r❡♥t ✑❡tt❡rs✒

✓✔t☞✑ ♥✉✕❜❡r ✔❢ ✖❡r✕✉t☞t✎✔♥s✗ ✘✻✙ ✼✚✛

◆✉✕❜❡r ✔❢ ✖❡r✕✉t☞t✎✔♥s ❜❡❣✎♥♥✎♥❣ ✜✎t❤

➅❏➆✗ ✢✺✣ ✶✥✦

✧ t✔t☞✑ ♥✉✕❜❡r ♥✔t ❜❡❣✎♥♥✎♥❣ ✜✎t❤ ➅❏➆✗

★ ✩✪✫✬ ✭✫✬ ✮✬✬

✸ ✯ ❈❤✔✔s❡ ✰ ❢r✔✕ ♠✱✗

✳✴✵

✽✾✿

❀❁

❂❀❃❁

❄ ❈❤✔✔s❡ ✰ ❢r✔✕ ❅❅❆ t❤❡♥ ❡①❝✑✉✍❡ t❤❡

❝❤✔✎❝❡s ✔❢ ☞✑✑ ❝❤✔❝✔✑☞t❡s✗ ✰ ❢r✔✕ ♠❅

❉❊❋

●❍■❇

❉❊❋

●❍❑ ■ ❑

▲▲

❖▲

▼❖P◗❘ ▲▲❘ ❖▼▲❘

❙ ✤❡r❡ ☞r❡ ❯ ✍✎✏❡r❡♥t ✑❡tt❡rs✒

✯ ✓✔t☞✑ ✖❡r✕✉t☞t✎✔♥s✗ ❱❲❳ ❨❩❬❩

❭❡r✕✉t☞t✎✔♥s ❜❡❣✎♥♥✎♥❣ ✜✎t❤ ➅❪❫➆✗

❴❵❛ ❞✐❥

✧ t✔t☞✑ ♥✔t ❜❡❣✎♥♥✎♥❣ ✜✎t❤ ➅❪❫➆✗

❦ ❧♣q✈q ✇②q ✈③②q

❄ ❭❡r✕✉t☞t✎✔♥s ❜❡❣✎♥♥✎♥❣ ✜✎t❤ ➅❪❫➆ ✔r

➅❫❪➆✗ ④ ⑤⑥ ⑦⑧ ⑥⑨⑩

✧ t✔t☞✑ ♥✔t ❜❡❣✎♥♥✎♥❣ ➅❪❫➆ ✔r ➅❫❪➆✗

❶ ❷❸❹❺❹ ❻❺❹ ❺❼❹❹

❽ ✓✔t☞✑ ❝✔✕✕✎tt❡❡ ✖✔ss✎❜✎✑✎t✎❡s✗

❿➀➁

➂➃➄

➇➈

➉➇➊➋➌➍

✓✔t☞✑ ☞✑✑➎✕☞✑❡ ❝✔✕✕✎tt❡❡s ✖✔ss✎❜✑❡✗➏

➐➑➒

➓➔→

➣↔

↕➙➛↔

✧ t✔t☞✑ ❝✔✕✕✎tt❡❡s t❤☞t ☞r❡ ♥✔t ☞✑✑ ✕☞✑❡✗➜ ➝➞➟➠➡➢ ➤➢➞ ➥➢➦➦➤

➧ ✓✔t☞✑ ✖✔ss✎❜✑❡ s❡✑❡❝t✎✔♥s ✔❢ ❯ t✎✑❡s✗

➩➫➭

➯➲➳

➵➸

➺➸➻➺➼➽➽

✓✔t☞✑ ✖✔ss✎❜✑❡ s❡✑❡❝t✎✔♥s ✜✎t❤ ♥✔ ➾✔✜❡✑s✗

➪➶➹

➘➴➷

➬➮

➱➮➮✃➬❐❒

✓✔t☞✑ ✖✔ss✎❜✑❡ s❡✑❡❝t✎✔♥s ✜✎t❤ ✔♥❡ ➾✔✜❡✑✗

❰ÏÐ

ÑÒÓ Ô

ÕÖ

×Ø ÕÙÖÚÕÛ

✧ t✔t☞✑ ✖✔ss✎❜✑❡ s❡✑❡❝t✎✔♥s ✜✎t❤ ☞t ✑❡☞st

t✜✔ ➾✔✜❡✑s✗Ü Ü ÝÞßàáââ ããÞäáâ äàã åäâ äàâäââ

æ ✓✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s✗ç

èéê

ëìí

îî

ïðñïòó

✓✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s ✜✎t❤ ♥✔ ✜✔✕❡♥✗ô

õö÷

øùú

ûü

ýþûü

✓✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s ✜✎t❤ ♥✔ ✕❡♥✗

ÿ

✣✢✜

✛✚✙

✶�

✻✾�✁

✓✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s ✜✎t❤ ❡①☞❝t✑② ✔♥❡ ✕☞♥✗

✂✄☎

✆✝✞✤

✂✄☎

✆✝✘

✟✠

✟✡

✟✼☛✠✡

✧ t✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s ✜✎t❤ ☞t ✑❡☞st ❅ ✕❡♥

☞♥✍ ♠ ✜✔✕☞♥✗☞✌ ✍✎✏ ✷✎✑ ✒✷✌ ☞✒✷✑ ✍✓✓✓✒✔ ✔ ✔ ✕

✽ ❆✕✔♥❣ t❤❡ ✎♥t❡❣❡rs ♠s♠✖❆ t❤❡r❡ ☞r❡

♠✱ ✔✍✍ ♥✉✕❜❡rs ☞♥✍ ✖ ❡➾❡♥ ♥✉✕❜❡rs✒

✯ ◆✔ ❡➾❡♥ ♥✉✕❜❡rs ✗ ❝❤✔✔s❡ ❯ ❢r✔✕ ♠✱✗

✦✧★

✩✪✫

✬✭

✮✬✯✭

❖♥❡ ❡➾❡♥ ♥✉✕❜❡r✗ ❝❤✔✔s❡ ✌ ❢r✔✕ ♠✱

☞♥✍ ♠ ❢r✔✕ ✖✗✰

✱✲✳

✴✵✸✰

✱✲✳

✴✵✹

✿❀

✿✿❃❂❀

✓✜✔ ❡➾❡♥ ♥✉✕❜❡rs ✗ ❈❤✔✔s❡ ❡ ❢r✔✕

♠✱ ☞♥✍ ❅ ❢r✔✕ ✖✗

❅❇❈

❉❊❋❄

❅❇❈

❉❊●

❍■

▲❑■▼▲

✧ ❝❤✔✎❝❡s ✜✎t❤ ☞t ✕✔st t✜✔ ❡➾❡♥

♥✉✕❜❡rs✗◆P◗ ◆❘❙◗ ❙◗❚P ◆◆◗❘P❯ ❯ ❱

❄ ✓✔t☞✑ ✖✔ss✎❜✎✑✎t✎❡s ✗ ❝❤✔✔s❡ ❯ ❢r✔✕ ♠✖✗

❳❨❩

❬❭❪

❫❴

❵❛❜❝❞❞

✧ ❝❤✔✎❝❡s ✜✎t❤ ☞t ✑❡☞st t✜✔ ❡➾❡♥

♥✉✕❜❡rs✗ ❢ ❢ ❣❤✐❥❦❦ ❧♠✐ ❧❦♥✐ ♦❦❥♣❦

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✺✶ �✁✂✄☎✆✄✝ ✞✟✆✄✠✆✞✡☛☞

✾ ❚♦t❛❧ ✌❡r♠✉t❛t✐♦♥s✍ ✤✻✎ ✼✏✑❚♦t❛❧ ✒❡❣✐♥♥✐♥❣ ✓✐t❤ ✔❉➆ ❛♥❞ ❡♥❞✐♥❣ ✓✐t❤✔▲➆✍ ✕✹✖ ✷✹✗ t♦t❛❧ ♥♦t ✒❡❣✐♥♥✐♥❣ ✓✐t❤ ✔❉➆ ♦r ♥♦t❡♥❞✐♥❣ ✓✐t❤ ✔▲➆✍ ✘ ✙✚✛✜ ✛✢ ✣✥✣

❊✦✧★❝✩✪✧ ✫✬

✭ ✮P ✯✯ ✰✱✲ ✲✳✳ ✴✵✸✽✿❀

❁ ❂P ❃❄❄ ❅❃❆❇❈❈

❋ ●P ❍■❏❑▼

◆ ❖P ◗◗❘❙❯

✾ ❱ ❲P P ❳❨❩❬❭❭❭❪❫❪

❴❵❜

❢❥❦ ♣q ✈

✇① ①

✇ ✇②P③

③ ③

③③

⑤ ④

⑤④④⑥

⑥ ⑦

⑧ ⑨⑩

❶⑩ ⑩ ⑩P

③ ③

③ ③③

❸❷

❹❷⑥

❺ ❻❼P P⑥⑥

⑥⑥❽

❢❢ ❾❡r❡ ❛r❡ ❿ ❞✐➀❡r❡♥t ❧❡tt❡rs➁ ➂♦♥s✐st✐♥❣ ♦➃➄ ➅♦✓❡❧s ❛♥❞ ➇ ➂♦♥s♦♥❛♥ts➈❚♦t❛❧ ♥✉♠✒❡r ♦➃ ✌❡r♠✉t❛t✐♦♥s✍ ➉P ➊➋➌➍

➎❚♦t❛❧ ♥✉♠✒❡r ♦➃ ✌❡r♠✉t❛t✐♦♥s ✓✐t❤ ♥♦➅♦✓❡❧s✍ ➏P ➐➑➒

➓✗ t♦t❛❧ ✌❡r♠✉t❛t✐♦♥s ✓✐t❤ ❛t ❧❡❛st ♦♥❡➅♦✓❡❧✍ ➔ →➣↔↕ ➣➙ ↔➛➜

❢➝ ➞t❤❡r t❤❛♥ ➟❛♠❡s➁ ➂❤♦♦s❡ ➠ r✉♥♥❡rs

➃r♦♠ ❿✍➡

➢➤➥

➦➧➨

➫➫➭

➯rr❛♥❣❡ ➄ r✉♥♥❡rs ✐♥ ♠❡❞❛❧ ✌♦s✐t✐♦♥s✍➲➳➵ ➸

➺♥ t✓♦➻t❤✐r❞s ♦➃ t❤❡s❡ ❛rr❛♥❣❡♠❡♥ts➁➟❛♠❡s ✐s ✐♥ ➼rst ♦r s❡➂♦♥❞➈

❚♦t❛❧ ➅❛❧✐❞ ❛rr❛♥❣❡♠❡♥ts✍ ➽ ➽ ➾➚➪ ➶➚

➹➘➴

❢➷ ➬❛s❡ ➮✍ ➱❛✃✐❞ ♥♦t s❡❧❡➂t❡❞➈➬❤♦♦s❡ ➄ st✉❞❡♥ts ➃r♦♠ ➮❿ ❛♥❞ ✌❡r♠✉t❡

t❤❡♠✍❐

❒❮❰

ÏÐÑ Ò

ÓÔ

ÕÕÖ ×ØÙØ

➬❛s❡ ➠✍ ➱❛✃✐❞ ✐s s❡❧❡➂t❡❞➈➬❤♦♦s❡ ➠ st✉❞❡♥ts ➃r♦♠ ➮❿➁ s❡❧❡➂t ♦♥❡✌♦st ➃r♦♠ t✓♦ ➃♦r ➱❛✃✐❞➁ t❤❡♥ ✌❡r♠✉t❡t❤❡ ♦t❤❡r t✓♦ ➂❛♥❞✐❞❛t❡s ✐♥ t❤❡ t✓♦

r❡♠❛✐♥✐♥❣ ✌♦sts✍Ú

ÛÜÝ

Þßà à á

âã

ää äå æçç

❚♦t❛❧ ➅❛❧✐❞ ❛rr❛♥❣❡♠❡♥ts✍

è éêëìë íêê êîïê

❢ð

ñ òñ òñ ò

ó

ô õ õ óõ

P P

③ ③ ③③

ö ö ÷ ö öøù

ø ù

⑥⑥

úû

ü

ý✐♥➂❡ P⑥þ ✐s ❛ r❡❛❧ ➅❛❧✉❡➁ ③ ÿ ✥

ý✐♥➂❡ P⑥✷ ✸ ✐s ❛ r❡❛❧ ➅❛❧✉❡➁ ③ � ➠❚r②✐♥❣ ❛❧❧ ✌♦ss✐✒❧❡ ➅❛❧✉❡s✍

✤ ✤ ✣ ✤③ P P✁ ✂ ✄ ☎✆✹✝

✻✞

✟ ✟ ✟③ P P✠ ✡ ✶☛☞✌✍

✌✍

✎ ✎ ✏ ✎③ P P✑ ✒ ✓✓✔ ✓✔✕✽✖

✗✘

✙ ✙ ✙③ P P✺ ✚ ✼✛✜✢✦✧

★✩

✪ ✪ ✫ ✪③ P P✬ ✭ ✮✯✰✱ ✲✰✱✳✴✵

✾✾

ý♦ ③✿➄ ♦r r

❊✦✧★❝✩✪✧ ✫❀

❢ ❚r❡❛t✐♥❣ ➟♦s❤✉❛ ❛♥❞ ➟♦❧❡♥❡ ❛s ♦♥❡ ✉♥✐t➁✌❡r♠✉t❡ ➮➄ ✉♥✐ts➁ t❤❡♥ ✐♥t❡r♥❛❧❧② ✌❡r♠✉t❡➟♦s❤✉❛ ❛♥❞ ➟♦❧❡♥❡✍

❁ ❂❃❄❅ ❆ ❃❆❇❈❇❉❇❃❊❉❉

➝ ➯rr❛♥❣❡ t❤❡ t❤r❡❡ ✒❧♦➂❦s ✐♥ ❋●❍ ■ ✓❛②s➈➺♥t❡r♥❛❧❧② ✌❡r♠✉t❡ t❤❡ ♠❡♠✒❡rs ♦➃ ❡❛➂❤➂❧❛ss ✐♥ ❏ ❏❑▲ ▼▲ ▼▲✓❛②s➈❚♦t❛❧ ❛rr❛♥❣❡♠❡♥ts✍◆ ◆ ◆ ❖P P◗ ❘◗ ❘◗ ❙❘❚❚❯❙❱

➷ ❚r❡❛t✐♥❣ t❤❡ ✌❤②s✐➂s ✒♦♦❦s ❛s ♦♥❡ ✉♥✐t➁✌❡r♠✉t❡ ❿ ✉♥✐tst t❤❡♥ ✐♥t❡r♥❛❧❧② ✌❡r♠✉t❡t❤❡ ➄ ✌❤②s✐➂s ✒♦♦❦s✍

❲ ❳❨❩ ❬❩ ❬❭❪❫❭

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✻ ▼�✁✂✄ ✂✁☎✆�✝☎✞�✟✝ ✠✡☎☛✞�☛✂ ☞

✹ ✤❡ ✌✍ ✎r✏②s ✏♥❞ ✎r❡❡♥s ✑✏♥ ✒❡ ✏rr✏♥❣❡❞✐♥ ✓✶✔✦ ✕✷✷✖✵✷✵✽✵✵ ✥✏②s✗

✤❡r❡ ✏r❡ ✌✘ s♣✏✑❡s ✐♥ ✙❤❡ ✚✐♥❡✛✜♣✭✐♥✑✚✜❞✐♥❣ ✙❤❡ ❡♥❞s✢✣ ✏♥❞ ✧♥❡ ★r✧✥♥ ✩✜s✙s✙✏♥❞ ✐♥ ❡✏✑❤ ✧❢ ✘ ✧❢ ✙❤❡s❡ s♣✏✑❡s✗

✤❡r❡ ✏r❡✪

✫✬✮

✯✰✱

✲✳

✳✲✴✴✲ ♣✧ss✐✒✚❡ s♣✏✑❡

s❡✚❡✑✙✐✧♥s✗✤❡♥ ♣❡r✩✜✙❡ ✙❤❡ ★r✧✥♥s✸ ✺✼✾ ✿✼❚✧✙✏✚ ♣✧ss✐✒✚❡ ✏rr✏♥❣❡✩❡♥✙s✸❀ ❀ ❁ ❀❂❃❄ ❂❅❅❂ ❆❄ ❂❇❈❅ ❂❅ ❉❃❙❊❋●❍

■ ❏ ❑P ▲◆❖▲◆❖◗◗❘❯❱

❜ ✤❡r❡ ✏r❡ ❲ ♣✧ss✐✒✚❡ s♣✏✑❡s ❢✧r ✙❤❡✚❡❧✩✧s✙ ❢r✐❡♥❞ ✙✧ s✐✙✗

❚✧✙✏✚ ✏rr✏♥❣❡✩❡♥✙s✸ ❳ ❨❩ ❬❭ ❪❫❴❵❛

❝ ❏ ❚r❡✏✙✐♥❣ ✙❤❡ ✩❡♥ ✏s ✧♥❡ ✜♥✐✙✣ ♣❡r✩✜✙❡❥ ✜♥✐✙s✣ ✙❤❡♥ ✐♥✙❡r♥✏✚✚② ♣❡r✩✜✙❡ ✙❤❡✘ ✩❡♥✸

❦ ♠♦q tq ✉✈✇①③

❜ ❚r❡✏✙✐♥❣ ✏✚✚ ✙❤❡ ✩❡♥ ✏s ✧♥❡ ✜♥✐✙ ✏♥❞✏✚✚ ✙❤❡ ✥✧✩❡♥ ✏s ✧♥❡ ✜♥✐✙✣ ♣❡r✩✜✙❡✙❤❡ ④ ✜♥✐✙s✣ ✙❤❡♥ ✐♥✙❡r♥✏✚✚② ♣❡r✩✜✙❡✙❤❡ ✘ ✩❡♥ ✏♥❞ ✐♥✙❡r♥✏✚✚② ♣❡r✩✜✙❡ ✙❤❡⑤ ✥✧✩❡♥✸

⑥ ⑥ ⑦⑧⑨ ⑩⑨ ❶⑨ ❶❷❸❹

❺ ❻❡r✩✜✙❡ ✙❤❡ ⑤ ✥✧✩❡♥✸ ❼❽❾ ❿➀➁

➂♥✙✧ ✘ ✧❢ ✙❤❡ ❥ s♣✏✑❡s✣ ✐♥s❡r✙ ✧♥❡ ✩✏♥✣✏♥❞ ✙❤❡♥ ♣❡r✩✜✙❡ ✙❤❡ ✩❡♥✸ ➃P ➄➅➆➇

➈❚✧✙✏✚ ♣✧ss✐✒✚❡ ✏rr✏♥❣❡✩❡♥✙s✸

➉ ➊➋➌➍ ➎➏➍ ➐➎➌➍➍

➑ ➒❡➓✜✐r❡ ✏ ➔→➔→➔→➔→➔✏rr✏♥❣❡✩❡♥✙✗

❻❡r✩✜✙❡ ✙❤❡ ⑤ ✥✧✩❡♥ ✏♥❞ ✙❤❡ ✘ ✩❡♥✸➣ ↔↕➙ ➛➙ ➜➝➝➞

➟➠➡➢➤ ➢➡➥➦➠➧➥➨➠➩➧ ➫➭➥➯➨➠➯➢ ➲

➳➵➸➺➻ ➼➽➾➚➻➪➸➶➚

➹ ➘❤✧✧s❡ ✍ ❢r✧✩ ➴ ✏♥❞ ♣❡r✩✜✙❡ ✙❤❡✩✸

➬➮➱

✃❐❒ ❮ ❮P

ÏÏÐ ÑÒÓÔ

Õ

Ö ❻❡r✩✜✙❡ ⑤ ✜♥✐✙s✸ ×ØÙ ÚÛÜ

Ý ❻❡r✩✜✙❡ ✍ ✏♥❞ ♣❡r✩✜✙❡ ➴✸ Þ ßàá âá àãäåã

✹ æç èéçê

■ ❚✧✙✏✚ ♣✧ss✐✒✚❡ ✑❤✧✐✑❡s ✥✐✙❤✧✜✙ r❡s✙r✐✑✙✐✧♥ ë

✑❤✧✧s❡ ✘ ❢r✧✩ ì✸í

îïð

ñòó

ô

õö÷

➘❤✧✐✑❡s ✥❤✐✑❤ ✑✧♥✙✏✐♥ ✙❤❡ ✙✥✧ ✧✚❞❡s✙ ë✑❤✧✧s❡ ④ ❢r✧✩ ✙❤❡ r❡✩✏✐♥✐♥❣ ❥✸

ø

ùúû

üýþ

ÿ

✷✶�

✁ ✑❤✧✐✑❡s ♥✧✙ ✑✧♥✙✏✐♥✐♥❣ ✒✧✙❤ ✧❢ ✙❤❡✧✚❞❡s✙✸ ✤ ✂✼✄ ☎✆ ✆✆

❝ ✝ ✞ ✝ ✞✢ ✟ ✛♥ ♥✠ ✡ ✸☛ ✠ ✡

☞ ✌☞ ✌☞ ✌ ☞ ✌

☞ ✌

☞ ✌☞ ✌

✍ ✎ ✏ ✎

✍ ✏

✍ ✎ ✏

✎ ✍ ✏

✚ ✏ ✏ ✎

♥ ♥ ♥ ♥

♥ ♥

♥ ♥

♥ ♥

♥ ♥

✑ ✑ ✦ ✒✵ ✑ ✦

✑ ✒✵

✒✵ ✵

✺ ✓ ✵

✺ ✔r✕❥✕✖✗ ✗t✕ ❡✕✘✙✗✜✣✕ ✥♦✧★✗✜♦❡ ✓✮

✪ ✤❡r❡ ✏r❡ ì ❞✐✫❡r❡♥✙ ✚❡✙✙❡rs✣ ✑✧♥s✐s✙✐♥❣ ✧❢✘ ✹✧✥❡✚s ✏♥❞ ✘ ✑✧♥s✧♥✏♥✙s✗➘❤✧✧s❡ ✌ ✧❢ ✙❤❡ ✘ ✑✧♥s✧♥✏♥✙s ✙✧ ✒❡ ✙❤❡✬rs✙ ✚❡✙✙❡r✣ ✌ ✧❢ ✙❤❡ ✧✙❤❡r ✍ ✑✧♥s✧♥✏♥✙s✙✧ ✒❡ ✙❤❡ ✚✏s✙ ✚❡✙✙❡r✣ ✏♥❞ ♣❡r✩✜✙❡ ✙❤❡r❡✩✏✐♥✐♥❣ ❥ ✚❡✙✙❡rs ❢✧r ✙❤❡ ✑❡♥✙r❡✸

✯✰✱

✲✳✴✭

✯✰✱

✲✳✴ ✻

✾❀❁ ❂❀✽❃

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✼✶ �✁✂✄☎✆✄✝ ✞✟✆✄✠✆✞✡☛☞

✽✤ ✌✢

✜✛✚

✙✘✗

✖✗

♥ ♥ ♥

✷✍✎✏

✑ ✒✑ ✒

✓ ✓ ✔

✓ ✕ ✔

✣ ✔

♥ ♥

♥ ♥

✥✦✧ ✧

✦★ ✦✩ ✧

✦★

✭✫❡✬❡❝t t❤❡ ✮❡❣❛t✐✈❡ ✯♦❧✉t✐♦✮ ✰ ✱♥ ✲✳✴

✾ ❚♦t❛❧ ✵♦✯✯✐❜❧❡ ❝❤♦✐❝❡✯ ✸✐t❤♦✉t ✫❡✯t✫✐❝t✐♦✮✿

❝❤♦♦✯❡ ✹ ✺✫♦♠ ✻✹✿❀

❁❂❃

❄❅❆

❇❈

❈❉❊❊❉

❋❤♦✐❝❡✯ ✸❤✐❝❤ ❛✫❡ ❛❧❧ ❣✐✫❧✯✿ ❝❤♦♦✯❡ ✹

✺✫♦♠ ●✿❍

■❏❑

▲▼◆

PP◗

❘ ❝❤♦✐❝❡✯ ✸❤✐❝❤ ❝♦✮t❛✐✮ ❛t ❧❡❛✯t ♦✮❡ ❜♦②✿

❙ ❯❱❲❲❱ ❳❨ ❩❬❭❪

❫❴ ❚♦t❛❧ ✵❡✫♠✉t❛t✐♦✮✯ ✸✐t❤♦✉t ✫❡✯t✫✐❝t✐♦✮✿

❵❞❢ ❥❦♣

q❡✫♠✉t❛t✐♦✮✯ ❝♦✮t❛✐✮✐✮❣ rs ♦✫ sr✿

✵❡✫♠✉t❡ ✹ ✉✮✐t✯ ❛✮✇ t❤❡✮ ✐✮t❡✫✮❛❧❧②

✵❡✫♠✉t❡ r ❛✮✇ s✿ ① ③④⑤ ⑥⑤ ⑥⑦⑧

❘ ✵❡✫♠✉t❛t✐♦✮✯ ✸✐t❤♦✉t r ❛✮✇ s ✮❡⑨t t♦

❡❛❝❤ ♦t❤❡✫✿ ⑩ ❶❷❸❹ ❸❺❹ ❺❻❹

❫❫ ❚♦t❛❧ ✵♦✯✯✐❜❧❡ ❝❤♦✐❝❡✯ ✸✐t❤♦✉t ✫❡✯t✫✐❝t✐♦✮✿

❝❤♦♦✯❡ ✹ ✺✫♦♠ ✻❼✿❽

❾❿➀

➁➂➃

➄➅

➆➇➈➅

❋❤♦✐❝❡✯ ✸❤✐❝❤ ❝♦✮t❛✐✮ t❤❡ t✸♦ ②♦✉✮❣❡✯t✿

❝❤♦♦✯❡ ➉ ✺✫♦♠ t❤❡ ✫❡♠❛✐✮✐✮❣ ✻➊✿

➌➍➎

➏➐➑

➒➓

➔➒→➓

❘ ❝❤♦✐❝❡✯ ✮♦t ❝♦✮t❛✐✮✐✮❣ ❜♦t❤ ♦✺ t❤❡

②♦✉✮❣❡✯t✿ ➣ ↔↕➙➛ ➜➛➝ ➞↕➛

❫➟ ❋❤♦♦✯❡ ❛✮✇ ✵❡✫♠✉t❡ ➉ ❧❡tt❡✫✯ ✺✫♦♠ ❼➠✿

➡➢➤➥➦

➢➧➨ ➦➩➩➫➭

➯➲➳

➵➸➺

➻➼➽ ➽

❋❤♦♦✯❡ ✹ ✇✐❣✐t✯ ✸✐t❤ ✫❡✵❡❛t✯ ❛❧❧♦✸❡✇✿

➾➚ ➪➚➶➹➚➘

❘ t♦t❛❧ ✵♦✯✯✐❜❧❡ ✫❡❣✐✯t✫❛t✐♦✮ ✮✉♠❜❡✫✯✿➴➷ ➬➮➮ ➷➱ ➮✃➱ ➱❐➴➴➬✃ ✃➮➮❒ ❮

❫❰ ❋❤♦♦✯❡ ✻ ✺✫♦♠ ✹ t♦ ❜❡ t❤❡ ✇✫✐✈❡✫ ❛✮✇

✵❡✫♠✉t❡ t❤❡ ✫❡♠❛✐✮✐✮❣ Ï✿

Ð

ÑÒÓ

ÔÕÖ ×

Ø

ÙÚÛ ÜØÜÝÝ

Þßààáâã

äåæçèé æêëæ æêé éìëèæ ëííëîïéðéîæ åñòéåòóé çî éëèê íåô åñ æêé åæêéí õéëæõ çõçííéóéöëîæ÷ õçîèé éëèê õéëæ çõ øîçùøéóúçûéîæçüéûý þêé ëîõôéí ôåøóû ÿé æêéõëðé ñåí ë öëî ôçæê õéëæõ çî ë s�s�s�sëííëîïéðéîæ÷ ñåí éìëðòóéý

❫✶ ✤❡ ✇✫✐✈❡✫✯ ❝❛✮ ❜❡ ❛✫✫❛✮❣❡✇ ✐✮ ❼ ✸❛②✯✁

t❤❡✮ ❝❤♦♦✯❡ ➉ ✺✫♦♠ ● t♦ ❣♦ ✐✮ t❤❡ ❝❛✫✿

✂✣

✢✜✛

✚✙✘✷

✸✄✄✷

Þßààáâã

þêé õòéèçüè òéåòóé æå ïå çî æêé öëî îééûîåæ ÿé èåîõçûéíéû÷ õçîèé ëñæéí èêååõçîïæêåõé æå ïå çî æêé èëí÷ æêé íéõæ ôçóó ïå çî

æêé öëîý téèëøõé☎

✆✝✞

✟✠✡☎

✆✝✞

✟✠☛

✺÷ çæ ðë✱éõ îå

ûçññéíéîèé ôêçèê öéêçèóé çõ èåîõçûéíéû æåêëöé æêé üíõæ òçè✱ åñ òëõõéîïéíõ ôêéîðë✱çîï æêé èëóèøóëæçåîý

▲✌✍✎ ✏✑✒✓✔✕✌✍✓

❫ ✥ ❋❤♦♦✯❡ ✻ ♦✺ ❼ ✵❧❛❝❡✯ ✺♦✫ r✮②❛② t❤❡✮

✵❡✫♠✉t❡ t❤❡ ♦t❤❡✫ ✖✿ ✗ ✦✧ ★✩ ★✪

❜ ❚♦t❛❧ ✵♦✯✯✐❜❧❡ ✵❡✫♠✉t❛t✐♦✮✯✿ ✫✬✭ ✮✯✰

q❡✫♠✉t❛t✐♦✮✯ ✸✐t❤ r✮②❛ ✮♦t ❛t ❛✮ ❡✮✇✿

✲ ✳✴✵✹ ✻✼ ✾✵

❝ ❲✐t❤ r✮②❛ ❛t t❤❡ ❧❡✿② ✵❡✫♠✉t❡ t❤❡

♦t❤❡✫ ✖✿ ❀❁❂ ❃❁

❲✐t❤ s❧❡✮❛ ♦✮ t❤❡ ✫✐❣❤t② ✵❡✫♠✉t❡ t❤❡

♦t❤❡✫ ✖✿ ❄❅❆ ❇❅

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✽ ▼�✁✂✄ ✂✁☎✆�✝☎✞�✟✝ ✠✡☎☛✞�☛✂ ☞

■❢ ✌♦t❤ ✍♦♥❞✎t✎♦♥s ❤♦❧❞✱ ✏❡r♠✉t❡ t❤❡

♦t❤❡r ✑✒ ✤✸✓ ✻

❚♦ ✔♥❞ t❤❡ t♦t❛❧ ♥✉♠✌❡r ♦❢ ✏♦ss✎✌❧❡

✏❡r♠✉t❛t✎♦♥s✱ r❡♠♦✈❡ t❤❡ ❞♦✉✌❧❡✲

✍♦✉♥t❡❞ ✍❛s❡s✒ ✕ ✖ ✢✷✗ ✷✗ ✘ ✗✷

✙ ✥ ❯♥r❡str✎✍t❡❞ ✚❛②s ♦❢ s❤❛r✎♥❣✒ ✛✜ ✣✜✺

❈❛s❡s ✚✎t❤ ❛❧❧ s✚❡❡ts t♦ ♦♥❡ ✏❡rs♦♥✒ ✦

❈❛s❡s ✚✎t❤ ❛t ❧❡❛st ♦♥❡ s✚❡❡t t♦ ❡❛✍❤

✏❡rs♦♥✒ ✧ ★✩✪ ✪ ✩✵

❜ ✫✬ ✭✬✮

❝ ❈❤♦♦s❡ ✯ s✚❡❡t ❢r♦♠ ✰ t♦ ✌❡

s✏❧✎t✒✳

✹✼✾

✿❀❁ ✰

❈❤♦♦s❡ ✦ s✚❡❡ts ❢r♦♠ ❂ ❢♦r t❤❡ ✔rst

✏❡rs♦♥✒❃

❄❅❆

❇❉❊

●❍

❏❑❡♥ t❤❡ s❡✍♦♥❞ ✏❡rs♦♥ ❣❡ts t❤❡ r❡st

♦❢ t❤❡ s✚❡❡ts▲◆

❚♦t❛❧ ✏♦ss✎✌❧❡ ✍❤♦✎✍❡s✒ ❖ P◗ ❘ ❙❱

❲ ✥ ❈❤♦♦s❡ ❳ s❡❛ts ❢r♦♠ ✯❂ ❛♥❞ ✏❡r♠✉t❡

t❤❡ ❳ ✏❡♦✏❧❡✒❨❩

❬❬❭ ❨❪❨❫❬❫ ❴❵❫

❥❦♣

q✇① ③

❜ ❈♦♥s✎❞❡r t❤❡ ✏❡♦✏❧❡ ♦❢ t❤❡ s❛♠❡ ❢❛♠✎❧②

❛s ❛ s✎♥❣❧❡ ✉♥✎t▲

❈❤♦♦s❡ ④ s❡❛ts ❢r♦♠ ✯✦ ❛♥❞ ✏❡r♠✉t❡

t❤❡ ④ ✉♥✎ts✱ t❤❡♥ ✎♥t❡r♥❛❧❧② ✏❡r♠✉t❡ t❤❡

✑ ✏❡♦✏❧❡ ♦❢ t❤❡ s❛♠❡ ❢❛♠✎❧②✒

⑥⑦⑧

⑨⑩❶ ❶ ❷

❸❹

❺❺❻ ❼❻ ❼❽❽❸❺❾❿

❝ ❑❡ ✏❡rs♦♥ ✚✎t❤ t❤❡ ✍♦✉❣❤ ✍❛♥ s✎t ❛t ❛♥

❡♥❞ s❡❛t ✚✎t❤ ♦♥❡ ❡♠✏t② s❡❛t ♥❡➀t t♦ ✎t

♦r ✎♥ ❛ ♥♦♥✲❡♥❞ s❡❛t ✚✎t❤ ❛♥ ❡♠✏t② s❡❛t

♦♥ ❡✎t❤❡r s✎❞❡▲

❈❛s❡ ✯✒ ❈❤♦♦s❡ ❛♥ ❡♥❞ ❢♦r t❤❡ ✍♦✉❣❤❡r✱

✍❤♦♦s❡ ➁ s❡❛ts ❢r♦♠ t❤❡ r❡♠❛✎♥✎♥❣ ✯✦

❛♥❞ ✏❡r♠✉t❡ t❤❡ ➁ ✏❡♦✏❧❡ ➂

➃➄

➅➆➇

➈➉➃ ➊➋

➌➋

➍➍➎ ➍➏➐➑➑➒➓

❈❛s❡ ✦✒ ❈❤♦♦s❡ ❳ s❡❛ts ❢r♦♠ ✯✦

❛♥❞ ✏❡r♠✉t❡ t❤❡ ❳ ➂

→➣↔

↕➙➛ ➜

➝➞

➟➟➠ ➝➡➡➢➟➤➥➥

➦❧t❡r♥❛t✎✈❡❧②✱ ✍❤♦♦s❡ ✯ s❡❛t ❢r♦♠

✯✦ ♥♦♥✲❡♥❞ s❡❛ts ❢♦r t❤❡ ✍♦✉❣❤❡r✱

✍❤♦♦s❡ ➁ s❡❛ts ❢r♦♠ t❤❡ r❡♠❛✎♥✎♥❣

✯✯ ❛♥❞ ✏❡r♠✉t❡ t❤❡ ➁ ✏❡♦✏❧❡ ➂

➨➩➫

➭➯➲➧

➨➩➫

➭➯➲ ➳

➵➸

➵➵

➺➺➻ ➵➼➼➽➾➚➪➪

❚♦t❛❧ ✏♦ss✎✌✎❧✎t✎❡s✒

➶ ➹➘➴➷➬➬➮➱ ✃➴➴❐➷❒➱➱ ❮➘➴❒✃➘➮➱

❰ ✥ ■❢ t❤❡ ❢♦✉r s❧♦ts ❛r❡ ❧❛✌❡❧❧❡❞ ✯✱ ✦✱ ✑

❛♥❞ ❂✱ ✍❤♦♦s❡ t✚♦ ♦❢ t❤❡s❡ t♦ ✌❡ t❤❡

✏♦s✎t✎♦♥s ❢♦r t❤❡ ÏÐÑs ❏t❤❡♥ t❤❡ ♦t❤❡r

t✚♦ ✚✎❧❧ ❛✉t♦♠❛t✎✍❛❧❧② ✌❡ t❤❡ ✏♦s✎t✎♦♥s

❢♦r t❤❡ ÏÒÑs◆Ó t❤✎s ✎s ❡Ô❡✍t✎✈❡❧② ✍❤♦♦s✎♥❣

✦ s❧♦ts ❢r♦♠ ❂✱ ✚❤✎✍❤ ❣✎✈❡sÕ

Ö×Ø

ÙÚÛ

Ü▲

❜ Ý② t❤❡ s❛♠❡ ❧♦❣✎✍ ❛s ✎♥ ❏❛◆✱ Þ ÏÐÑs ❛♥❞ Þ

ÏÒÑs ✍❛♥ ✌❡ ❛rr❛♥❣❡❞ ✎♥ß

àáâ

ãäÞ

Þ

å✚❛②s▲

❝ ❚♦ ❣❡t ❢r♦♠ t♦✏ ❧❡æ t♦ ✌♦tt♦♠ r✎❣❤t

✎♥ ❛ ❂ ç ❂ ❣r✎❞✱ t❤❡ ♠✎♥❡r ♠✉st ♠❛è❡

✑ ♠♦✈❡s t♦ t❤❡ r✎❣❤t ❏Ð◆ ❛♥❞ ✑ ♠♦✈❡s

❞♦✚♥ ❏Ò◆✱ ✎▲❡▲ ✎♥ s♦♠❡ ♦r❞❡r✎♥❣ ♦❢ ✑ ÏÐÑs

❛♥❞ ✑ ÏÒÑs▲

Ý② ❏✌◆✱ t❤❡r❡ ❛r❡é

êëì

íîï

ð

ñòó ✚❛②s▲

ô Ý② t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❛s ✎♥ ❏❛◆✱ õ ö÷Þ ø

ÏÐÑs ❛♥❞ ù úûü ý ÏÒÑs ✍❛♥ ✌❡ ❛rr❛♥❣❡❞ ✎♥

þ ÿ

ÿ

✜✛✚

✙✘Þ ü

Þ

✶✚❛②s▲

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Cambridge University Press978-1-107-57937-8 — Mathematics for the IB Diploma Higher Level Solutions ManualPaul Fannon , Vesna Kadelburg , Ben Woolley , Stephen Ward ExcerptMore Information

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✾✶ �✁✂✄☎✆✄✝ ✞✟✆✄✠✆✞✡☛☞

✺ ✥ ❈❤♦♦s❡ ✌ ✍❡♦✍❧❡ ✎r♦♠ ✏✑ ✎♦r ♦♥❡ ✒❡❛♠✓✤

✣✢✜

✛✚✔✕

✻✇❛②s✖

❍♦✇❡✈❡r✱ ✒❤✐s ✇✐❧❧ ✗♦✉❜❧❡✘✙♦✉♥✒✱ ❜❡✙❛✉s❡ ✙❤♦♦s✐♥❣ ✌ ✍❡♦✍❧❡ ✎♦r ♦♥❡ ✒❡❛♠ ✐s ❡q✉✐✈❛❧❡♥✒ ✒♦✙❤♦♦s✐♥❣ ✒❤❡ ♦✒❤❡r ✌ ✍❡♦✍❧❡ ✎♦r ✒❤❡ ♦✒❤❡r ✒❡❛♠✱ s✐♥✙❡ ✒❤❡ ✒✇♦ ✒❡❛♠s ♦✎ ❡q✉❛❧ s✐✦❡ ❛r❡ ♥♦✒s✍❡✙✐❝❡✗ ❛s ✧ ❛♥✗ ❞✖★❡ ✒♦✒❛❧ ♥✉♠❜❡r ♦✎ ✍♦ss✐❜❧❡ ✒❡❛♠s ✐s ✒❤❡r❡✎♦r❡

✪✫✬

✭✮✯

✯✷

✰✖

✲ ❈❤♦♦s❡ ✏ ✎r♦♠ ✏✑✱ ✑ ✎r♦♠ ✏✑✱ ✳ ✎r♦♠ ✏✑✱ ✴ ✎r♦♠ ✏✑✱ ✵ ✎r♦♠ ✏✑ ♦r ✒❤❡ ✍r❡✈✐♦✉s ❛♥s✇❡r✓✸

✹✼✽

✿❀❁✸

✹✼✽

✿❀❁✸

✹✼✽

✿❀❁✸

✹✼✽

✿❀❁✸

✹✼✽

✿❀❁

✹✼✽

✿❀❂

❃❄

❃❄

❃❄

❃❄

❃❄

❇❃

❃❄

❉❄❊❆❋

● ❈❤♦♦s❡ ✴ ✎r♦♠ ✏✑✱ ✒❤❡♥ ✴ ✎r♦♠ ■✖ ❞✉✒ ✒❤✐s ✇✐❧❧ ♦✈❡r✘✙♦✉♥✒ ❜② ✒❤❡ ♥✉♠❜❡r ♦✎ r❡❛rr❛♥❣❡♠❡♥✒s♦✎ ✒❤❡ ✳ ❣r♦✉✍s✱ ✐✖❡✖ ✳❏✱ s♦❑

▲▼◆

❖P◗❑

▲▼◆

❖P❘ ❙

❚❯

❱❳❨ ❩❬❬❩

❭ ✥ ❈❤♦♦s❡ ✳ ✎r♦♠ ✳✏✓❪

❫❴❵

❢❥❦

♣t

♣①①③④

✲ ❈❤♦♦s❡ ✳ ✎r♦♠ ⑤✓⑥

⑦⑧⑨

⑩❶❷

❸❹❺❻❼

❽ ❾❽ ❾❿ ❿➀

❿ ➁ ❿ ➀

⑤ ⑤ ⑤

⑤ ⑤ ⑤

➂ ➃

➄➅➂➆➇➈

➄ ➃ ➉➃➇➈ ➈➊ ➋

➌r♦♠ ➍➎❈✓ ➏⑤ ➐➐

● ❈❤♦♦s❡ ✳ ✎r♦♠ ⑤✓➑

➒➓➔

→➣↔

⑤⑤

↕➙➛➛

➜ ➝

➞ ➟➞ ➟

➞ ➟➞ ➟

➠ ➠➡

➠ ➢ ➡

➠ ➠ ➡ ➡

➠ ➢ ➡

➤ ➡ ➡ ➠

⑤ ⑤ ⑤⑤

⑤ ⑤ ⑤ ⑤

⑤ ⑤ ⑤

⑤ ⑤

⑤ ⑤

➥ ➦

➧➨➥➩➩

➧ ➦ ➫➩➩

➧ ➭➯➲ ➩ ➳➵➸➺➸➻➼ ➩➽

➦➫ ➦➧ ➩

➦➫ ➳➵➸➺➸➻➼ ➾➸➚➪➼➶➹➸ ➘➴➷➬➼➶➴➾ ➦➧➽