The following diagram shows a circle with the centre O2z4(2kn, + scj3dsikzdc s+ kc and radius 8 cm.



Points $\mathrm{A}$, $\mathrm{B}$ lie on the circle and $\mathrm{AO}$ $\mathrm{B}=1.4$ radians.
1. Find the length of the minor arc A B .    $cm^2$
2. Find the area of the shaded region.    $cm^2$

  The following diagram shows a circle with ce krcw-8k( b7lentre O and radius 10 cm.


Points $ \mathrm{A}$, $\mathrm{B}$ lie on the circle and $\mathrm{AO}$ $\mathrm{B}=2.2$ radians.
1. Find:
1. the length of the minor arc A B ;    cm
2. the perimeter of the shaded region.    cm
2. Find the area of the shaded region.    $cm^2$

  The diameter of a spherical ob5e eeg,3apb)vt 5xcn) w*hy j3ject is 3 $\times 10^{8} \mathrm{~cm}$ .
1. Write down the radius of the object.

The volume of the object can be expressed in the form $ \pi\left(a \times 10^{k}\right) \mathrm{cm}^{3} $ where $1 \leq a<10$ and k$ \in \mathbb{Z}$ .    $\times 10^8$
2. Find the value of a and the value of k . a =    k =   

  The following diagram shows a czxp)hs,mh/s3mh50. yuon g os7)ff p-v rwg,,ircle with centre O and radius 8 cm.



Points $\mathrm{A}$, $\mathrm{B}$ lie on the circle and $\mathrm{AO} \mathrm{B}=2.7$ radians.
1. Find:
1. the length of the major arc A B ; ≈    cm
2. the perimeter of the shaded region. ≈    cm
2. Find the area of the shaded region. ≈    cm

  In a triangle $ \mathrm{ABC}$, $\mathrm{AB}=6 \mathrm{~cm}$, $\mathrm{BC}=10 \mathrm{~cm}$ and $\mathrm{CBA}=70^{\circ}$ .
1. Find the area of the triangle. ≈    $cm^2$
2. Find AC.

Give your answers correct to 3 significant figures.≈    cm

  In a triangle $ \mathrm{ABC}$, $\mathrm{AB}=4 \mathrm{~cm}$, $\mathrm{BC}=10 \mathrm{~cm}$ and $ \mathrm{CB} A=105^{\circ}$ .
1. Find the area of the triangle A B C .≈    $cm^2$
2. Find AC. ≈    cm

  The following diagram shows t9:7gk 2.4erqptukfdtm0 qf z8riangle ABC.




$\mathrm{AC}=14 \mathrm{~cm}, $$\mathrm{CBA}=115^{\circ}$, $\mathrm{BA} \mathrm{C}=38^{\circ} $.

1. Find BC. ≈    cm
2. Find the area of the triangle A B C .≈    $cm^2$

  A bronze sphere has a radiui3wv05;wzyv xi0 sw05v9 vygqf:l8(5 anmyucs of 10.5 $\mathrm{~cm}$ .
1. Find the volume of the sphere, expressing your answer in the form a $\times 10^{k}$, $1 \leq a<10 $ and $k \in \mathbb{Z}^{+}$ .

The sphere is to be melted down and remoulded into the shape of a cone with a height of 11.9 $\mathrm{~cm}$ .≈    $\times 10^3$cm
2. Find the radius of the base of the cone, giving your answer correct to 3 significant figures.≈    cm

  A sphere made out of gold has a radius ofx z6o 1seqc.ov m rg-2wkx+8.y 13.2 $\mathrm{~cm}$ .
1. Find the volume of the sphere, expressing your answer in the form a $\times 10^{k}$, 1 $\leq a<10$ and k $\in \mathbb{Z}^{+} $.

The sphere is to be melted down and remoulded into the shape of a cylinder with a height of 18.4 $\mathrm{~cm}$ .≈    $\times 10^3$cm
2. Find the radius of the base of the cylinder, giving your answer correct to 2 significant figures. ≈    cm

  The following diagram shows -rc) qazes yaoz9u-w (ydb4 e;+5w*c mtriangle $ \mathrm{ABC}$ , with $\mathrm{AB}=7 \mathrm{~cm}$,$ \mathrm{AC}=5 \mathrm{~cm} $, $ 0^{\circ}<\mathrm{BA} \mathrm{C}<90^{\circ}$ and $ \sin (\mathrm{BA} \mathrm{A})=\frac{4}{5}$



1. Find the area of triangle A B C . ≈    $cm^2$
2. Find $\cos (\mathrm{BA} C)$ .   
3. Find BC. $a\sqrt{b}$ ≈   

  The following diagram shows a right-an. hmtpo s9d5k(krkj o-)3 p7kbgled triangle $\mathrm{ABC}$ and a sector ABD of a circle with centre $\mathrm{A}$ . The point $\mathrm{E}$ lies on [AC] such that [BE] is perpendicular to [AC]. The region R is bounded by [DE], [BE] and arc BFD.




$\mathrm{AB}=5 \mathrm{~cm}, \mathrm{AC}=13 \mathrm{~cm}, \mathrm{BC}=12 \mathrm{~cm}$

1. Find B A C , giving your answer in radians.≈   
2. Find the area of R .≈    $cm^2$

[/BE][/BE]

  The following diagram shows tx/afng; b /pm2riangle ABC.


1. Find BC.≈    cm
2. Find CBA, given that it is obtuse.≈    $^{\circ}$
3. Find the area of the triangle A B C .≈    $cm ^2$

  The following diagram shows trian ojo+g1k *e/+jc) qqrngle ABC.



1. Find A $\hat{C}$ B , given that it is acute.≈    $^{\circ}$
2. Find BC.≈    cm

  The following diagram sh6jn (woyoqrx6--z idnx bj0.)ows triangle $ \mathrm{PQR}$ , with $\mathrm{PQ}=10 \mathrm{~cm}$, $\mathrm{PR}=6 \mathrm{~cm}$ , $0^{\circ}<\mathrm{Q} \hat{\mathrm{P}} \mathrm{R}<90^{\circ}$ and $ \cos (\mathrm{Q} \hat{\mathrm{P}} \mathrm{R})=\frac{4}{5}$ .



1. Find $\sin (\mathrm{QP} R) $.   
2. Find the area of triangle P Q R .    $cm^2$
3. Find QR.a$\sqrt{b}$ ≈   

  In a triangle $ \mathrm{ABC}$, $\mathrm{AB}=12 \mathrm{~cm}$ and $ \mathrm{AC}=15 \mathrm{~cm}$ . The area of the triangle is 80 $\mathrm{~cm}^{2}$ .
1. Find the two possible values for BÂC.≈      
2. Given that BÂC is obtuse, find BC.≈    cm

  $\mathrm{ABCD} \text { is a quadrilateral where } \mathrm{AB}=9 \mathrm{~cm}, \mathrm{BC}=12 \mathrm{~cm}, \mathrm{CD}=11 \mathrm{~cm}, \mathrm{DA}=8.5 \mathrm{~cm} \text { and } \mathrm{AB} \mathrm{C}=90^{\circ} \text {. Find } \mathrm{AD} \hat{\mathrm{D}} \text {, giving your answer correct to the nearest degree. }$ ≈    $^{\circ}$

  The following diagram shows a circle with centre Oys: v5yidur 7k x rim+e5 sz2-u2l*6tr and radius 5 cm.


Points $ \mathrm{A}$,$ \mathrm{B} $ lie on the circle and $ \mathrm{AO} \mathrm{B}=2.1 $ radians.
1. Find the length of:
1. minor arc A B ;    cm
2. chord A B .≈    cm
2. Find the area of the shaded region.≈    $cm ^2$

  The following diagram shows hunft( s*futw5qo,r m uq0320 a circle with centre O and radius 7.2 cm.


Points $\mathrm{A}, \mathrm{B}$ lie on the circle and $\mathrm{AO} \mathrm{B}=1.3$sv radians.
1. Find the length of:
1. minor arc A B ;   
2. chord A B .≈    cm
2. Find the area of the shaded region.≈    $cm ^2$

  $\text { The app logo, for a mobile arcade game, is a sector of a circle of radius } 3 \mathrm{~cm} \text {, shown as shaded in the diagram below. The area of the logo is } 6 \pi \mathrm{cm}^{2} \text {. }$



1. Find, in radians, the measure of the angle AOB.   
2. Find the total length of the perimeter of the logo.   

  Cabin boy Jim is at the top of a cliff of:uv;ukc(, +(wyvkhfr ,c t 24kzjm(j n Tortuga Island, standing 100 $\mathrm{~m} $ above the sea level, and observes two ships in the Caribbean sea. The Black Pearl $(\mathrm{P})$ is at an angle of depression of $25^{\circ} $ and the Hispaniola $(\mathrm{H}) $ is at an angle of depression of $50^{\circ} $.
The following three dimensional diagram shows the foot of the cliff at $\mathrm{O}$ , Jim at $\mathrm{J}$ , two ships at $\mathrm{P}$ and $\mathrm{H}$ , and the angle $\mathrm{POH}=75^{\circ}$ .



Find the distance between the two ships, giving your answer correct to 3 significant figures. PH =    m

  $\text { Consider a triangle with sides } \mathrm{AB}=10 \mathrm{~mm} \text { and } \mathrm{AC}=8 \mathrm{~mm} \text {. Given that the area of the triangle is } 24 \mathrm{~mm}^{2} \text {, find the possible values for the length of }[B C] \text {. }$      
[/B C]

  The following diagrama;rg3 ie25lh;j rtct3 shows triangle ABC.


The area of the triangle A B C is 22 $\mathrm{~cm}^{2} $.
1. Find A B .≈    cm
2. Find B C .≈    cm
3. Find CBA.≈    $^{\circ}$

  The following diagram shows a cr h(o/0xnl4kq *bix1p-m jy lf1(dpvl x +:d6cell tower $ \mathrm{AB} 6.4 \mathrm{~m} $ tall on the roof of a commercial building. The angle of depression from $\mathrm{A} to a point \mathrm{C} $ on the horizontal ground is $49^{\circ}$. The angle of elevation of the top of the building from $\mathrm{C} is 45^{\circ}$ .


Find the height of the building. ≈    m

  The following diagram shows part of a circle wi fvm1 d2 js,u-b8ysn;bth centre O and radius 5 cm.


Points $\mathrm{A}, \mathrm{B}$ lie on the circle, chord $\mathrm{AB}$ has a length of 8 $\mathrm{~cm}$ and $\mathrm{AO} \mathrm{B}=\theta$ .
1. Find the value of $ \theta$ , giving your answer in radians.≈   
2. Find the area of the shaded region.≈    $cm ^2$

  A loaf pan is made in the shape ofl:lzix, (mg(6pdxy) ch d/wp . a cylinder. The pan has a diameter of 24 cm and a height di(x x:6p ,( )yhpzlwgldm.c/of 5 m.



1. Calculate the volume of this pan.

Gloria prepares enough bread dough to exactly fill the pan. The dough was in the shape of a sphere.≈    $cm ^3$
2. Find the radius of the sphere in $\mathrm{cm}$ , correct to one decimal place.

The bread was cooked in a hot oven. Once taken out of the oven, the bread was left in the kitchen. The temperature, T , of the bread, in degrees Celsius, ${ }^{\circ} \mathrm{C}$ , can be modelled by the function

$T(t)=a \times(1.51)^{-\frac{t}{3}}+21, \quad t \geq 0$,

where a is a constant and t is the time, in minutes, since the bread was taken out of the oven. When the bread was taken out of the oven its temperature was 205^{\circ} \mathrm{C} .≈    cm
3. Find the value of a .   
4. Find the temperature that the bread will be 10 minutes after it is taken out of the oven.

The bread can be eaten once its temperature drops to 35^{\circ} \mathrm{C} .≈    $C^{\circ}$
5. Calculate, to the nearest minute, the time since the bread was taken out of the oven until it can be eaten.
6. In the context of this model, state what the value of 21 represents.≈   

  The following diagram z.cnbbl .*whzhw 9r 3.shows a right-angled triangle ABC and a sector CBE of a circle with centre C. The point D lies on [AC] such that [BD] is perpendicular to [AC]. The region R is bounz nwrbbw.c.*9z lh3h.ded by [BD], [DE] and arc BFE.


Find the area of the shaded region R.≈    $cm ^2$[/BD][/BD]

  The following diagram shows a circle w2);kop mmt mg yjo.,4eith centre O and radius r cm.The circle is divided.o4ommtkp;yg e,2)m j into three equal sectors.


Points $\mathrm{A}, \mathrm{B}$ lie on the circle and $\mathrm{AO} \mathrm{B}=\theta$ radians.
1. Find the exact value of $\theta$ , giving your answer in terms of $\pi$ .

The area of the shaded sector $\mathrm{AOB} $ is $ 3 \pi \mathrm{cm}^{2}$ .   
2. Find the radius of the circle, r .   cm
3 . Find the length of the chord A B . ≈    cm

  $\text { The triangle } A B C \text { is equilateral of side } 5 \mathrm{~cm} \text {. The point } D \text { lies on }[B C] \text { such that } D C=2 \mathrm{~cm} \text {. Find } \cos (C \hat{D A}) \text {. }$   

[/B C]

  In a triangle $ \mathrm{ABC}$, $\mathrm{AB}=3 \mathrm{~cm}$, $\mathrm{BC}=5 \mathrm{~cm}$ and $\mathrm{ACB}=\frac{\pi}{6}$
1. Find, to three significant figures, the two possible lengths of [AC]      
2. Find the difference between the areas of the two possible triangles A B C .   

  The following diagram shows triangle ABC and sector ACD of a circgp:tcnoy(*b2;h pe/ a le with centre 2a*p en opc g:;y(h/tbC.


$\mathrm{AC}=5 \mathrm{~cm}, \mathrm{BC}=8 \mathrm{~cm} $, the area of triangle $\mathrm{ABC}=10 \sqrt{3} \mathrm{~cm}^{2}$ .

1. Find AĈB.   
2. Find the exact area of sector ACD.   

  $\text { The following diagram shows triangle } \mathrm{ABC} \text {, with } \mathrm{AB}=9 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm} \text {, and } \mathrm{ABC}=\frac{\pi}{3} \text {. }$




1. Show that $\mathrm{AC}=3 \sqrt{7} \mathrm{~cm}$ .   
2. The shape in the following diagram is formed by adding a semicircle with diameter [A C] to the triangle.



Find the exact perimeter of this shape.   

  $\text { The following diagram shows triangle } \mathrm{ABC} \text {, with } \mathrm{AB}=9 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm} \text {, and } \mathrm{ABC}=\frac{\pi}{3} \text {. }$




1. Show that $\mathrm{AC}=3 \sqrt{7} \mathrm{~cm}$ .   
2. The shape in the following diagram is formed by adding a semicircle with diameter [A C] to the triangle.



Find the exact perimeter of this shape.   

  $\text { A triangle } \mathrm{ABC} \text { has } a=10.2 \mathrm{~cm}, b=17.5 \mathrm{~cm} \text { and area } 32 \mathrm{~cm}^{2} \text {. Find the largest possible perimeter of triangle } \mathrm{ABC} \text {. }$   

  The following diagram shows a radioactivity warning symbol made out oy:z p gf0/ g5 oqn;klujhp4m83f a circle in the centre and three ejf/n 0pqm8 o5 lh3 4p;zggky:uqual blades.

$\text { Given that } \mathrm{OA}=2 \mathrm{~cm}, \mathrm{AB}=1 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm} \text {, and } \mathrm{CO} \mathrm{D}=30^{\circ} \text {, find the area of the symbol. }$   

  The diagram below shows a sector of a circle with radiusysud+4ao 4tj0,hn (rxouu y8s unv az*9mi534 r where $\mathrm{AO} \hat{\mathrm{B}}=x $ radians and the length of the $\operatorname{arc}$ A B is $\frac{4}{x} \mathrm{~cm}$ .


$\text { Given that the area of the sector is } 27 \mathrm{~cm}^{2} \text {, find the length of the arc } A B \text {. }$   

  The following diagram shows three cities n uf0que -/6 ,/y iz2uckhv/xg$\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ . City $\mathrm{B}$ is 70 $\mathrm{~km}$ from $\mathrm{A}$ , on a bearing of $130^{\circ}$ . City $ \mathrm{C}$ is 40 $\mathrm{~km} $ from City $\mathrm{B}$ , on bearing of $75^{\circ}$ .



1. Find CBA.   
2. Find the distance from City A to C . ≈   
3. If you wanted to travel from city A directly to City C, find the bearing you would need to travel. ≈   
4. Find the area enclosed by connecting the three cities in a triangle ABC.   

  The following diagram showdtvsv9t c(b36w j5r3ps quadrilateral ABCD.



$\mathrm{AB}=12 \mathrm{~cm}, \mathrm{AD}=4 \mathrm{~cm}, \mathrm{AC} B=85^{\circ}, \mathrm{CBA}=38^{\circ}$ .

1. Find A C .≈   
2. Find the area of triangle A B C .

The area of triangle A B C is three times bigger than the area of triangle A C D .≈   
3. Find the acute angle CÂD.   
4. Find C D .≈   

  The following diagram shows tri ls913t do8robzut1f +angle $ \mathrm{ABC}$ . Point $\mathrm{D}$ lies on [AC] so that [DB] bisects CBEA. The area of the triangle A B C is 3 $\mathrm{~cm}^{2}$ .



$\mathrm{AB}=2 \sqrt{7} \mathrm{~cm}, \mathrm{BC}=x \mathrm{~cm}$ , and $\mathrm{CB} \mathrm{D}=\theta$ , where $\sin \theta=\frac{3}{4}$
Find the value of x in the form of $\frac{a}{b}$ where a and b are positive integers.   

$\text { A right cylinder has height } h \mathrm{~mm} \text { and diameter } x \mathrm{~mm} \text {. The volume of this cylinder is equal to } 45 \mathrm{~mm}^{3} \text {. }$



The total surface area, A , of the cylinder can be expressed as $A=\frac{\pi}{2} x^{2}+\frac{k}{x} $.
1. Find the value of k .
2. Find the value of x that makes the total surface area a minimum.

  The diagram below shows a circle of radius 9 cm. The points A, B, C and 7jg9m lom8w9 hD lie on t7m8o wj9 lmhg9he circle.



1. Find A C . ≈   
2. 1. Find DĈA.≈   
2. Hence find A C $\hat{B}$ .≈   
3. Find the area of triangle A C D .≈   
4. Hence, or otherwise, find the total area of the shaded region.   

  Consider the following d (ku+fweyaa 18x hr w2xsz62p(e6kv0o0w cm5tiagram.




The sides [A B] and [B C] of the isosceles triangle A B C have lengths 5 $\mathrm{~cm}$ and the third side [A C] has length 6 $\mathrm{~cm}$ . The midpoint of [A C] is denoted by $\mathrm{M}$ . The circular arc $\mathrm{AC}$ has centre, G , the midpoint of [BM].
1. 1. Find A G .   
2. Find MGA in radians.≈   
2. Find the area of the shaded region.≈   

[/BM][/B C]

  Jack sets out for a horse ride from his fam*hb:;b zmw bs/ily ranch at point $\mathrm{A} $. He rides at an average speed of 6.3$ \mathrm{~km} / \mathrm{h} $ for 28 minutes, on a bearing of $40^{\circ}$ from the ranch, until he stops for a break near a spring at point B.
1. Find the distance from point $\mathrm{A}$ to point $\mathrm{B}$ .   

$\text { Jack leaves point } \mathrm{B} \text { on a bearing of } 112^{\circ} \text { and continues to ride for a distance of } 5.8 \mathrm{~km} \text { until he reaches a river at point } \mathrm{C} \text {. }$


2. 1. Show that $A \hat{B}$ C is $108^{\circ}$ .   
2. Find the distance from point A to point C . ≈   
3. Find BĈA.

Jack's brother John wants to ride a horse directly from the ranch to meet Jack at point $\mathrm{C} $.   
4. Find the bearing that John must take to point $\mathrm{C} $.

John rides at an average speed of 12.9 $\mathrm{~km} / \mathrm{h}$ .   
5. Find, to the nearest minute, the time it takes for John to reach point $\mathrm{C}$ .≈    minutes

  The diagram below shows the cross section of a cylindrical q).(s 8xiu).zzltkmc pipe, 80 cm in length, carrying water. The pipe has az )k.its)x (uczlq.8m radius of 15 cm.


The pipe is not at full capacity, such that the chord length of the water level [AB] is20 cm. Find the volume of water in the pipe.   

  The following diagram shows a circular crop fiel91w oh (8dn- qvezle0ld.


The circle has centre O and a radius of 400 $\mathrm{~m}$ , and the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ lie on the circle. The angle $\mathrm{AOB}$ is 1.6 radians.
1. Find the length of chord A B .≈   
2. Find the area of triangle A O B .

The angle BOC is 2.5 radians.≈   
3. Find the length of the minor arc AC.≈   
4. Find the area of the shaded region.

The shaded region is to be planted with corn. Corn seeds are sold in bags which cost 140 dollars each. One bag is enough for seeding 8960 $\mathrm{~m}^{2}$ .≈   
5. Find the cost of the corn seeds.   

  1. Given that $\theta=18^{\circ}$ satisfies the equation 4 $\sin ^{2} \theta+2 \sin \theta-1=0$ , find the value of $ \sin 18^{\circ}$ .   
2. Hence find the value of $\cos 36^{\circ}$ .   
The following diagram shows the triangle $\mathrm{ABC}$ where $ \mathrm{AB}=4 \mathrm{~cm}, \mathrm{BC}=5 \mathrm{~cm}$ and CBA=$36^{\circ}$ .

$\text { 3. Find } \mathrm{AC} \text { in the form } \sqrt{a+b \sqrt{5}} \mathrm{~cm} \text { where } a, b \in \mathbb{Z} \text {. }$

  The diagram below shows a fenced triangular enloz)/:tpjv6 yhy ao q pausa8/68 :z*lclosure in the middle of a grassy field6t8)ao/oyzs v a8:ppz l:h6auqljy*/ where $\mathrm{AC}=3 \mathrm{~m}, \mathrm{DC}=2 \mathrm{~m}, \mathrm{CB} \mathrm{A}=\alpha $ radians and $ \mathrm{AC} \hat{\mathrm{CB}}=\frac{\pi}{3} $ radians. One end of a rope is attached at point D on the outside edge of the enclosure, and the other end is attached to a goat G . Given that the rope is 6 $\mathrm{~m}$ long and the area of field outside the enclosure that the goat is able to graze is 74 $\mathrm{~m}^{2}$ , find the value of $\alpha$ .≈   

  1. Expand and simpli 485ddg a4gz(ssus*g-q2lj lcfy $(1+\sqrt{3})^{2} $.  
2. By writing $75^{\circ} $ as $30^{\circ}+45^{\circ}$ , find the value of $\cos 75^{\circ} $.

The following diagram shows the triangle $\mathrm{ABC}$ where $\mathrm{BC}=\sqrt{6}$, $\mathrm{CA}=2 $ and $ \mathrm{A} \hat{\mathrm{C} B}=75^{\circ}$ .   

$\text { 3. Find } \mathrm{AB} \text { in the form } a+\sqrt{b} \text { where } a, b \in \mathbb{Z} \text {. }$   

  In a triangle $ \mathrm{ABC}, \mathrm{AB}=2 \mathrm{~cm}, \mathrm{CBA}=\frac{\pi}{4}$ and$ \mathrm{BA} \hat{\mathrm{A}} \mathrm{C}=\theta$.
1. Show that $\mathrm{AC}=\frac{2}{\cos \theta+\sin \theta} $. AC =  (代数式) 
2. Given that $\mathrm{AC} $ has a minimum value, find the value of $ \theta $ for which this occurs.   

In a triangle $\mathrm{ABC}, \mathrm{BA} \mathrm{A} C=60^{\circ}, \mathrm{AB}=(1-x) \mathrm{cm}$, $\mathrm{AC}=(x+3)^{2} \mathrm{~cm}$,$-3\lt x\lt1$
1. Show that the area, A $\mathrm{~cm}^{2}$ , of the triangle is given by

A=$\frac{\sqrt{3}}{4}\left(9-3 x-5 x^{2}-x^{3}\right)$ .

2. 2a Calculate $ \frac{\mathrm{d} A}{\mathrm{~d} x} $.
2b Verify that $\frac{\mathrm{d} A}{\mathrm{~d} x}=0 $ when $x=-\frac{1}{3}$ .
3. 3a Find $\frac{\mathrm{d}^{2} A}{\mathrm{~d} x^{2}} $ and hence verify that $x=-\frac{1}{3}$ gives the maximum area of triangle A B C .
3b Calculate the maximum area of triangle A B C .
3c Find the length of [B C] when the area of triangle A B C is a maximum.

[/B C]

Peggy owns a rectangular fiewy2hzyvd5t/p:7ss no7ci 55 cld, 12 $\mathrm{~m} $ by 5 $\mathrm{~m}$ . Peggy attaches a rope to a metal post at the upper left corner of her field, and attaches the other end to her cow Daisy.
1. Given that the rope is 6 $\mathrm{~m} $ long, calculate the percentage of Peggy's field that Daisy is able to graze. Give your answer correct to the nearest percent.
2. Peggy replaces Daisy's rope with another one of length $\alpha \mathrm{m}, 5<\alpha<12$ , so that Daisy can now graze exactly three quarters of Peggy's field.
Show that $\alpha$ satisfies the equation

5 $\sqrt{\alpha^{2}-25}+\alpha^{2} \arcsin \left(\frac{5}{\alpha}\right)=90$

3. Find the value of $\alpha$ .

In a triangle $ \mathrm{XYZ}, \mathrm{XY}=9 \mathrm{~cm}, \mathrm{YZ}=x \mathrm{~cm}, \mathrm{XZ}=y \mathrm{~cm} $ and $\mathrm{XY} \mathrm{Y}=45^{\circ}$ .
1. Using the cosine rule, show that $x^{2}-9 \sqrt{2} x+81-y^{2}=0$.

Consider the possible triangles with $\mathrm{XZ}=7 \mathrm{~cm}$ .
2. Calculate the two corresponding values of Y Z .
3. Hence find the area of the smaller triangle.

Consider the case where y , the length of [X Z] , is not fixed at $ 7 \mathrm{~cm}$.
4. Find the range of values of y for which it is possible to form two triangles.

The diagram below shows two circles with cenu z9hfdw4 zt.7o0ci(dtres at the points A and B and radii 3r and 2r, respectively. The point B9zwozddi.fu0c7 (4 th lies on the circle with centre A. The circles intersect at the points C and D.



Let $\alpha$ be the measure of the angle CAD and $\theta $ be the measure of the angle CBD in radians.
1. Find an expression for the shaded area in terms of $ \alpha, \theta$ and r .
2. Show that $ \alpha=4 \arcsin \frac{1}{3}$ .
3. Hence find the value of r given that the shaded area is equal to 16 $\mathrm{~cm}^{2}$ .

Farmer Edward owns a triangula)i70sfnky7b 0 l ,tvlmr field with ratio of the sides 15: 15: 15 $\sqrt{2}$ . Edward attaches a rope to a wooden post at the right angle corner of his field, and attaches the other end to his cow Gertie.
1. Given that the rope is 10 $\mathrm{~m}$ long, calculate the percentage of Edward's field that Gertie is able to graze. Give your answer correct to the nearest percent.

Edward replaces the rope with another one, this time of length b metres, 102. Show that b satisfies the equation

$b^{2}\left[\frac{\pi}{4}-\arccos \left(\frac{15 \sqrt{2}}{2 b}\right)\right]+\left(\frac{15 \sqrt{2}}{2}\right) \sqrt{b^{2}-\frac{225}{2}}=101.25$ .

3. Find the value of b . Give your answer correct to two decimal places.

A water truck tank which is 3 metres long has alyvbhotl+0 n884zu;o uniform cross-section in the shape of a major segment. The tank is divided into two equal parts and is partially filled with water as shown in the following diagram of the cross-secb0o4+n otv8y zl8h lu;tion. The centre of the circle is O, the angle AOB is α radians, and the angle AOF is β radians.

1. Given that $\alpha=\frac{\pi}{4}$ , calculate the amount of water, in litres, in the right part
of the water tank. Give your answer correct to the nearest integer.
2. Find an expression for the volume of water V , in $\mathrm{m}^{3}$ , in the left part of the water tank in terms of $\beta$ .

The left part of the tank is now being filled with water at a constant rate of 0.001 $\mathrm{~m}^{3}$ per second.
3. Calculate $\frac{\mathrm{d} \beta}{\mathrm{d} t}$ when $\beta=\frac{3 \pi}{5}$. Round your answer to 3 significant figures.
4. Calculate the amount of time it will take for the left part of the tank to be fully filled with water. Give your answer in minutes and correct to the nearest integer.

In a triangle $\mathrm{ABC}$ ,

$\begin{array}{l}
5 \sin (\mathrm{ABC})-6 \cos (\mathrm{B} \hat{\mathrm{CA}})=7 \\
6 \sin (\mathrm{B} \hat{\mathrm{C}} \mathrm{A})-5 \cos (\mathrm{ABC})=\sqrt{2}
\end{array}$

1. Show that $\sin (\mathrm{ABC}+\mathrm{B} \hat{\mathrm{BA}})=\frac{1}{6}$ .

James claims that C A B can have two possible values.
2. Show that James is wrong by proving that CÂB has only one possible value.

This question ask you to investigate the relationship betweent -p3n, vihae0nte (v4rrd g3- the number of sides andnvtv3( -r,e p4g3n- art 0iedh the area of an enclosure with a given perimeter.
A farmer wants to create an enclosure for his chickens, so he has purchased 28 meters of chicken coop wire mesh.
1. Initially the farmer considers making a rectangular enclosure.
1. Complete the following table to show all the possible rectangular enclosures with sides of at least $4 \mathrm{~m} $ he can make with the 28$ \mathrm{~m} $ of mesh. The sides of the enclosure are

2. What is the name of the shape that gives the maximum area?

The farmer wonders what the area will be if instead of a rectangular enclosure he uses an equilateral triangular enclosure.
2. Show that the area of the triangular enclosure will be $\frac{196 \sqrt{3}}{9}$ .

Next, the farmer considers what the area will be if the enclosure has the form of a regular pentagon. The following diagram shows a regular pentagon.

Let O be the centre of the regular pentagon. The pentagon is divided into five congruent isosceles triangles and angle $\mathrm{A} \widehat{O}$ B is equal to $ \theta $ radians.
3. 1. Express $\theta$ in terms of $\pi$ .
2. Show that the length of $\mathrm{OA}$ is $\frac{14}{5} \operatorname{cosec}\left(\frac{\pi}{5}\right) \mathrm{m}$.
3. Show that the area of the regular pentagon is $\frac{196}{5} \cot \left(\frac{\pi}{5}\right) \mathrm{m}^{2} $.

Now, the farmer considers the case of a regular hexagon.
4. Using the method in part (c), show that the area of the regular hexagon is

$\frac{196}{6} \cot \left(\frac{\pi}{6}\right) \mathrm{m}^{2}$

The farmer notices that the hexagonal enclosure has a larger area than the pentagonal enclosure. He considers now the general case of an n -sided regular polygon. Let $ A_{n} $ be the area of the n -sided regular polygon with perimeter of 28 $\mathrm{~m} $.
5. Show that $A_{n}=\frac{196}{n} \cot \left(\frac{\pi}{n}\right)$ .
6. Hence, find the area of an enclosure that is a regular 14 -sided polygon with a perimeter of 28 $\mathrm{~m}$ . Give your answer correct to one decimal place.
7. 1. Evaluate $\lim _{n \rightarrow \infty} A_{n}$ .
2. Interpret the meaning of the result of part $(\mathrm{g}) (i)$.