Mary (M) sits on the top of the Eastern side of the Grand/aa+p 4:e1 jjhw5q8*pvkw.m z moty 4l Canyon. Below her stands Peter (P), at the bottom of the canyon. It is known that the depth of the Grand Canyon is 1850 meters at this point. Nathan (N) stands on the Western side of the bottom of the Canyon, with a jatph4op*:1zke a4jml w5y+v q.wm/8distance of 2.5 km to Peter.


1.Write down the depth of the Grand Canyon, MP, in kilometers.MP=  km
2.On the diagram, label the angle of elevation from N to M with an x.
3.Find the size of the angle of elevation from N to M.  $^{\circ}$
4.Find MN, the distance in kilometers between Mary and Nathan.  km

  A ladder 3.7 m long leans against a be8iztf fv t+b1hjp8 r :89tb)vertical wall. The distance from the top of the ladder toief+t8 vtbfbz8r:t)jph 9 81b the ground is 3.1 m.


1.Represent this information on a diagram in the space provided above. Show the ground, the ladder and the wall as the labelled line segments.
2.Find the distance from the bottom of the ladder to the wall.  m
3.Write down the acute angle made by the ladder with the wall.  $^{\circ}$

  The diagram below shows a triangle ABC. The s7gutymosp j./8 fyo)qb qt 7.+hrio*6ize of angle $\mathrm{A}\hat{\mathrm{M}}\mathrm{C}$ is $150^{\circ}$, where M is the midpoint of AB. MBC is an isosceles triangle, with MC = MB = 4 cm.

1.Write down the length of AM.  cm
2.Find the size of angle $\textrm{C}\hat{\textrm{M}}\textrm{B}$.  $^{\circ}$
3.Find the size of angle $\textrm{A}\hat{\textrm{B}}\textrm{C}$.  $^{\circ}$
4.Find the length of CB.  cm

  A lawn sprinkler watering a garden sprak8)tlqc:gue:o. -+adymw/u ul q 0f8iys at an arc of $140^{\circ}$ with a maximum spray distance of 3.5 metres, as shown in the diagram below.

1.Calculate the area of lawn that is watered by the sprinkler.  $m^2$
2.Calculate the length of the outer arc, where the maximum spray
distance lands.  m

  In a school playground, the grass is watered by lawn sprink/r,p5dj 0gdlhk1 acc -ii+ d:klers that are set to spray at an r1d:ki gjdkd acph/-5lci0 +,arc of $220^{\circ}$ with a maximum spray distance of 55 metres, as shown in the diagram below.


1.Calculate the area of grass that is watered by this sprinkler.  $m^2$
2.Calculate the length of the sprinkler arc edge, where the maximum
spray distance lands.  m

  A windscreen wiper blade b smmg5de )f-7i-yn1vf*7j: l6wgmpj has a length of 0.45 m. When it's raining and the blade is in mo17 esl7ffn- j)gd-mj mb 6vw5*iymg:ption, it moves through an arc angle of $\dfrac {3\pi} 4$ radians.


1.Determine the area of the windscreen wiped by the blade.  $m^2$
2.Determine the arc length travelled by the outer edge of the blade.  m

  There are two traffic lights on a road below a 120 m himii0vgv t2 mp3 75bvkl0g:da+ gh skyscraper. From the top of the skyscraper, the traffic lights ca0v3+:vbpii 07 tm5a2mg dlgkvn be seen at angles of depression of $35^{\circ}$ and $65^{\circ}$ on each side of the skyscraper, as illustrated in the diagram below.


Find the distance between the two traffic lights, correct to the nearest whole metre.  m

  A triangular plot of farming land is shown in the diagram below. The longest sibb .nrke298ly de is 120120 m, with the two adjoining angles to nrl9b. 2kyeb8this length being $35^{\circ}$ and $25^{\circ}$.


1.Find the lengths of two other sides of this plot of land.a$\approx$  m,b$\approx$  m
2.Find the area of the plot of farming land.  $m^2$

  A national park is in the form of a triangle, ABC, as shown in the follo2 f:, e4hefoogh q8(0yvsi ;kvwing diagram. Side length AB is 49 km and side leg(hy v 2o:;fq ifksv8o4e,0e hngth BC is 66 km. Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $58^{\circ}$.


1.Calculate the side length AC in km.  km
2.Calculate the area of the park.  $km^2$

  John is building furniture using cylindrical logs ;bjsbc6ol3 f + lg6*p v3vluct9*f-67iaqu hlof length 1.8 m and radius 9.2 cm. + cobb l3*a9*tvg;jcvu - lluf6366qifl 7phsA wedge is cut from one log and the cross-section of this log is illustrated in the following diagram.


1.Find the length of the wedge arc, ABC.  cm
2.Find the area of the empty sector, OABC.  $cm^2$
3.Find the volume of each log.  $cm^3$

  A farmer has a field in the form of a triangle, Ahkfgi2:f, x*rBC, as shown in the following diagram. Side length AB is 150 m and side length BC is 200 m. An2:g khfrfxi* ,gle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $62^{\circ}$.


1.Calculate the side length AC.  m
2.Calculate the area of the field, rounding your answer to the nearest square meter.  $m^2$

  During a hiking trip, Jim stopped and checked his compass for directimo0h09 p 3rxjwons. The compass has a circular face with centre at O andj m0h 9row3x0p the diameter of 20 cm. The needle of the compass is at an angle of $\theta$ to the East of North, covering an arc length of $l$, as illustrated in the diagram below.

1.Determine the circumference of the face of the compass.  cm
2.Using the diagram, write down the size of the angle $\theta$ in degrees.  $^{\circ}$
3.Calculate the arc length, $l$.  cm
4.Calculate the total area of the shaded sectors on the diagram.  $cm^2$

  A rotating sprinkler waters part of a lawn, shown as the shaded areapppa*p+. ktmn(jn3 )i in the diagram below. The sprin*n+3 a )pmppktp(.nij kler covers a distance of 13m, reaching a maximum distance of 17m and covering an angle of $130^{circ}$.

1.Calculate the length of the outer arc the sprinkler can reach, the arc from point A to point B.  m
2.Find the area the sprinkler can cover. Give your answer to the nearest square metre.  $m^2$

  A surveillance camera records the arehvv*(txfrv8m 4 6sm.u a in front of a warehouse, shown as the shaded area in the following diagr *8vxusvfmr.t4m6 ( vham. The camera's field of view has a radial distance of 15m, reaching a maximum distance of 21m away from the camera, and covering an angle of $150^{\circ}$.




1.Calculate the length of the outer arc the camera can record, the arc from point A to point B.  m
2.Find the area the camera can record.  $m^2$

  Cylindrical logs of length 2.7 m and radius 7.5 cm are used to buiyap:gua dpr6 lgk1q12z 7z6j 1ld a wooden cabin. A wedge is cut from one log and the cross-seg rja171 6y zul:6pz2ad1qg pkction of this log is illustrated in the following diagram.


1.Convert 108 degrees into radians, leaving your answer in exact values.
2.Find the length of the missing arc ABC.  cm
3.Find the area of the missing sector ABCO.  $cm^2$
4.Find the volume of this log.  $cm^3$

  A right pyramid has apex V and square base ABCk 8;*n74dzb yjynt 8cx;rsj c*D. The vertical height of the pyramid, VM, is 5 cm. The sloping edges arb;7d*y rx* s z8t;8nnk4cjcyje 8 cm long.


1.Calculate the length of MC.  cm
2.Calculate the size of the angle that sloping edge VC makes with the
vertical height of the pyramid.  $^{\circ}$
3.Calculate the area of the triangle ABC.  $cm^2$

  A right pyramid has apex V and square base ABCD with the base diagonal rz 6 w-t5ckcas fev-65AC=4cm. The sloping edges z5 6-6 sakfwt-rvce5care 7 cm long.



1.Calculate the size of the angle that sloping edge VC makes with
the pyramid base.  ${\circ}$
2.Calculate the vertical height of the pyramid, VO.  cm
3.Calculate the volume of the pyramid.  $cm^3$

  Triangle Island is a small island located off the north-western tip of Britip9bopl4tnp:c0 16 u yfsh Columbia, Canada. Two side lengths of Triangle Island are equal to 1600 and 1800 metres and the angle between these two sides is equal t lto4 61p9:p b0pnfcuyo $78^{\circ}$.

1.Calculate the length of the third side of Triangle island.  m
2.Find the area of a Triangle Island in $m^2$. Give your answer in the form $a\times 10^k$, where $1 \leq a < 10$ and $k \in \mathbb Z$.  $\times10^6m^2$

  The quadrilateral ABCD has side lengths AB =17 cm, BC = 12cm, and CD = 11cm. The izq(tb23r4ax(a t6q8fc 7xe r size qacf72r(3trbqzi 8x6 atex(4 of angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ is $105{\circ}$ and the size of angle $\mathrm{C}\hat{\mathrm{A}}\mathrm{D}$ is $25^{\circ}$.


1.Find the length of AC in centimetres.  cm
2.Find the size of angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$

  The diagram below shows quadrilateral ABCD with sj1s 7pg 8rws38ojwt (t7 2qt5uljqc k0ide lengths of AD = 12 m, DC = 7 m and BC 7ktlwo0 1p8r(85tjj cjq sw3gu72 tsq= 5 m. Angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}=50^{\circ}$ and angle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}=100^{\circ}$.


A new line is drawn from B to D direct.
1.Calculate the length of this new line, BD.  m
2.Calculate the size of angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{D}$.  $^{\circ}$

  Cylindrical logs of length 2.2 m and radius 7.6 cm are )7f8 v qse-0hm1wdnrt used to construct a fence line. Part of the logs are removed, as illuf vn-1r0w qetsm)7dh 8strated in the diagram below.

Find the volume of each log.  $cm^3$

  Consider the analogue clock below. The cloc3kzdp :lgelt - ad35m;k has a diameter of 30 cm and is shaded kpa:m dt5zell3gd -;3 between 1 and 6.


1.Find the length of the arc between 1 and 6, rounded to the nearest cm.  cm
2.Find the area of the sector between 1 and 6, rounded to the nearest
integer.  $cm^2$

  A bakery has designed g6smefyi1hdavkb*;5 5ze 9 v- a box to deliver their speciality birthday cakes. The box has a rectangular prism base, with a rectangular pyramid toag9*5bmfi-;dvky1z 6 sehev 5p, as illustrated in the diagram below with dimensions shown.
After the cake is put into the box, the outside of the box is completely wrapped in colourful gift paper.

Point O shown on the diagram is the midpoint of the base of the pyramid.
1.Find the value of a shown in the diagram; the slant height of one of the pyramid faces.  cm
2.Find the total outer surface area of the box to be wrapped in gift paper.  $cm^2$

  Joe creates a small basyrf twp*c+ 1b6ueq-n8eball field in his backyard to play with his children. The field is covered by grass in the shape of a sector of a circle with radius 8 f+ t n*u-p68yebqcwr1metres, as shown in the following diagram. The angle at the centre of the sector is $\theta^{\circ}$


A total of $48 m^2$ of grass was used to cover the field.
1.Find the value of $\theta$.  $^{\circ}$
Joe asks his son, Lancer, to run around the field twice to warm up before playing a game.
1.Find the total distance that Lancer runs.d=  m

  Kindergarten students are creating cone-shaped birthday hats from pieces of ci7fj9v5 kgv-(/u8rq i ibnn c9olorful paper circles. The first step is to cut out a sector of the paper, as illustrated by sector OABC in the diagram below. The paper circles have centre O and . The length of -b grin/i85cknjv7vfq( 99i uarc ABC is $4\pi$ cm and angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{C}= \dfrac {2\pi} 7$

1.Find the radius of the paper circles, r.  cm
2.Find the area of the sector the students cut out, OABC.  $cm^2$

  A right pyramid has apex V and squarq pky.xrtda;2 ++bu9,iijt- e e base ABCD, with AB = 3 cm. The vertical height of the pyramid, VM, is 12 cm. k+ i -2.xa 9eqjpdr,ib;+tuyt


1.Calculate the length of VA.  cm
2.Calculate the volume of the pyramid.  $cm^3$

  The sides of a triangle ABC have the following lengths 3 )k:jdrvts vqv*8ionw- q.h* AB = 40 mm, BC = 30 mm, and CAd r:-sin.j*)ktvv qwohv*83 q = 55 mm.



1.Calculate the size of the angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$
2.Calculate the area of the triangle ABC.  $mm^2$
A point D is located outside of the triangle, such that angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$ is $80^{\circ}$ and angle $\mathrm{D}\hat{\mathrm{B}}\mathrm{A}$ is $10^{\circ}$.
3.Show that the angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{B}$ is $90^{\circ}$.  $^{\circ}$
4.Find the distances AD and BD.  mm
5.Find the shortest path which starts at point B and passes through all points A, C, D (in any order).  mm

  Two hot air balloons, located at points A and*e6bg2s,fl hah8n u. cu2(myv B, float in the sky 50 m apart from each other on the same height above a flat (horizontal) ground. Two straight ropes, AP and BP, are attached to the same point, P, on the ground. The length of A2sg(.un 8ymfuvea 2bl6* c ,hhP is
36m and angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{P}$ is $45^{\circ}$.


1.On the diagram, draw and label with an $\alpha$ the angle of elevation of A Angle APB is acute.
2.Find the size of angle $\mathrm{A}\hat{\mathrm{P}}\mathrm{B}$.  $^{\circ}$
3.Find the size of the angle of elevation of A from P.  $^{\circ}$

  A plot of farmland is shown by the shaded1 eg*)/rf az:dr pafg:ohiqnj +:b+*7hn im a7 region in the diagram below. The plot has a curved border which can7h g+f/ai*nrmoq:ae ra):*bi pg1d:+fj zhn 7 be described as an arc, PQ, of a circle with centre O and radius,$r=500m$, such that $PÔQ =80^{\circ}$ The straight border of the farmland is defined by the chord PQ.


The farm owner wishes to construct a fence along the curved border of the plot.
1.Find the length of the fence to be built.  m
2.Find the area of the farmland, rounding your answer to the nearest $m^2$.  $m^2$

  A large triangular garden, ABC, is shown inr 0z7rt9z-y:rt7rqkm the diagram below. A walking path passes through the garden at pointr77mq tr kzzyrt9 r:0- D and leaves at point E, such that DE is parallel to the garden boundary BC.


Triangles ABC and ADE are found to be similar (same shape, but side lengths can vary). Angles $AD̂E = AB̂C =\theta$ and $AÊD = AĈB =\alpha$.
1.Find the value of
1.x.  m
2.y.  m
2.Find the value of
1.$\alpha$.  $^{\circ}$
2.$\theta$  $^{\circ}$
3.$\gamma$  $^{\circ}$
3.Find the area of the triangle ABC, rounding your answer to the nearest $m^2$.  $m^2$

  Two concentric circles with centre O are shown in the diagram below.8g,q7sy61osq apj ms .rb qz8) The radius of the smaller circle is 5 cm and the radius of the larger circle is m bq.r7 )ssq8j6zs1pg8oqa,y 7 cm. Points A and B lie on the circumference of the larger circle such that angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$ is $\frac {\pi}{5}$ radians.


1.Find the length of the short arc AB.  cm
2.Find the area of the shaded region.  $cm^2$

  In a triangle ABC,AB=3 cm,BC=5 cmb2l3ffnciv- ybx t t,g+z*chx )8( y4x and $\mathrm{A}\hat{\mathrm{C}}\mathrm{B} = \dfrac{\pi}{6}A$.
1.Find, to three significant figures, the two possible lengths of [AC].  cm   cm
2.Find the difference between the areas of the two possible triangles ABC.  $cm^2$

  In a triangle ABC,AB=12 cm and AC=15 cm. The area of the triangle i,tl cvpkqcq/ :a. xt9+s $80cm^2$.
1.Find the two possible values for angle BÂC.  $^{\circ}$,  $^{\circ}$
2.Given that BÂC is obtuse, find the length of side BC.  cm

  A triangle ABC has side lengths a = 10.2 cm, b= 17.5 cm and an are4ofst2ir-8 mn a of $32 cm^2$. Find the largest possible perimeter of triangle ABC.$P{_\Delta ABC}$=  CM

  A toy house has a roof in the shape of a right u gkr)x/;ftt2-angled isosceles triangle, ABC.AB = grk u)tft;2 /x6 cm and angle $\textrm{A}\hat{\textrm{C}}\textrm{B}=90^{\circ}$.


1.Find the length of AC.  cm
2.Calculate the area of triangle ABC.  $cm^2$
The garage of the toy house also has a roof in the shape of right-angled triangle, LMN. , angle $\textrm{L}\hat{\textrm{M}}\textrm{N}=56^{\circ}$ and angle $\textrm{N}\hat{\textrm{L}}\textrm{M}=90^{\circ}$.
3.Calculate the length of MN.  cm

  From a sailboat moored at sea, the angle of .qc+eah jobb::b,yuiv tp1- /elevation to the top of a 20 m high cliff 1.:v ube +b/ qb:yptco jiha-,is $22^{\circ}$. On the cliff top there is a coconut tree. The angle of elevation to the coconut tree is $29^{\circ}$.
1.Draw a diagram representing the situation.
2.Find the height of the coconut tree.  m

  The quadrilateral ABCD shown below represents a spyv31 u : jdrny)ut7f:dorts complex, where AB = 210 m, BC = 90 m and CD = 130 m. Af3td ru1d 7j)yuynv::ngle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $70^{\circ}$ and angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A} is $105^{\circ}$.


A straight path through the sports complex joins points A and C.
1.Find the length of the path AC.  m
2.Show that the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$ is $39.4^{\circ}$ , correct to three significant figures.  $^{\circ}$
3.Find the area of the sports complex.  $m^2$

A new path, DE, is to be built such that E is the point on AC closest to D.

4.Find the length of the path DE.  m

The section of the sports complex represented by triangle CDE will be an outdoor running track. A track will be marked along the sides of this section.

5.Calculate the total length of the track.  m

  The quadrilateral ABCD is shown below, where AB = 56mrm8 :2 br2s3i7xfj.ye lqq4et, AD =122 m, CD =40 m, and the angl7xb3yiq.8e 2jr :lseqtmf24res $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ and $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ are $68^{\circ}$ and $80^{\circ}$ respectively.


1.Find the length of line AC.  m
2.Find the area of triangle ACD.  $m^2$
3.Find the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$. Give your answer in degrees correct to one
decimal place.  $^{\circ}$
4.Find the shortest distance between point B and the line AC.  m
5.Find the length of the perimeter of triangle ABC.  m

  A flagpole stands at the top of a building. The angle of elevation from a p9uf c6kl6l c:ioint on the ground 40m from the base of the buildil9f6u l:k 6cicng to the bottom of the flagpole is $29^{\circ}$, and to the top of the flagpole is $31^{\circ}$.
1.Draw a diagram representing the situation.
2.Find the length of the flagpole.  m

  Consider the triangle below, which hae: r kcjs(u2 5 -bw(4fbc1ilncs side lengths AB = 25 cm and BC = 35 cm. The angle adjacent t(lj5 cncci4bs 1fk:euw2(b- ro these side lengths, AB̂C, is $70^{\circ}$.

1.Find the area of the triangle, rounding your answer to the nearest $cm^2$.  $cm^2$
When drawing the triangle, George measured each side length correct the nearest centimetre, and angle AB̂C correct to the nearest degree.
2.Calculate the maximum possible area of the triangle, rounding your answer to the nearest $cm^2$.  $cm^2$

  The angle of depression from a lookout at the top of a lighta nj mglpm67 9.xv k0jafnga4c) -r-n*house (A) down to a boat locan0-a*c7pan)9vl4. -gj fmrnja 6 k xmgted at point C is $30^{\circ}$.
The boat travels towards the lighthouse and after 1 minute has travelled a distance of 50 m and is now located at point B. The angle of elevation from the boat at B up to the lighthouse lookout is $60^{\circ}$.


1.Write down the value of $x^{\circ}$ in the diagram.  $^{\circ}$
2.Find the height of the lighthouse.  m
3.Find the speed of the boat, in metres per second, from C to B.  $ms^{-1}$

  Melody is camping with her friends in a forest. After setting up the tent, hb :mkhg7u b0/( 9pek 2yhigrc1bcu92 she goes for ahkc9cm 9hgbge: uu2 72h(0y rpib1bk/ walk to see the nearby surroundings. She walks 200 m on a bearing of $280^{\circ}$, changes direction, then walks another 150 m on a bearing of $035^{\circ}$.
1.Find the distance Melody is from the tent at this point.  m
2.Determine the bearing Melody should walk on to return to the tent.  $^{\circ}$

  On a trip to Morocco, a groupv* tyw :oylntv(i-aguhir*j m 4,)5(w of tourists experience a horse trek through the Saharan desert.(,tavwy*r5ygh n v wt*ju-4mi :o(li) They start from a horse farm, moving on a bearing of $075^{\circ}$ for 3.2km to reach an oasis. After taking a rest there, they take another ride on a bearing of $210^{\circ}$ for 2.1km to visit the Draa valley.
1.Find the distance that they need to cover to get back to the farm.  km
2.Determine the bearing they should ride on to get back to the farm.  $^{\circ}$

  Consider a triangle with sidesAB=10 mm and AC=8 mm. Given that the2*g8 vbbidhev94+gkk area of the trianglhvkgv+b g4k9dbe2*8ie is $24 mm^2$, find the possible values for the length of side BC.BC$\approx$  mm and  mm

  Daniela (D) stands near the window on the second flc)2hnb;h s4f9f c4; fltjs +uhoor of the six-story apartment building and can see Andrea (A), who is 14 m from the base of the building. The angle of depress bhhln9s;f) c442jch t;+ffs uion from Daniela to Andrea is $20^{\circ}$.


1.Calculate the distance from Daniela to Andrea, AD.  m
Clara (C) is standing at the window of the sixth floor, three times higher from the ground than Daniela.
2.Calculate the angle of depression from C to A.  $^{\circ}$

  A triangular farm is shown inx;p7.w9fw)c ) 8 p4x/gv sd)mr 3ucqq.nucgte the diagram below. A river runs along the edge from A to B and the farm house is located at C, a safe distance from the river. The distancedf.uenpcup8qxw c 7q4.))3sxw; /9 gmcgt v)r from the farm house to point A is 270 m and to point B is 360 m. The angle $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}$ is $130^{\circ}$.



1.Calculate the distance along the river from A to B.  m

The cost of fencing is 130130 Australian dollars (AUD) per metre.

2.alculate the total cost of fencing the whole perimeter of the farm, providing your answer to the nearest dollar.  
3.Find the sizes of angels $\mathrm{C}\hat{\mathrm{A}}\mathrm{B}$ and $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$
4.Calculate the area of the farm. Give your answer correct to the nearest whole square metre.  $m^2$
5.Calculate the shortest distance from the farm house to the river.  m

A tower is located at point B. The angle of elevation to the top of the tower, H, from point C is measured to be $1^{\circ}$.

6.Calculate the vertical height, BH, of the tower, correct to the nearest centimetre.  m
7.Calculate the distance between the top of the tower, H, to point A.  m

  In the shaded region shf. 0ry.9xdgrj own below, the radius of the smaller arc is 66 cm and the distance between the two 0rxyf 9.r d.gjarcs is 44 cm.

1.Find the perimeter of the shaded region.  cm
2.Find the area of the shaded region.  $cm^2$

  Consider the shape shown below (shaded region). The radius of t0-a6nkfr1(y) xruwfc he longer arc is 12 cm and theu x)c 61rywfk0arn (f- distance between the arcs is 3 cm.


1.Find the perimeter of the shape.  cm
2.Find the area of the shape.  $cm^2$

  ABC is a triangle of arev0d0x0h(3u : xjwnu*w;uwt .dv gqvm 8a $32 cm^2$ . The sides AB and BC have lengths 7 cm and 12 cm respectively. Find the two possible lengths of the side AC, giving your answers correct to 3 significant figures.AC$\approx$  cm and   cm

  Paper party cups in the shape of a cone are formed from a sector of papem( :cmvrq 8w2nr material. The diagram below left shows the finished paper cup. The diagram below right shows the original: (w crmq2mn8v paper sector used to create the cup. When the cones are formed, the straight edges of the sector join without any overlap.


1.Find the slant height of the cups, L.  cm
2.Find the arc length of the sector.  cm
3.Find the angle $\theta$ in radians.  radians

  BT represents the height of a sk3foj e8g)k )qzyscraper, with its base at B and top at T. Jack has an apartment onfzj)eg) 8oq3k the 32nd floor at C, three fifths of the way to the top of the building.


From a point A, on horizontal ground 200 m from B, the angle of elevation to C is $38^{\circ}$.
1.Calculate BC, the height of Jack's apartment above the ground.  m
2.Calculate the angle of depression from T to A.  $^{\circ}$

  A lawn sprinkler is se;xya1v ataw/* gszl3k v+n h ;d-n(3mttup in the corner of a rectangular lawn which is enclosed by a fence. The sprinklera(3v*/txl zw+ayvnash;; 3t dgk -nm1 has a maximum spray distance of 5 m. The lawn has side lengths 4 m and 10 m. This situation is illustrated in the diagram below, where the sprinkler is positioned in the top right corner.


1.Find the angle $\alpha$ in the diagram, leaving your answer in degrees.$\alpha$=  $^{\circ}$
2.Find the area of grass that is currently being watered by the sprinkler
in this position.  $m^2$

  A motion sensing security light is inz2- 1ccy0 hogdkcp w9+stalled in the corner of a front garden. The sensor can detect any movement up to 7 meters. The garden has boundaries of two 12 m long parallel walls, and the house facade and a wooden fence which are 66 m long. This creates to a rectangular area as shown ink-o0 dc2hz yg+1pcwc9 the follow diagram, where the motion light is positioned in the top right corner.


1.Find the angle \alphaα in the diagram, leaving your answer in degrees.  $^{\circ}$
2.Find the area of the front garden that is currently being monitored by the motion sensing system.  $m^2$

  Bus-1 and Bus-2 leave a terminal, T, at the.fxhmd 89rry( sdvz5- same time. Bus-1 travels 65 km on a true- .dyz5x8 r(9vhsd rmf bearing of $140^{\circ}$ to point U, then a further 40 km on a true bearing $235^{\circ}$ to point V. Bus-2 travels directly from the terminal to point V.


1.Find the distance, d, travelled by Bus-2.  km
2.Find the true bearing on which Bus-2 travels.  $^{\circ}$

Bus-1 averages a speed of 55 km/h and Bus-2 averages a speed of 36 km/h

3.Determine which bus arrives at point V first. Justify your answer. ----待选择----Bus-1Bus-2 

  A camping tent in the shape of a cone is constructed from a large circ2w92 2byib) pxr rf mwq,.b3wyular sheet x., b2qmw)fyi9 r3rw22ybpbw of polyester with a sector cut out, as illustrated in the diagram below. The circular polyester sheet has a centre O and radius rr. The area of sector cut-out, OABC, is $\piπ m^2$ and the length of the arc ABC is $\dfrac {2\pi} 3m$.

1.Determine the slant height of the tent, r.  m
2.Determine the angle of the sector OABC,$\theta$, leaving your answer in radians with exact values.

  The side lengths of a qu w.hs8roa2 -wjadrilateral ABCD are AB =6 mm, BC =5 mm, CD =8 mm, and DA = 9 mm. The angle 2-howrw.aj8 s$\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is equal to $90^{\circ}$.


1.Calculate the length of AC.  mm
2.Calculate the size of angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$.  $^{\circ}$
3.Calculate the area of triangle ACD.  $mm^2$
4.Calculate the area of quadrilateral ABCD.  $mm^2$

A point P is chosen on the side CD in a such a way that the area of triangle ACP is equal to the area of the triangle ABC.

5.Calculate the length of CP.  mm
6.Calculate the length of AP.  mm

  A picnic area is shaped like a quadrilateral, ABCD, whero4e8s k08cz uzt38qn ue AB = 10 m andCD = 17.5 m. Three of the internal angles have been measured; angle zus ktu38e 8onqcz80 4$\mathrm{A}\hat{\mathrm{D}}\mathrm{C}=51^{\circ}$, angle $\mathrm{C}\hat{\mathrm{A}}\mathrm{D}=70^{\circ}$ and angle $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}= 42^{\circ}$. This information is represented in the following diagram.



1.Calculate the distance AC.  m
2.Calculate the angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$

There is a lamp post at A, perpendicular to the ground. The angle of elevation to the top of the lamp post from B is $32^{\circ}$.

3.Calculate the height of the lamp post.  m

Patrick's family is having a barbecue party on the picnic area. The family brings a 1.5 litre bottle of lemonade and cone-shaped paper cups.
Each cup has a vertical height of 88 cm and the top of the cup has a diameter of 5 cm.

4.Calculate the volume of one paper cup.  $cm^3$
5.Calculate the maximum number of cups that can be completely filled with the 1.5 litre bottle of lemonade.  completely filled cups

  The diagram below shows a triangular backyard ABC. A dog is on a leash attachw2sfl2 b/q iu0wgm p.ucr) 6v6ed to a pole at point A. The lewmf uqv66rwsu/2l.2 b)ic pg0ngth of the leash is 10 metres.

1.Find the area of the backyard that the dog cannot reach.  $m^2$

The owner of the dog estimates that the area the dog cannot reach is $50 m^2$.

2.Calculate the percentage error in the dog owner's estimate.  %

  Two drones are being flown in a park. Drone A starts at poie/s,p* jm cd, v.cugt.nt P and travels towards point Q in a straid sp v ,/m.cj*c.,eugtght line on a true bearing of $115^{\circ}$ at a speed of $2.5ms^{-1}$. The distance from P to Q is 800m.
Drone B starts at point Q and travels towards point R in a straight line on a true bearing of $215^{\circ}$ at a speed of $7.5ms^{-1}$ .
These two paths are shown on the diagram below.



1.Find the acute angle $\mathrm{P}\hat{\mathrm{Q}}\mathrm{R}$.  $^{\circ}$
2.Find the distance between Drone A and Drone B after 4 minutes, rounding your answer to the nearest metre.  m

  Consider the analogue clock below, which has a citk )j+b*7 c(btx+seclb*7 fzb rcular face with centre at point es+ *t)xjc7zt cbbf(lb+ b*k7O. The minute hand of the clock, OP, has a length of 12 cm and the hour hand, OQ, has a length of 10 cm.


In Figure:1 the time on the clock is 7:00,pm.
1.For Figure: 1, find
1.the size of the reflex angle,$\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}$. , in degrees.  $^{\circ}$
2.the distance between the ends of the hour and minute hands.  cm

In Figure: 2, the time is now 7:18pm, and as such, the end point of the minute hand has rotated through an angle,$\theta$, covering an arc length of $l$.

2.For figure: 2, find
1.the size of the angle $\theta$ in degrees.  $^{\circ}$
2.the arc length $l$.  cm
3.the area of the shaded sector.  $cm^2$

Another circular analogue clock, shown below, has a face radius of 14 cm, and minute and hour hands of length 14 cm. The current time shown on the clock is 6:00\,pm. The bottom of the clock face is 4cm above the base of the frame holding the clock.


The height, hh, of the end point of the minute hand above the base of the frame is modelled by the function
$h(\theta)=14\cos \theta + 18h$
where $\theta$ is the angle rotated by the minute hand after 6:00pm.

3.Find the value of h when $\theta =170^{\circ}$.  cm

The height, gg, of the end point of the hour hand above the base of the frame is modelled by the function

$g(\theta)=-14 \cos (\frac{\theta}{12}) + 18$

where $\theta$ is the angle rotated by the minute hand after 6:00pm.
The end points of the minute and hour hands have the same height from the base of the frame for the first time when $\theta=k$.

4.Find the value of k.  $^{\circ}$

  An airplane leaves Doha air kk7 nz5me2a;)qqv n1xport bound for Paris Charles de Gaulle airport. There are lights located at the end of the runway at points A(0, 4) and B(6, 8), relative to a terminal at the or 2ve7xmqkq) z5nkan ;1igin. The takeoff path of the airplane is the perpendicular bisector of line AB.

1.Find the equation of the takeoff path in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$.

The airplane travels at an average speed of $570km\:hr^{-1}$ in a straight line. Once the airplane has reached cruising altitutde, an air traffic controller at the top of a 200m high air traffic control tower at C(7,0) observes that the angle of elevation to the airplane is $40^{\circ}$. Five minutes later, the controller observes that the angle of elevation is $10^{\circ}$.

2.Find the cruising altitude of the airplane in metres.  m

As the airplane is about to land at the Paris Charles de Gaulle airport, the pilot is asked to delay the landing due to a traffic issue. The pilot is instructed to turn the airplane on a bearing of $045^{\circ}$ for 10km until reaching point P, then travel on a bearing of $165^{\circ}$ for 30km to point Q before flying back to the original point O for landing.

3.Find the angle OP̂Q.  $^{\circ}$
4.Find the shortest distance from Q back to O for landing.  km
5.Find the bearing the airplane must travel on to get back to O from Q.  $^{\circ}$

  The temperature in New York ran3 fvtr:vvsx u98 mjt1mfv83 k2ges from $-4^{\circ}C$ to $22^{\circ}C$ throughout the course of a year. The temperature, T, can be modelled by the function
$T(t) = -a\cos(bt)+d$,
Where t is time measured in months from the beginning of the year, and a, b and $d \in \mathbb{Z}^{+}$.

1.Show that
1.a=13;
2.b=30;
3.d=9.
2.Sketch the function T(t) for $0 \leq t \leq 12$, clearly labelling the coordinates of the maximum and minimum points.
3.Find the temperature when t=4.  $^{\circ}C$
4. 1.Determine the number of months in a year that the temperature is above $9^{\circ}C$.
2.Find the probability that on a randomly chosen day of the year, the temperature is above $9^{\circ}C$.
5.If the temperature in New York ranged from $-7^{\circ}C$ to $25^{\circ}C$ instead of $-4^{\circ}C$ to $22^{\circ}C$, describe how this would affect the probability found in (d) part (ii).

  A large plot of land is shaped like a tra :outayrsgn sk 8o )s9g+t5g6we wt2+*pezoid, ABCD, where AB is parallel to CD, AB = 12 km and 5+6k8tg*u)2gy9 sas tw et:o gwnr+osCD =8 km. The diagonal BD of the trapezoid is equal to 10 km. The internal angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{D}$ is equal to $58^{\circ}$.

Let a new point, H, be a point on AB closest to D.

1.Calculate the distance from H to D.  km
2.Calculate the length of AD.  km
3.Calculate the size of the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$.  $^{\circ}$

A landowner estimates that the length of AD is equal to 11.5 km.

4.Calculate the percentage error in the landowner's estimate.  %

The landowner decides to install cone-shaped concrete bollards along the perimeter of the plot of land. The bollards are to be installed at a distance of 100 m from each other. The radius of the base of each bollard is 20 cm and the height of each bollard is 40 cm.

5.Calculate the volume of one of the bollards.  $cm^3$
6.Calculate the total volume of concrete needed to install all the bollards.  $cm^3$

  The quadrilateral ABCD represents a recreational park, where AB =250 mxa1* gwd,bd9ux i*3cnym +x-p y,l zc8, BC =110 m andimu8d lpc9*,yb,3x- *x +xd1a ycgzw n CD =130 m. Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $80^{\circ}$ and angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A}$ is $115^{\circ}$. This information is shown in the following diagram.


A straight path through the park joins the points A and C.

1.Find the length of the path AC.  m
2.Show that the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$ is $27.5^{\circ}$, correct to three significant figures.  $^{\circ}$
3.Find the total area of the recreational park.  $m^2$

A new path, DE, is to be built such that E is the point on AC closest to D.

4.Find the length of the path DE.  m

The section of the park represented by triangle CDE will be used for a school cross country carnival. A track will be marked along the sides of this section.

5.Calculate the total length of the track CDE.  m

  The diagram below shows a triangularhwu 92:tspno 2 park ABC. Alice is in the park with her dog. She stops at the corner B of the park to make a phone call. The dog's leash i :nwthosup292s 12 metres long.


1.Find the area of the park that the dog cannot reach when attached to its leash.  $m^2$
2.Determine the percentage of total park area that the dog can reach while attached to its leash  %

  On the diagram below, points X anio- sc,e v4v ry.-xz7zd Y are the centres of two circles containing the two arcs from A e,zros4 y-xcv-i 7z v.to B.

1.Find the length from point A to point Y.  mm
2.Find the perimeter of the shaded region.  mm
3.Find the area of the shaded region.  $mm^2$