P1: FXS/ABE P2: FXS CHAPTER 13 - Cambridge … · P1: FXS/ABE P2: FXS ... 13.1 Multiple-choice questions 1 A ladder 2.6 m long rests with one end on horizontal ground while the other

P1: FXS/ABE P2: FXS

9780521740494c13.xml CUAU033-EVANS September 9, 2008 11:9

C H A P T E R

Revision13

Revision of chapters8–12

13.1 Multiple-choice questions1 A ladder 2.6 m long rests with one end on horizontal ground

while the other end rests against a vertical wall at a point which

is 2.1 m from the ground. The angle between the ladder and the

wall, to the nearest degree, is

A 36◦ B 39◦ C 51◦ D 54◦ E 63◦

2.6 m2.1 m

2 The graph shown has amplitude

A 2 B 3 C 4

D 6 E 2� x

y

2

–4

0 2π3 Which one of the following equations gives the

correct value for c?

A c = 58 cos 38◦

cos 130◦ B c = 58 sin 38◦

sin 130◦

C c = 58 sin 38◦ D c = 58 cos 130◦

cos 38◦

E c = 58 sin 130◦

sin 38◦58 cm

12° 38°130°

B

A

C

4 A map is drawn so that a wall 17.1 m long is represented by a line 45 mm long. The scale

is

A 1 : 3.8 B 1 : 38 C 1 : 380 D 1 : 3800 E 1 : 38000

5 The point (5, −2) is reflected in the line y = x . The coordinates of its image are

A (5, −2) B (−5, 2) C (2, −5) D (−2, 5) E (−5, −2)

6 If sin A = 5

13, sin B = 8

17where A and B are acute, then sin (A − B) is given by

A140

221B

−21

221C

34 209

23 560D

−107

140E

107

140

363

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Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

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n364 Essential Advanced General Mathematics

7 In triangle ABC, c = 5, b = 9 and A = 43◦. Which of the following statements are

correct?

i With the information we can find the area of triangle ABC.

ii With the information given we can find angle B.

iii With the information given we can find side a.

A i and ii only B i and iii only C ii and iii only

D i, ii and iii E none of these

8 In the figure, AB = 15, CD = 5, BF = 6, GD = 6,

EG = 9. x is equal to

A 3 B 4 C 4.5

D 4.75 E 5

156

6

9

D

A

B

CE

F

G 5x

9 The point (2, −6) is reflected in the line y = −x . The coordinates of its image are

A (2, −6) B (−2, 6) C (6, −2) D (−6, 2) E (−2, −6)

10 The graph shown is best described by

A y = sin (a) B y = 2 cos (a)

C y = sin (a) + 1 D y = cos (2a)

E y = cos (a) + 1

y

2

1

0 ππ2

2π3π2

a

11 If sin A = 5

13, sin B = 8

17where A and B are acute, then tan (A + B) is given by

A140

221B

−21

221C

34 209

23 560D

−171

140E

171

140

12 P is the point (5, −4). After translation by

[2

−3

]and reflection in the line y = 1, the

coordinates of the image of P are

A (7, 7) B (7, 9) C (−5, −7) D (7, 11) E (7, 10)

13 A model car is 8 cm long and the real car is 3.2 m long. The scale factor is

A 1 : 8 B 1 : 32 C 1 : 24 D 1 : 400 E 1 : 40

14 If 2 sin(

x − �

6

)= √

3 and 0 ≤ x ≤ 2�, then x is equal to

A�

3or

2�

3B

�

6or

5�

6C

�

6or

�

2D

�

2or

5�

6E

�

3or �

15 Which one of the following expressions will give the area of triangle ABC?

A1

2× 6 × 7 sin 48◦ B

1

2× 6 × 7 cos 48◦

C1

2× 6 × 7 sin 52◦ D

1

2× 6 × 7 cos 52◦

E1

2× 6 × 7 tan 48◦

A B

C

6 cm

7 cm48° 52°

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nChapter 13 — Revision of chapters 8–12 365

16 Given that cos � = c and that � is acute, cot � can be expressed in terms of c as

A c√

1 − c2 B√

1 − c2 C1√

1 − c2D

c√1 − c2

E 2c√

1 − c2

17 A trigonometric graph has the following characteristics:� period of 120◦

� amplitude is 3

� range is [−4, 2] � y = −1 when a = 0

This graph would be described by the equation

A y = 3 sin (a)◦ + 1 B y = 3 cos (3a)◦ − 1 C y = −3 sin (3a)◦ − 1

D y = 3 sin (3a)◦ + 1 E y = cos (3a)◦ − 2

18 The point (a, b) is reflected in the line with equation x = m. The image point has

coordinates

A (2m − a, b) B (a, 2m − b) C (a − m, b)

D (a, b − m) E (2m + a, b)

19 A child on a swing travels through an arc of length 3 m. If the ropes of the swing are 4 m

in length, the angle which the arc makes at the top of the swing (where the swing is

attached to the support) is best approximated by

A 135◦ B 75◦ C 12◦ D 75c E 42◦58′

20 Compared with the graph of y = sin �, the graph of y = sin

(1

2�

)has

A the same amplitude but double the period

B the same amplitude but half the period

C double the amplitude but the same period

D half the amplitude but the same period

E the same amplitude but shifted1

2a unit to the left.

21 The image of the line {(x, y) : x + y = 4} after a dilation of factor1

2from the y axis

followed by a reflection in x = 4 is

A {(x, y) : y = 2x} B {(x, y) : y + 2 = 0} C {(x, y) : y + 2x − 16 = 0}D {(x, y) : x + y = 0} E {(x, y) : y = 2x − 12}

22 If A + B = �

2, the value of cos A cos B − sin A sin B is

A −2 B 1 C −1 D 0 E 2

23 Given that sin A =√

5

3and that A is obtuse, the value of sin 2A is

A16

√5

243B −1

9C −8

√5

27D

5

9E −4

√5

9

24 A ladder rests against a wall, touching the wall at a height of 5.6 m. The bottom of the

ladder is 2 m from the wall. The distance (to the nearest centimetre) that a person, of

height 1.6 m, must be from the wall to just fit under the ladder is

A 1.43 m B 0.57 m C 1.75 m D 0.25 m E 1.2 m

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25 A possible equation of the graph shown is

A y = sin 2(

x − �

12

)B y = cos 2

(x − �

12

)C y = sin 2

(x + �

12

)D y = cos 2

(x + �

12

)E y = − sin 2

(x − �

12

) x

y

0

1

π2

π4

π12

3π4

26 Let �ABC and �DEF be similar triangles such that AB = 4 cm and DE = 10 cm. If the

area of �ABC is 24 cm2, then the area of �DEF, in cm2, is

A 60 B 240 C 150 D 96 E none of these

27 Which of the following statements is true for f (x) = −2 tan (3x)◦?

i The period is 60. ii The amplitude is 2. iii The period is 120.

iv The graph is a reflection of the graph of h(x) = 2 tan (3x)◦ in the x axis.

v The graph is a reflection of g(x) = tan (x)◦ in the y axis.

A i and iv only B i, ii and iv only C i, iv and v only

D ii and iv only E iii and iv only

28 The image of {(x, y) : y = x2} under a translation determined by the vector

[3

2

]followed

by a reflection in the x axis is

A {(x, y) : y = (x − 3)2 + 2} B {(x, y) : −(x − 3)2 = y + 2}C {(x, y) : y = (x + 3)2 + 2} D {(x, y) : −y + 2 = (x − 3)2} E none of these

29 The area, in cm2 correct to two decimal places, of a sector with included angle of 60◦ in a

circle of diameter 10 cm is

A 104.72 cm2 B 52.36 cm2 C 13.09 cm2 D 26.16 cm2 E 750 cm

30 KLMN is a parallelogram and OQ is parallel to KL.

If O divides KN in the ratio of 1 : 2,

the ratioarea�KOP

areaKLMNis equal to

A1

4B

1

9C

1

12D

1

18E

1

20

K

N

L

M

PO Q

31 VABCD is a right, square pyramid with base length 80 mm and

perpendicular height 100 mm. The angle between a sloping

face and the base ABCD, to the nearest degree, is

A 22◦ B 29◦ C 51◦

D 61◦ E 68◦

A B

CD

V

Eθ

O

32 Given that cos � = c and that � is acute, sin 2� can be expressed in terms of c as

A c√

1 − c2 B√

1 − c2 C1√

1 − c2D

c√1 − c2

E 2c√

1 − c2

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33 The image of {(x, y) : y = 2x} after a dilation of factor 2 from the x axis followed by a

dilation of factor1

3from the y axis is

A y = 1

3× 23x B y = 3 × 2

x2 C y = 2 × 23x

D y = 2 × 2x3 E none of these

34 The angles between 0◦ and 360◦ which satisfy the equation 4 cos x − 3 sin x = 1, given

correct to two decimal places, are

A 53.13◦ and 126.87◦ B 48.41◦ and 205.33◦ C 41.59◦ and 244.67◦

D 131.59◦ and 334.67◦ E 154.67◦ and 311.59◦

35 In the figure, the volume of the shaded solid B is 49 cm3.

The volume of A is

A 19.5 cm3 B 17.3 cm3 C 13.5 cm3

D 12.5 cm3 E 10.5 cm3A

B

3 cm

2 cm

36 The area of the shaded region in the diagram, in cm2

(to the nearest cm2), is

A 951 B 992 C 1944

D 2895 E 110 424

110° 45 cm

37 The expression 8 sin � cos3 � − 8 sin3 � cos � is equal to

A 8 sin � cos � B sin 8� C 2 sin 4� D 4 cos 2� E 2 sin 2� cos 2�

38 A possible equation for the graph shown is

A y = tan

(1

2x − �

4

)+ 3

B y = tan

(1

2x + �

4

)− 3

C y = 3 tan

(1

2x − �

4

)

D y = 3 tan

(1

2x + �

4

)E y = tan 3x

x

y

–3

3

–π2

2ππ 3π

2

39 If the ratio volume of the hemisphere : volume of the right circular cone equals 27 : 4

where r is the radius of the base of the cone and R is the radius of the hemisphere,

then R : r is equal to

R

r

r

A 1 : 2 B 2 : 3 C 3 :√

2 D 27 : 8 E 3 : 2

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40 Let T be the translation determined by the vector

[2

3

]and S the transformation, reflection

in the line with equation x = 2. The rule for the composition TS is given by

A TS(x, y) = (2 − x, y + 3) B TS(x, y) = (−x, y + 3)

C TS(x, y) = (x + 2, y + 3) D TS(x, y) = (6 − x, y + 3) E none of these

41 The square shown is subject to successive transformations.

The first transformation has matrix

[−1 0

0 1

]and the second

transformation has matrix

[0 −1

−2 1

].

x

y

1

1

0

(1, 1)

Which one of the following shows the image of the square after these two

transformations?

A

x

y

1 3

–1

20

B

1–1

3

2

1

2

0x

y C

2–1 1

2

0

1

3

x

y

D

x

y

1–1

–2

–1

1

0

E

x

y

2

1

3

–1

1

0

13.2 Extended-response questions1 a Find the rule of the transformation which

maps triangle ABC to triangle A′ B ′C ′.b On graph paper, draw triangle ABC and its

image under reflection in the x axis. The

coordinates of A, B and C are (−4, 1), (−2, 1)

and (−2, 5) respectively.

c On the same set of axes draw the image of

�ABC under a dilation of factor 2 from

the y axis.

d Find the image of the parabola y = x2

under a dilation of factor 2 from the x axis

followed by a translation defined by the vector

[−3

2

].

y

2 40

1

8

45

–4 –2

A B

C

C'

A' B'

x

(cont’d.)

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e Find the rule for the transformation which maps the graph of

y = x2 to y = −2(x − 3)2 + 4

f If f (x) = x3 − 2x use a graphics calculator to help sketch the graph of

y = 3 f (x − 2) + 4

2 In �ABC, A = 30◦, a = 60 (i.e., for the diagram

BC = BC ′ = 60) and c = 80.

a Find the magnitudes of angles

i ABC and BCA ii ABC′ and BC′Ab Find the length of line segment

i AC ii AC ′ iii CC ′A

CC'

B

80

30°

6060

c i Show that the magnitude of ∠CBC′ is 96.38◦ (correct to two decimal places).

Using this value,

ii find the area of triangle BCC′ iii the area of the shaded sector

iv the area of the shaded segment.

3 a Find the image of the point (1, 1) under a dilation D, of factor 4 from the y axis.

b i Describe the image of the square with vertices A(0, 0), B(0, 1), C(1, 1), E(1, 0)

under the dilation D.

ii Find the area of the square ABCE.

iii Find the area of the region defined by the image of ABCE.

iv If the dilation had been of factor k, what would the area of this region be?

c State the rule for the dilation.

d i Find the equation of the image of the curve with equation y = x2 under the

dilation D.

ii Find the equation of the image of the curve with equation y = x2 under the

dilation D followed by the translation defined by the vector

[2

−1

].

iii Sketch the graph of y = x2 and of its image defined in ii on the one set of axes.

State the coordinates of the vertex and of the axes intercepts.

e State the rule for the transformation which maps the curve with equation

y = 5(x + 2)2 − 3 to the curve with equation y = x2.

4 A transformation is represented by the matrix M =

3

5

4

5

−4

5

3

5

a Describe the transformation.

b Let C be the circle which passes through the origin and which has as its centre the

point (0, 1).

i Find the equation of C.

ii Find the equation of C′, the image of C under the transformation determined

by M.

c Find the coordinates of the points of intersection of C and C′.

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5 A transformation is defined by the matrix M =[

4 1

2 3

].

a Find the image of (−2, 5) under this transformation. b Find the inverse of M.

c Given that the point (11, 13) is the image of the point (a, b), find the values of

a and b.

d Find in terms of a, the image of the point (a, a).

e If M

[a

b

]=

[�a

�b

], find the possible values of � and the relationship between

a and b in each of these cases.

6 Let R �4

be the matrix of the transformation, rotation of�

4in an anticlockwise direction.

a Give the 2 × 2 matrix associated with this transformation.

b Find the inverse of this matrix.

c If the image of (a, b) is (1, 1), find the value of a and b.

d If the image of (c, d) is (1, 2), find the value of c and d.

e i If (x, y) → (x ′, y′) under this transformation, use the result of b to find x and y in

terms of x ′ and y′.ii Find the image of y = x2 under this transformation.

7 A particle oscillates along a straight line. Its displacement x (m), at time t(s), from a point

O is given by x = 5 + 3 sin(�

6t)

.

a Find its displacement at time

i t = 0 ii t = 3

b Sketch the graph of x against t for t ∈ [0, 24], labelling clearly all turning points.

c State

i the maximum distance of the particle from O

ii the minimum distance of the particle from O.

d At what times (t ∈ [0, 24]) is the particle

i 5 m from O ii 6 m from O, correct to two decimal places?

8 A logo for a Victorian team is as shown here. O is the

centre of the circle and A, B and C are points on the

circle. OC = OA = OB = 10 cm.

a i Convert 30◦ to radians.

ii Find the length of the minor arc AB.

b The magnitude of ∠AOC is 167◦ and the

magnitude of ∠BOC is 163◦.

Find the length of chord BC, correct to two

decimal places.

167° 163°

30°

A

O

C

B

c Find, correct to two decimal places,

i the area of triangle BOC ii the area of triangle AOC

iii the shaded area of the logo.

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9 Triangle LMN is an isosceles, right-angled triangle.

P, Q and R are midpoints of LM, MN and LN respectively.

a Prove �PRQ ∼ �LMN. b State the scale factor.

c Find the area of triangle PQR in terms of a.

X, Y and Z are the midpoints of PR, PQ and RQ

respectively.

�XYZ is similar to �LMN.

a units

a units

R

L

XP

Y Z

NQM

d State the scale factor.

e Find the area of triangle XYZ in terms of a.

f Let A1 be the area of triangle LMN.

Let A2 be the area of triangle PQR.

Let A3 be the area of triangle XYZ.

The process of forming triangles by joining midpoints of the previous triangle is

continued to form a sequence of triangles, �1, �2, �3, . . . , �n, . . . and associated

areas A1, A2, A3, . . . , An, . . .

i Find An in terms of a and n.

ii Find in terms of a, the sum to infinity of the series A1 + A2 + · · · + An + · · ·

10 It is known that y varies partly as x and partly as1

x2; i.e. there exist constants k1 and k2

such that y = k1x + k2

x2.

a When x = −1, y = 1 and when x = 1, y = 5.

Find the values of k1 and k2.

b The graph of y against x is as shown.

i Sketch the graph of the image of

y = k1x + k2

x2under the transformation

determined by reflection in the x axis

followed by a translation determined

by the vector

[3

0

].

(The answers to parts ii, iii and iv

below may help you answer this.)

ii Find the value of c and hence the

x axis intercept of the image.

iii The image of the point with coordinates

(e, f ) is (e′, f ′). Find e′ and f ′ in terms

of e and f.

x

y

y = k1x +x2

k2

y = k1x

(c, 0)

(e, f )

0

iv Find the equation of the image of the curve with equation y = k1x + k2

x2under

this transformation.

11 Let M be the transformation ‘reflection in the line y = x ′.

a i Find the coordinates of the image of the point A (1, 3) under this transformation.

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ii Find the coordinates of the triangle which is the image of the triangle with vertices

A (1, 3), B (1, 5), C (3, 3).

iii Illustrate triangle ABC and its image on a set of axes scaled from −5 to 5 on both

axes.

b i Show that the equation of the image of the curve with equation y = x2 − 2 under

the transformation M is x = y2 − 2.

ii Find the coordinates of the points of intersection of the curve y = x2 − 2 with the

line y = x .

iii Show that the x coordinates of the points of intersection of y = x2 − 2 and its

image may be determined by the equation x4 − 4x2 − x + 2 = 0.

iv Two solutions of the equation x4 − 4x2 − x + 2 = 0 are x = 1

2(−1 +

√5) and

x = 1

2(−1 −

√5).

Use this result and the result of b ii to find the coordinates of the points of

intersection of y = x2 − 2 and its image under M.

12 In the figure, AE = BE = BD = 1 unit.

∠BCD is a right angle.

a Show that the magnitude of ∠BDE is 2�.

b Use the cosine rule in triangle BDE to show that

DE = 2 cos 2�.A

B

D

E

C

1 11

3θθ

c Show that

i DC = sin 3�

ii AD = sin 3�

sin �d Use the results of b and c to show sin 3� = 3 sin � − 4 sin3 �

13 a Adam notices a distinctive tree while orienteering on a flat horizontal plane. He

discovers that the tree is 200 m from where he is standing on a bearing of 050◦. Two

other people, Brian and Colin, who are both standing due east of Adam, claim the tree

is 150 m away from them. Given that their claim is true and that Brian and Colin are

not standing in the same place, how far apart are they? Give your answer to the nearest

metre.

b From the top of a vertical tower of height 10 m,

standing in the corner of a rectangular courtyard,

the angles of depression to the nearest corners

(B and D) are 32◦ and 19◦ respectively.

Find

i AB, correct to two decimal places

T

A

B C

D

ii AD, correct to two decimal places

iii the angle of depression of corner C diagonally opposite the tower from T, correct

to the nearest degree.

c Two circles, each of radius length 10 cm, have their centres 16 cm apart. Calculate the

area common to both circles, correct to one decimal place.

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14 A satellite travelling in a circular orbit 1600 km

above the Earth is due to pass directly over a

tracking station at noon. Assume that the

satellite takes two hours to make an orbit and

the radius of the Earth is 6400 km.

a If the tracking station antenna is aimed

at 30◦ above the horizon, at what time will

the satellite pass through the beam of the antenna?

satellite

direction of travel

6400 km8000 km

30°

S

O

T

b Find the distance between the satellite and the tracking station at 12.06 p.m.

c At what angle above the horizon should the antenna be pointed so that its beam will

intercept the satellite at 12.06 p.m.?

15 An athlete in a gymnasium is training on an exercise bike.

At time t = 0, the position of the pedal is as shown.

The height of the pedal, h cm, from the floor at time

t seconds, is given byh(t) = a + b cos (�(t + c))

where a and b are in centimetres.

a Find the values of a, b and c.

b i Find the times at which the height of the pedal

above the floor is 60 cm, 0 ≤ t ≤ 4.

direction of movement

Pedal

Floor

35 cm

25 cm60°

ii Find the times at which the height of the pedal above the floor is 37.5 cm,

0 ≤ t ≤ 4.

c Sketch the graph of h against t for 0 ≤ t ≤ 4.

16 ABCD is a parallelogram whose diagonals

intersect at angle �◦ at the point E.

Let AB = CD = x, AD = BC = y, BD = p,

AC = q.

a Apply the cosine rule to triangle DEC to

find x in terms of p, q and �.

A

B C

E

D

θ

b Apply the cosine rule to triangle DEA to find y in terms of p, q and �.

c Use the results of a and b to show that 2(x2 + y2) = p2 + q2

d A parallelogram has sides 8 cm and 6 cm and one diagonal of 13 cm. Find the length

of the other diagonal.

17 The figure shows the circular cross section of a

uniform log of radius 40 cm floating in water. The

points A and B are on the surface of the water

and the highest point X is 8 cm above the surface.

a Show that the magnitude of ∠AOB is

approximately 1.29 radians.

b Calculate

i the length of arc AXB

X

B A

O

40 cm

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ii the area of the cross section below the surface

iii the percentage of the volume of the log below the surface.

18 In �ABC, AB = 7 cm , BC = 9 cm and ∠ABC = �.

a Show that AC2 = 130 − 126 cos �.

D is the point on the opposite side of AC from B such that ABCD is a cyclic

quadrilateral in which CD = 6 cm and DA = 5 cm.

b Obtain another expression for AC2 in terms of � and prove that cos � = 23

62.

c Calculate

i the length of AC ii the area of quadrilateral ABCD.

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P1: FXS/ABE P2: FXS CHAPTER 13 - Cambridge … · P1: FXS/ABE P2: FXS ... 13.1 Multiple-choice questions 1 A ladder 2.6 m long rests with one end on horizontal ground while the other