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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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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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nChapter 13 — Revision of chapters 8–12 369
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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nChapter 13 — Revision of chapters 8–12 371
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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nChapter 13 — Revision of chapters 8–12 373
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
8 cmSAMPLE
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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n374 Essential Advanced General Mathematics
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.
SAMPLE
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