Pack B — Fluency

Mathematics

Year 11 · Surveying & Navigation

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–30

1.
In a right-angled triangle, the hypotenuse is 12 and one angle is $40^\circ$ . Find the side opposite this angle, to 3 s.f.
2.
In a right-angled triangle, opposite is 7 and hypotenuse is 25. Find the angle, to 3 s.f.
3.
In a right-angled triangle, the two shorter sides are 8 and 15. Find the hypotenuse (exact).
4.
State the exact value of $\sin 30^\circ$ .
5.
A right-angled triangle has legs 6 cm and 8 cm. Find its area.
6.
A ship sails on bearing $060^\circ$ . State the angle it makes with North (clockwise).
7.
Find the area of a triangle with sides $a = 10$ , $b = 14$ and included angle $C = 50^\circ$ , to 3 s.f.
8.
In triangle $ABC$ , $\angle A = 40^\circ$ and $\angle B = 75^\circ$ . Find $\angle C$ .
9.
In triangle $ABC$ , write the sine rule equation relating $a, b, \angle A, \angle B$ .
10.
State the cosine rule for $a$ in terms of $b, c, A$ .
11.
Convert $135^\circ$ to radians, as an exact multiple of $\pi$ .
12.
Convert $\dfrac{ {a}\pi}{ {b}}$ radians to degrees.
13.
Find the exact value of $\sin\!\left(\dfrac{\pi}{ {n}}\right)$ .
14.
Find the exact value of $\cos\!\left(\dfrac{\pi}{ {n}}\right)$ .
15.
Find the exact value of $\tan\!\left(\dfrac{\pi}{ {n}}\right)$ .
16.
A sector has radius 8 cm and angle $1.5$ rad. Find the arc length.
17.
A sector has radius 10 cm and angle $0.6$ rad. Find its area.
18.
A sector has radius 8 cm and angle $45^\circ$ . Find the arc length, to 3 s.f.
19.
A sector has radius 10 cm and angle $60^\circ$ . Find the sector area, to 3 s.f.
20.
Using a 45-45-90 triangle, give the exact value of $\sin 45^\circ$ and $\cos 45^\circ$ .
21.
State the amplitude of $y = 7\sin x$ .
22.
State the period of $y = \sin(Bx)$ when $B = \pi/3$ .
23.
For $y = 2\sin x + 7$ , state the maximum and minimum values.
24.
For $y = 2\sin x + 7$ , find $y(0)$ .
25.
A sinusoid has amplitude 4, mean line $y = 1$ , period $2\pi$ , no phase shift. Write its equation in the form $y = A\sin(Bx) + C$ .
26.
State the period of $y = \cos\!\left(\dfrac{\pi x}{ {n}}\right)$ .
27.
State the phase shift of $y = \sin(x - \pi/4)$ relative to $y = \sin x$ .
28.
The depth of water in a harbour rises and falls. State the kind of function (sine, cosine, linear, exponential) that best models this.
29.
A periodic process has max value 14 and min value 4. State its amplitude and mean (vertical shift) for a $\sin$ -based model.
30.
Describe the graph of $y = 3\sin(2x)$ : amplitude, period, and any vertical shift.

SilverQuestions 31–60

31.
In triangle $ABC$ , $\angle A = 35^\circ$ , $\angle B = 80^\circ$ , side $a = 6$ . Find $b$ to 3 s.f.
32.
In triangle $ABC$ , $a = 10$ , $b = 13$ and $\angle A = 35^\circ$ . Find $\angle B$ (acute case), to 3 s.f.
33.
Find side $a$ when $b = 6$ , $c = 10$ and $A = 75^\circ$ , to 3 s.f.
34.
In a triangle with sides $a = 5$ , $b = 6$ , $c = 7$ , find angle $A$ , to 3 s.f.
35.
Triangle $ABC$ has $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$ . Find (a) area, (b) $BC$ , to 3 s.f.
36.
From a point on the ground 50 m from the base of a tower, the angle of elevation of the top is $32^\circ$ . Find the height, to 3 s.f.
37.
A chord subtends an angle of $100^\circ$ at the centre of a circle of radius 10 cm. Find the chord length, to 3 s.f.
38.
A ship sails 80 km on bearing $130^\circ$ from port. How far north and east is the ship? To 3 s.f.
39.
In triangle $ABC$ , $AB = 4$ , $AC = 6$ , $\angle BAC = 60^\circ$ . Find the area exactly.
40.
From the top of a cliff 80 m high, the angle of depression of a boat at sea is $15^\circ$ . How far is the boat from the foot of the cliff, to 3 s.f.?
41.
Find the exact value of $\cos\!\left(\dfrac{ {a}\pi}{ {b}}\right)$ .
42.
Find the exact value of $\sin\!\left(\dfrac{ {a}\pi}{ {b}}\right)$ .
43.
Solve $2\cos x = -1$ for $0 \leq x \leq 2\pi$ .
44.
Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ exactly.
45.
If $\sin\theta = \dfrac{3}{5}$ and $\cos\theta = \dfrac{4}{5}$ , find $\tan\theta$ .
46.
Find the perimeter of a sector with radius 5 cm and angle $\dfrac{ {a}\pi}{ {b}}$ rad, to 3 s.f.
47.
State the value of $\sin\!\left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)$ exactly.
48.
Simplify $\sin(-\theta)$ and $\cos(-\theta)$ .
49.
Solve $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$ .
50.
A sector has radius 9 cm and angle $\dfrac{4\pi}{9}$ rad. Find (a) the arc length, (b) the sector area, exactly.
51.
For $y = A\sin(Bx) + C$ with $A = 3$ , $B = \pi/6$ , $C = 4$ , state (a) amplitude, (b) period, (c) maximum.
52.
Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for the smallest positive $t$ .
53.
The depth (m) of water in a harbour is $d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5$ ( $t$ in hours after midnight). Find $d(0)$ and the maximum depth.
54.
Using $d(t) = 3\sin(\pi t/6) + 5$ , find the first time after midnight when the depth is exactly 8 m.
55.
A Ferris wheel has radius 12 m and centre 15 m above ground. The lowest point is at $t = 0$ . Write a height function $h(t)$ if the period is 4 minutes.
56.
A temperature in a city is modelled by $T(t) = 5\sin\!\left(\dfrac{\pi(t - 6)}{12}\right) + 18$ ( $T$ in °C, $t$ in hours after midnight). What is the temperature at $t = 6$ ?
57.
A periodic process has values 12, 18, 12, 6, 12, 18 at $t = 0, 1, 2, 3, 4, 5$ seconds. State the period and amplitude.
58.
For $y = A\sin(Bx) + C$ , the graph oscillates between $y = 2$ and $y = 12$ . State $A$ and $C$ .
59.
A sinusoid passes through $(0, 4)$ with maximum, has amplitude 3, and period 8. Write its equation in the form $y = A\cos(Bx) + C$ .
60.
In $h(t) = -15\cos\!\left(\dfrac{\pi t}{2}\right) + 18$ , state the meaning of the values 18 and 15 in the context of a Ferris wheel.

GoldQuestions 61–90

61.
In triangle $ABC$ , $AB = 12$ , $AC = 9$ , $\angle BAC = 75^\circ$ and $BC \approx 13.0$ . Find $\angle ABC$ to 3 s.f.
62.
From port $P$ , a ship sails 50 km on bearing $060^\circ$ to $A$ , then 80 km on bearing $150^\circ$ to $B$ . Find the angle at $A$ in triangle $PAB$ , and $PB$ to 3 s.f.
63.
A triangle has sides 7, 9, 11. Find its largest angle to 3 s.f.
64.
Triangle $XYZ$ has $XY = 14$ , $\angle X = 50^\circ$ , $\angle Z = 70^\circ$ . Find $YZ$ to 3 s.f.
65.
A triangle has sides 10 and 14 and area 50 m $^2$ . Find the included angle, giving the acute case to 3 s.f.
66.
A box has dimensions $5 \times 4 \times 3$ cm. Find the length of its space diagonal exactly.
67.
A box has dimensions $5 \times 4 \times 3$ . Find the angle the space diagonal makes with the $5 \times 4$ base, to 3 s.f.
68.
A hiker walks 5 km on bearing $050^\circ$ from $C$ to $V$ . State the bearing of $C$ from $V$ (return bearing).
69.
In triangle $ABC$ , $a = 8$ , $b = 11$ , $\angle A = 35^\circ$ . Find both possible values of $\angle B$ , to 3 s.f.
70.
A boat sails 6 km on bearing $080^\circ$ from $X$ to $Y$ , then 8 km on bearing $160^\circ$ from $Y$ to $Z$ . Find $XZ$ to 3 s.f.
71.
Find an angle between $0$ and $2\pi$ coterminal with $\dfrac{13\pi}{4}$ .
72.
Find the exact value of $\sin\!\left(\dfrac{11\pi}{6}\right) + \cos\!\left(\dfrac{5\pi}{6}\right)$ .
73.
Solve $2\sin^2 x - 1 = 0$ for $0 \leq x \leq 2\pi$ .
74.
A chord subtends $\dfrac{2\pi}{3}$ rad at the centre of a circle of radius 10 cm. Find the area of the minor segment, to 3 s.f.
75.
Verify the identity $\dfrac{1 - \cos^2\theta}{\sin\theta} = \sin\theta$ .
76.
If $\sin\theta = -\dfrac{4}{5}$ and $\theta$ is in QIII, find $\cos\theta$ and $\tan\theta$ .
77.
Solve $\sin(2x) = \dfrac{1}{2}$ for $0 \leq x < 2\pi$ .
78.
Use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ and a half-angle / difference identity to find $\sin\dfrac{\pi}{12}$ exactly.
79.
Convert $\dfrac{7\pi}{12}$ rad to degrees.
80.
Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x < 2\pi$ .
81.
A Ferris wheel of radius 15 m has its centre 18 m above ground. Period 4 minutes; starts at the lowest point. Find the first time (to 3 s.f.) the capsule is 25 m above ground.
82.
Solve $3\sin\!\left(\dfrac{\pi t}{6}\right) + 5 = 7$ for all $t$ in $[0, 12]$ .
83.
Daylight in a city varies sinusoidally between 9 h (winter solstice) and 15 h (summer solstice). Write a model $L(t) = A\cos(B(t - C)) + D$ where $t$ is months after January 1.
84.
The temperature in a town is $T(t) = 8\sin\!\left(\dfrac{\pi(t - 9)}{12}\right) + 20$ for $t$ in hours. (a) When is the temperature highest? (b) What is the highest temperature?
85.
The depth model $d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5$ (m) for the harbour: for how many hours per cycle is the depth above 7 m? To 3 s.f.
86.
A sinusoid $y = A\sin(Bx) + C$ has amplitude 4, period 6 and passes through $(0, 2)$ . Find $A$ (assuming positive) and $C$ .
87.
A periodic process has minimum at $t = 0$ , max value 10 and min value 2, period 8. Write a model in the form $y = A\cos(Bt) + C$ with $A < 0$ .
88.
Rewrite $y = \sin x$ in terms of cosine using a phase shift.
89.
For $y = 5\sin(2x) + 8$ , find the average value of $y$ over one full period.
90.
A pendulum swings between $-15$ cm and $15$ cm from rest, with period 2 s. Write a position model $x(t) = A\cos(Bt)$ assuming $x(0) = 15$ .

PlatinumQuestions 91–120

91.
A hiker walks 5 km on bearing $050^\circ$ from base camp $C$ to viewpoint $V$ . From $V$ they walk 7 km on bearing $145^\circ$ to lake $L$ . Find (a) the interior angle at $V$ , (b) $CL$ to 3 s.f.
92.
A vertical flagpole stands at $F$ on horizontal ground. Surveyor $S_1$ is 80 m due south of $S_2$ . From $S_1$ , $F$ is on bearing $045^\circ$ ; from $S_2$ , $F$ is on bearing $115^\circ$ . The angle of elevation of the top from $S_2$ is $28^\circ$ . Find the height to 3 s.f.
93.
A boat sails 6 km on bearing $080^\circ$ , then 9 km on $150^\circ$ , then 4 km on $250^\circ$ . Find its straight-line distance from the start to 3 s.f.
94.
In triangle $ABC$ , $a = 8$ , $b = 11$ , $\angle A = 35^\circ$ . For each ambiguous-case solution, state $\angle C$ and $c$ , to 3 s.f.
95.
In triangle $ABC$ , $AB = 4$ , $AC = 6$ , $\angle BAC = 60^\circ$ . Find (a) $BC$ exactly, (b) area exactly.
96.
A cuboid has a square base of side 4 and height 6. A diagonal is drawn from one base vertex to the opposite top vertex. Find the angle this diagonal makes with the **base diagonal**, to 3 s.f.
97.
In triangle $ABC$ , $\angle A = 40^\circ$ , $\angle B = 75^\circ$ , $AB = 12$ cm. Find the area to 3 s.f.
98.
A ship sails 30 km on bearing $090^\circ$ from $P$ to $Q$ . From $Q$ it changes course to bearing $030^\circ$ and sails to $R$ . The total straight-line distance $PR$ is 50 km. Find the distance $QR$ , to 3 s.f.
99.
Two stations $A$ and $B$ are 200 m apart on level ground. The angles of elevation of a mountain top $T$ from $A$ and $B$ are $25^\circ$ and $35^\circ$ respectively, with $B$ closer. Find the height of the mountain to 3 s.f., assuming $A$ , $B$ , and the foot of $T$ are collinear.
100.
Two aircraft leave the same airport at the same time. Aircraft A flies at 400 km/h on bearing $060^\circ$ ; aircraft B flies at 350 km/h on bearing $130^\circ$ . Find the distance between them after 2 hours, to 3 s.f.
101.
A chord of length 12 cm subtends an angle of $\dfrac{\pi}{3}$ rad at the centre of a circle. Find the radius and the area of the minor segment, to 3 s.f.
102.
A circle has radius 8 cm and a chord whose arc has length 12 cm. Find the angle subtended at the centre, exact and to 3 s.f.
103.
Prove the identity $\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = \dfrac{2}{\cos\theta}$ .
104.
Solve $\cos 2x = \sin x$ for $0 \leq x \leq 2\pi$ .
105.
A sector has perimeter 20 cm. Find the radius that maximises the area.
106.
Find the exact value of $\sin\!\left(\dfrac{7\pi}{12}\right)$ using $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$ .
107.
A circle of radius 6 cm has a chord of length 6 cm. Find the angle subtended at the centre and the area of the minor segment, exact and to 3 s.f.
108.
Solve $\sin x + \cos x = 1$ for $0 \leq x \leq 2\pi$ .
109.
An annular sector has inner radius 4 cm, outer radius 7 cm, and angle $\dfrac{\pi}{3}$ rad. Find its area exactly.
110.
Show that if $\sin\theta + \cos\theta = \dfrac{1}{2}$ , then $\sin\theta \cos\theta = -\dfrac{3}{8}$ .
111.
Use the Ferris wheel $h(t) = 18 - 15\cos\!\left(\dfrac{\pi t}{2}\right)$ . For how long during one revolution is the capsule above 25 m?
112.
The depth model $d(t) = 3\sin(\pi t/6) + 5$ — find the first time after midnight that the tide is rising and depth reaches 6 m.
113.
A pendulum has displacement at $t = 0$ of 8 cm (max). Period is 2 s. Find (a) amplitude, (b) $B$ , (c) the position model $x(t) = A\cos(Bt)$ .
114.
Daylight in a town varies from 8.5 h (winter, $t = 0$ ) to 15.5 h (summer, $t = 6$ months). Write a model $L(t) = A\cos(Bt) + D$ , and use it to predict $L(3)$ (i.e. at the spring equinox).
115.
A signal is modelled by $V(t) = 5\sin(120\pi t)$ volts. Find (a) the frequency in Hz, (b) the time of the first peak after $t = 0$ .
116.
A motorboat's vertical bob is observed: at $t = 0$ it is at the **mean** height and rising, reaches max at $t = 2$ s. Write a model $y(t) = A\sin(Bt)$ with $A > 0$ and amplitude 4.
117.
The function $f(t)$ describes a combined model: $f(t) = 2\sin(t) + \sin(2t)$ . State whether $f$ is periodic; if so, give its period.
118.
A sinusoid $y = A\sin(Bt) + C$ with $A, B > 0$ passes through $(0, 2)$ and reaches its first maximum value 6 at $t = 3$ . Find $A, B, C$ .
119.
For $T(t) = 6\sin(\pi t/12) + 18$ (°C, $t$ in hours, daily cycle 24 h), find the fraction of one day during which the temperature exceeds 21°C.
120.
A voltage $V(t) = 10\sin(120\pi t)$ V drives a 5 Ω resistor. The average power is $\frac{V_{\text{rms}}^2}{R}$ where $V_{\text{rms}} = \frac{|A|}{\sqrt{2}}$ for a sinusoid. Find the average power, exact and to 3 s.f.