1  (IB/s1/2017/November/Paper2/q10)
[Maximum mark: 17] Note: In this question, distance is in millimetres. Let $f(x)=x+a \sin \left(x\frac{\pi}{2}\right)+a$, for $x \geq 0$
The graph of $f$ passes through the origin. Let $\mathrm{P}_{t}$ be any point on the graph of $f$ with $x$coordinate $2 k \pi$, where $k \in \mathbb{N} .$ A straight line $L$ passes through all the points $\mathrm{P}_{k}$.
Diagram 1 shows a saw. The length of the toothed edge is the distance $\mathrm{AB}$. 
2  (Continue: 1)
The toothed edge of the saw can be modelled using the graph of $f$ and the line $L$. Diagram 2 represents this model. The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of $f$ and the line $L$, between $\mathrm{P}_{t}$ and $\mathrm{P}_{z+1}^{}$

3  (IB/s1/2017/May/paper2tz1/q7)
[Maximum mark: 7] A particle $\mathrm{P}$ moves along a straight line. Its velocity $v_{\mathrm{p}} \mathrm{ms}^{1}$ after $t$ seconds is given by $v_{\mathrm{P}}=\sqrt{t} \sin \left(\frac{\pi}{2} t\right)$, for $0 \leq t \leq 8 .$ The following diagram shows the graph of $v_{\mathrm{p}}$.

4  (IB/s1/2017/May/paper2tz1/q8)
[Maximum mark: 14] At Grande Anse Beach the height of the water in metres is modelled by the function $h(t)=p \cos (q \times t)+r$, where $t$ is the number of hours after $21: 00$ hours on 10 December 2017 . The following diagram shows the graph of $h$, for $0 \leq t \leq 72$. The point $\mathrm{A}(6.25,0.6)$ represents the first low tide and $\mathrm{B}(12.5,1.5)$ represents the next high tide.

5  (IB/sl/2017/May/paper2tz2/q4)
[Maximum mark: 6] The depth of water in a port is modelled by the function $d(t)=p \cos q t+7.5$, for $0 \leq t \leq 12$, where $t$ is the number of hours after high tide. At high tide, the depth is $9.7$ metres. At low tide, which is 7 hours later, the depth is $5.3$ metres.

6  (IB/s1/2016/November/Paper2/q10)
[Maximum mark: 15] The following diagram shows the graph of $f(x)=a \sin b x+c$, for $0 \leq x \leq 12$. The graph of $f$ has a minimum point at $(3,5)$ and a maximum point at $(9,17)$.
The graph of $g$ is obtained from the graph of $f$ by a translation of $\left(\begin{array}{l}k \\ 0\end{array}\right)$. The maximum point on the graph of $g$ has coordinates $(11.5,17)$.
The graph of $g$ changes from concaveup to concavedown when $x=w$.

7  (IB/s1/2016/May/paper1tz1/q3)
[Maximum mark: 7] Let $f(x)=3 \sin \left(\frac{\pi}{2} x\right)$, for $0 \leq x \leq 4$.

8  (IB/s1/2016/May/paper2tz1/q9)
[Maximum mark: 14] A particle P moves along a straight line so that its velocity, $v \mathrm{~ms}^{1}$, after $t$ seconds, is given by $r=\cos 3 t2 \sin t0.5$, for $0 \leq t \leq 5$. The initial displacement of P from a fixed point $O$ is 4 metres.
The following sketch shows the graph of $v$.

9  (IB/sl/2016/May/paper2tz2/q4)
[Maximum mark: 8] The height, $h$ metres, of a seat on a Ferris wheel after $t$ minutes is given by $$ h(t)=15 \cos 1.2 t+17, \text { for } t \geq 0 $$

10  (IB/sl/2019/May/paper1tz2/q7)
[Maximum mark: 7] Consider the graph of the function $f(x)=2 \sin x, 0 \leq x < 2 \pi$. The graph of $f$ intersects the line $y=1$ exactly twice, at point $\mathrm{A}$ and point $\mathrm{B}$. This is shown in the following diagram.
Consider the graph of $g(x)=2 \sin p x, 0 \leq x < 2 \pi$, where $p>0$.

11  (IB/s1/2018/November/Paper1/q7)
[Maximum mark: 6] Given that $\sin x=\frac{1}{3}$, where $0 < x < \frac{\pi}{2}$, find the value of $\cos 4 x$. 
12  (IB/s1/2018/May/paper2tz1/q10)
[Maximum mark: 15] Let $f(x)=12 \cos x5 \sin x,\pi \leq x \leq 2 \pi$, be a periodic function with $f(x)=f(x+2 \pi)$. The following diagram shows the graph of $f$. There is a maximum point at $\mathrm{A}$. The minimum value of $f$ is $13$.

13  (IB/s1/2018/May/paper2tz1/q10b)
A ball on a spring is attached to a fixed point $O$. The ball is then pulled down and released, so that it moves back and forth vertically. The distance, $d$ centimetres, of the centre of the ball from $\mathrm{O}$ at time $t$ seconds, is given by $$d(t)=f(t)+17,0 \leq t \leq 5$$

14  (IB/s1/2018/May/paper2tz2/q6)
[Maximum mark: 8] At an amusement park, a Ferris wheel with diameter 111 metres rotates at a constant speed. The bottom of the wheel is $k$ metres above the ground. A seat starts at the bottom of the wheel. The wheel completes one revolution in 16 minutes.
After $t$ minutes, the height of the seat above ground is given by $h(t)=61.5+a \cos \left(\frac{\pi}{8} t\right)$. for $0 \leq t \leq 32$,

15  (IB/s1/2016/November/Paper1/q2)
[Maximum mark: 5] Let $\sin \theta=\frac{\sqrt{5}}{3}$, where $\theta$ is acute.

16  (IB/sl/2019/November/Paper2/q6)
[Maximum mark: 6] The diagram below shows a triangularbased pyramid with base ADC. Edge $\mathrm{BD}$ is perpendicular to the edges $\mathrm{AD}$ and $\mathrm{CD}$. $\mathrm{AC}=28.4 \mathrm{~cm}, \mathrm{AB}=x \mathrm{~cm}, \mathrm{BC}=x+2 \mathrm{~cm}, \mathrm{ABC}=0.667, \mathrm{BAD}=0.611$ Calculate AD. 
17  (IB/s1/2019/November/Paper1/q6)
[Maximum mark: 8] Let $f(x)=4 \cos \left(\frac{x}{2}\right)+1$, for $0 \leq x \leq 6 \pi$. Find the values of $x$ for which $f(x)>2 \sqrt{2}+1$ 
18  (IB/sl/2019/May/paper1tz1/q3)
[Maximum mark: 7] The following diagram shows a right triangle $\mathrm{ABC}$. Point $\mathrm{D}$ lies on $\mathrm{AB}$ such that $\mathrm{CD}$ bisects ACB.

19  $(\mathrm{IB} / \mathrm{s} 1 / 2019 /$ May $/$ paper $2 \mathrm{tz} 2 / \mathrm{q} 3)$
[Maximum mark: 7] The following diagram shows the quadrilateral $\mathrm{ABCD}$. $\mathrm{AB}=6.73 \mathrm{~cm}, \mathrm{BC}=4.83 \mathrm{~cm}, \mathrm{BC} \mathrm{D}=78.2^{\circ}$ and $\mathrm{CD}=3.80 \mathrm{~cm}$

20  (IB/s1/2018/November/Paper2/q7)
[Maximum mark: 6] A communication tower, $T$, produces a signal that can reach cellular phones within a radius of $32 \mathrm{~km}$. A straight road passes through the area covered by the tower's signal. The following diagram shows a line representing the road and a circle representing the area covered by the tower's signal. Point $R$ is on the circumference of the circle and points $S$ and $R$ are on the road. Point $\$$ is $38 \mathrm{~km}$ from the tower and RST $=43^{\circ}$,

21  (IB/sl/2018/May/paper2tz1/q6)
[Maximum mark: 7] Triangle $\mathrm{ABC}$ has $a=8.1 \mathrm{~cm}, b=12.3 \mathrm{~cm}$ and area $15 \mathrm{~cm}^{2}$. Find the largest possible perimeter of triangle $\mathrm{ABC}$. 
22  (IB/s1/2018/May/paper2tz2/q2)
[Maximum mark: 6] The following diagram shows quadrilateral $\mathrm{ABCD}$. $$\mathrm{AB}=11 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm}, \mathrm{BAD}=59^{\circ}, \mathrm{ADB}=100^{\circ} \text {, and } \mathrm{CBD}=82^{}$$

23  (IB/s1/2017/November/Paper2/q1)
[Maximum mark: 6] The following diagram shows a triangle $\mathrm{ABC}$. $$\mathrm{AB}=5 \mathrm{~cm}, \mathrm{CAB}=50^{\circ} \text { and } \mathrm{ACB}=112^{\circ}$$

24  $(\mathrm{IB} / \mathrm{sl} / 2017 /$ May $/$ paper $1 \mathrm{tz} 1 / \mathrm{q} 3)$
[Maximum mark; 6] The following diagram shows triangle $\mathrm{PQR}$. $$\mathrm{PQR}=30^{\circ}, \mathrm{QR} \mathrm{P}=45^{*} \text { and } \mathrm{PQ}=13 \mathrm{~cm} $$ Find PR. 
25  (IB/sl/2017/May/paper2tz2/q9) [Maximum mark: 15] A ship is sailing north from a point $A$ towards point $D$. Point $C$ is $175 \mathrm{~km}$ north of $A$. Point $\mathrm{D}$ is $60 \mathrm{~km}$ north of $\mathrm{C}$. There is an island at $\mathrm{E}$. The bearing of $\mathrm{E}$ from $\mathrm{A}$ is 055 . The bearing of $\mathrm{E}$ from $\mathrm{C}$ is $134^{\prime \prime}$. This is shown in the following diagram.

26  (IB/s1/2016/May/paper1tz1/q6)
[Maximum mark: 7] The following diagram shows triangle $\mathrm{ABC}$. The point $\mathrm{D}$ lies on [BC] so that [AD] bisects BAC $AB= 2\sqrt 5$ cm, $AC=x$ cm, and $D\hat A C=\theta,$ where $\sin\theta=\frac 23.$ The area of triangle $\mathrm{ABC}$ is $5 \mathrm{~cm}^{2}$. Find the value of $x .$ 
27  (IB/s1/2016/May/paper2tz1/q3)
[Maximum mark: 7 ] The following diagram shows three towns $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$. Town $\mathrm{B}$ is $5 \mathrm{~km}$ from Town $\mathrm{A}$, on a bearing of $070^{\circ}$. Town C is $8 \mathrm{~km}$ from Town B, on a bearing of $115^{\circ}$.

28  (IB/s1/2016/May/paper2tz2/q2)
[Maximum mark: 6] The following diagram shows a quadrilateral $\mathrm{ABCD}$. $\mathrm{AD}=7 \mathrm{~cm}, \mathrm{BC}=8 \mathrm{~cm}, \mathrm{CD}=12 \mathrm{~cm}, D\hat A B=1.75$ radians, $A\hat BD=0.82$ radians.

29  (IB/s1/2015/May/paper1tz1/q5)
[Maximum mark: 7 ] Given that $\sin x=\frac{3}{4}$, where $x$ is an obtuse angle, find the value of

30  (IB/sl/2015/May/paper1tz2/q2)
[Maximum mark: 6] Let $f(x)=a \sin b x$, where $b>0 .$ The following diagram shows part of the graph of $f .$

31  (IB/sl/2015/May/paper2tz1/q8)
[Maximum mark: 13] The following diagram shows a straight shoreline, with a supply store at $\mathrm{S}$, a town at $\mathrm{T}$, and an island L. A boat delivers supplies to the island. The boat leaves $\mathrm{S}$, and sails to the island. Its path makes an angle of $20^{4}$ with the shoreline.
It is decided to change the position of the supply store, so that its distance from $\mathrm{L}$ is $5 \mathrm{~km}$. The following diagram shows the two possible locations $\mathrm{P}$ and $\mathrm{Q}$ for the supply store.

32  (IB/s1/2015/May/paper2tz2/q1)
[Maximum mark: 6] The following diagram shows triangle $\mathrm{ABC}$. $\mathrm{BC}=10 \mathrm{~cm}, \mathrm{ABC}=80^{\circ}$ and $\mathrm{BAC}=35^{\prime}$

33  (IB/sl/2015/November/Paper2/q8)
[Maximum mark: 14] The following diagram shows the quadrilateral $\mathrm{ABCD}$. $\mathrm{AD}=6 \mathrm{~cm}, \mathrm{AB}=15 \mathrm{~cm}, \mathrm{ABC}=44^{\circ}, \mathrm{ACB}=83^{*}$ and $\mathrm{DAC}=\theta$
The area of triangle $\mathrm{ACD}$ is half the area of triangle $\mathrm{ABC}$.

Answer (Trigo)
1 (a) Show (b)(i) $P_{0}=(0,0), P_{1}(2 \pi, 2 \pi)$ (ii) $y=x$ (c) Show
2 (d) 33
3 (a) (i) $t=2$ (ii) $9.65$ (b) $k=5.94$
4 (a)(i) $6.25$ (ii) $0.9$ (b) (i) $p=0.45$ (ii) $q=\frac{4 \pi}{25}$ (iii) $r=1.05$ (d) $23: 00$
5 (a) $\quad p=2.2$ (b) $q=\frac{\pi}{7}$ (c) $7.01$
6 (a)(i) $\quad c=11$ (ii) $b=\frac{\pi}{6}$ (iii) $a=6$ (b) (i) $k=2.5$ (ii) $g(x)=6 \sin \left(\frac{\pi}{6}(x2.5)\right)+11$ (c)(i) $w=8.5$ (ii) $\pi$
7 (a)(i) 3 (ii) 4 (b) Graph
8 (a) $0.284$ (b) $t=0.180$ (c) 2 (d) $0.744$ (e) $3.28$
9 (a) $h(0)=2$ (b) $k=1.48$ (c) $5.2$
10 (a) $x=\frac{7\pi}{6},\frac{11\pi}{6}$ (b) $ p=\frac{7}{12} $
11 $\frac{11}{81}$
12 (a) $A(0.395,13)$ (b) (i) $13$ (ii) $2 \pi$ (c) $f(x)=13 \cos (x+0.395)$
13 (d) 13 (e) $1.02$
14 (a) $k=6$ (b) $a=55.5$ (c) $t=18.5$
15 (a) $\quad \cos \theta=\frac{2}{3}$ (b) $\cos 2 \theta=\frac{1}{9}$
16 34.6
17 $0\le x<\frac{\pi}{2}, \frac{7\pi}{2} <x<\frac{9\pi}{2}$
18 (a) $\frac{4}{5}$ (b) $\frac{7}{25}$ (c) 50
19 (a) $5.50$ (b) $\theta=92.0^{\circ}$ or $1.61$
20 (a) Show (b) $37.5$
21 perimeter $=40.6$
22 (a) $D B=9.57$ (b) $D C=10.6$
23 (a) $B C=4.13$ (b) Area $=3.19$
24 $P R=\frac{13 \sqrt{2}}{2}$
25 (a) $235$ (b) 146 (c) 193 (d) $27.3$
26 $x=\frac{9}{4}$
27 (a) $\quad A \hat{B} C=135^{\circ}$ (b) $12.1$ (c) $A\hat{ C} B=17.0^{\circ}$
28 (a) $B D=9.42$ (b) $B \widehat{D} C=1.51$
29(a) $\cos x=\frac{\sqrt{7}}{4}$ (b) $\cos 2 x=\frac{1}{8}$
30(a)(i) $\pi $ (ii) 3 (b) (i) $a=3$ (i) $b=2$
31(a) $9$ (b) $S\hat{ Q} L=142^{\circ}, \quad S \hat{P} L=38.0^{\circ}$ (c) (i) new store is at Q (ii) $4.52$
32(a) $A C=17.2$ (b) area $=77.8$
33(a) $\quad A C=10.5$ (b) area $=62.9$ (c) $\theta=86.7,93.3$ (d) $12.4$
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