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Date	May 2017	Marks available	3	Reference code	17M.2.SL.TZ1.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term		Question number	S_8	Adapted from	N/A

Question

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 $0 \leqslant t \leqslant 72$ .

M17/5/MATME/SP2/ENG/TZ1/08

The point ${\text{A}}(6.25,{\text{ }}0.6)$ represents the first low tide and ${\text{B}}(12.5,{\text{ }}1.5)$ represents the next high tide.

How much time is there between the first low tide and the next high tide?

[2]

a.i.

Find the difference in height between low tide and high tide.

[2]

a.ii.

Find the value of $p$ ;

[2]

b.i.

Find the value of $q$ ;

[3]

b.ii.

Find the value of $r$ .

[2]

b.iii.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

[3]

Markscheme

attempt to find the difference of $x$ -values of A and B (M1)

eg $\,\,\,\,\,$ $6.25 - 12.5{\text{ }}$

6.25 (hours), (6 hours 15 minutes) A1 N2

[2 marks]

a.i.

attempt to find the difference of $y$ -values of A and B (M1)

eg $\,\,\,\,\,$ $1.5 - 0.6$

$0.9{\text{ (m)}}$ A1 N2

[2 marks]

a.ii.

valid approach (M1)

eg $\,\,\,\,\,$ $\frac{{{\text{max}} - {\text{min}}}}{2},{\text{ }}0.9 \div 2$

$p = 0.45$ A1 N2

[2 marks]

b.i.

METHOD 1

period $= 12.5$ (seen anywhere) (A1)

valid approach (seen anywhere) (M1)

eg $\,\,\,\,\,$ ${\text{period}} = \frac{{2\pi }}{b},{\text{ }}q = \frac{{2\pi }}{{{\text{period}}}},{\text{ }}\frac{{2\pi }}{{12.5}}$

0.502654

$q = \frac{{4\pi }}{{25}},{\text{ 0.503 }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)$ A1 N2

METHOD 2

attempt to use a coordinate to make an equation (M1)

eg $\,\,\,\,\,$ $p\cos (6.25q) + r = 0.6,{\text{ }}p\cos (12.5q) + r = 1.5$

correct substitution (A1)

eg $\,\,\,\,\,$ $0.45\cos (6.25q) + 1.05 = 0.6,{\text{ }}0.45\cos (12.5q) + 1.05 = 1.5$

0.502654

$q = \frac{{4\pi }}{{25}},{\text{ }}0.503{\text{ }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)$ A1 N2

[3 marks]

b.ii.

valid method to find $r$ (M1)

eg $\,\,\,\,\,$ $\frac{{{\text{max}} + {\text{min}}}}{2},{\text{ }}0.6 + 0.45$

$r = 1.05$ A1 N2

[2 marks]

b.iii.

METHOD 1

attempt to find start or end $t$ -values for 12 December (M1)

eg $\,\,\,\,\,$ $3 + 24,{\text{ }}t = 27,{\text{ }}t = 51$

finds $t$ -value for second max (A1)

$t = 50$

23:00 (or 11 pm) A1 N3

METHOD 2

valid approach to list either the times of high tides after 21:00 or the $t$ -values of high tides after 21:00, showing at least two times (M1)

eg $\,\,\,\,\,$ ${\text{21:00}} + 12.5,{\text{ 21:00}} + 25,{\text{ }}12.5 + 12.5,{\text{ }}25 + 12.5$

correct time of first high tide on 12 December (A1)

eg $\,\,\,\,\,$ 10:30 (or 10:30 am)

time of second high tide = 23:00 A1 N3

METHOD 3

attempt to set their $h$ equal to 1.5 (M1)

eg $\,\,\,\,\,$ $h(t) = 1.5,{\text{ }}0.45\cos \left( {\frac{{4\pi }}{{25}}t} \right) + 1.05 = 1.5$

correct working to find second max (A1)

eg $\,\,\,\,\,$ $0.503t = 8\pi ,{\text{ }}t = 50$

23:00 (or 11 pm) A1 N3

[3 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations

Show 89 related questions

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22M.2.SL.TZ2.8b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
EXM.3.AHL.TZ0.5a.iii:
Write down the period of this graph
18M.2.SL.TZ1.S_10a:
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18M.2.SL.TZ1.S_10b.i:
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19M.2.AHL.TZ1.H_10a:
Write down the maximum and minimum value of $v$ .
17M.2.SL.TZ2.S_4c:
Use the model to find the depth of the water 10 hours after high tide.
19M.2.SL.TZ1.S_8b:
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17N.2.SL.TZ0.S_10a:
Show that $f(2\pi ) = 2\pi$ .
19M.2.AHL.TZ1.H_10c:
Sketch the graph of $y = p\left( t \right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.
EXM.3.AHL.TZ0.5f.ii:
Show that $\frac{a}{b} = {\text{tan}}\,\theta$ .
17M.2.SL.TZ1.S_8a.i:
How much time is there between the first low tide and the next high tide?
EXM.3.AHL.TZ0.5c.ii:
Comment on your answer to part (c)(i).
EXM.3.AHL.TZ0.5e.i:
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EXM.3.AHL.TZ0.5b.ii:
Find the value of $C$ .
17N.2.SL.TZ0.S_10d:
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EXM.3.AHL.TZ0.5e.iii:
$C$ .
18M.2.SL.TZ1.S_10e:
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19M.1.AHL.TZ2.H_9a:
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19M.2.AHL.TZ1.H_10d:
Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.
17M.2.SL.TZ1.S_8a.ii:
Find the difference in height between low tide and high tide.
17N.2.SL.TZ0.S_10b.ii:
Find the equation of $L$ .
18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
19M.2.AHL.TZ1.H_10g:
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EXM.3.AHL.TZ0.5a.i:
Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

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(iii) Find the value of $a$ .
19M.1.SL.TZ2.S_7b:
Consider the graph of $g\left( x \right) = 2\,{\text{sin}}\,px$ , 0 ≤ $x$ < $2\pi$ , where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y = - 1$ .
EXM.3.AHL.TZ0.5c.i:
Find ${\text{arctan}}\frac{3}{4}$ , giving the answer to 3 significant figures.
EXM.3.AHL.TZ0.5g:
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17N.2.SL.TZ0.S_10b.i:
Find the coordinates of ${{\text{P}}_0}$ and of ${{\text{P}}_1}$ .
22M.1.SL.TZ1.6b:
The $y$ -intercept of the graph of $g$ is at $(0, r)$ .

Given that $g (x) \geq 7$ , find the smallest value of $r$ .
EXN.1.SL.TZ0.9b:
Find the minimum height of the ball above the ground.
EXN.1.SL.TZ0.9d:
For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8 + 0.2 \sqrt{2}$ metres above the ground.
17N.2.SL.TZ0.S_10c:
Show that the distance between the $x$ -coordinates of ${{\text{P}}_k}$ and ${{\text{P}}_{k + 1}}$ is $2\pi$ .
19M.2.AHL.TZ1.H_10b:
Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .
22M.2.AHL.TZ2.10b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
20N.2.SL.TZ0.S_8a.iii:
Hence find the number of rotations the water wheel makes in one hour.
EXM.3.AHL.TZ0.5b.i:
Use your answers from part (a) to write down the value of $A$ , $B$ and $D$ .
20N.2.SL.TZ0.S_8b:
Find $k$ .
20N.2.SL.TZ0.S_8d:
Determine the rate of change of $h$ when the top of the bucket is at $B$ .
EXM.3.AHL.TZ0.5f.i:
Show that $\frac{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta$ .
19M.2.SL.TZ1.S_8c.ii:
Find the period of $g$ .
19M.2.SL.TZ1.S_8d:
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18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
19M.1.AHL.TZ2.H_9c:
At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$ -coordinate of A on the graph of $g$ .
17M.2.SL.TZ2.S_4a:
Find the value of $p$ .
SPM.2.SL.TZ0.9b.ii:
Hence, find the value of $f$ (6).
21M.1.SL.TZ2.8a:
Find the $x$ -coordinates of $A$ and $B$ .
19M.1.AHL.TZ2.H_9b:
Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .
17M.2.SL.TZ1.S_8b.ii:
Find the value of $q$ ;
SPM.2.SL.TZ0.9b.i:
Find the value of $b$ .
EXN.1.SL.TZ0.9a:
Find the height of the ball above the ground when it is released.
EXN.1.SL.TZ0.9c:
Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.
20N.2.SL.TZ0.S_8c:
Find $A \hat{O} B$ .
19M.2.AHL.TZ1.H_10e:
Find ${p_{av}}$ (0.007).
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Calculate the number of seconds it takes for the water wheel to complete one rotation.
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Find the height of point $A$ above the ground.
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17M.2.SL.TZ2.S_4b:
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21M.2.SL.TZ2.7d:
Find the value of $p$ and the value of $q$ .
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
21M.2.SL.TZ2.7e:
If the observer stands directly under point $C$ for one minute, point $D$ will pass over his head $n$ times.

Find the value of $n$ .
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.1.SL.TZ1.6a:
Describe these two transformations.
21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
EXM.3.AHL.TZ0.5a.ii:
Write down the amplitude of this graph
EXM.3.AHL.TZ0.5d:
By considering the graph of $y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x$ , find the value of $A$ , $B$ , $C$ and $D$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
17M.2.SL.TZ1.S_8b.i:
Find the value of $p$ ;
EXM.3.AHL.TZ0.5e.ii:
$B$ .
EXM.3.AHL.TZ0.5e.iv:
$D$ .
17M.2.SL.TZ1.S_8b.iii:
Find the value of $r$ .
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
SPM.2.SL.TZ0.9d:
Find the value of $x$ for which the functions have the greatest difference.
SPM.2.SL.TZ0.9c:
Find the value of $p$ and the value of $q$ .
21M.2.SL.TZ1.4:
A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $P$ is a point on the wheel. The wheel starts moving with $P$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $P$ above the ground after $t$ minutes is given by $h (t) = a \cos (b t) + c$ , where $a, b, c \in ℝ$ .

Find the values of $a$ , $b$ and $c$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

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Topic 3— Geometry and trigonometry

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Examiners report

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