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Date	May 2022	Marks available	3	Reference code	22M.2.SL.TZ2.8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	8	Adapted from	N/A

Question

A scientist conducted a nine-week experiment on two plants, $A$ and $B$ , of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant $A$ was given fertilizer regularly, while Plant $B$ was not.

The scientist found that the height of Plant $A, h_{A} cm$ , at time $t$ weeks can be modelled by the function $h_{A} (t) = \sin (2 t + 6) + 9 t + 27$ , where $0 \leq t \leq 9$ .

The scientist found that the height of Plant $B, h_{B} cm$ , at time $t$ weeks can be modelled by the function $h_{B} (t) = 8 t + 32$ , where $0 \leq t \leq 9$ .

Use the scientist’s models to find the initial height of

Plant $B$ .

[1]

a.i.

Plant $A$ correct to three significant figures.

[2]

a.ii.

Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .

[3]

For $0 \leq t \leq 9$ , find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$ .

[6]

Markscheme

$32$ (cm) A1

[1 mark]

a.i.

$h_{A} (0) = \sin (6) + 27$ (M1)

$= 26.7205 \dots$

$= 26.7$ (cm) A1

[2 marks]

a.ii.

attempts to solve $h_{A} (t) = h_{B} (t)$ for $t$ (M1)

$t = 4.00746 \dots, 4.70343 \dots, 5.88332 \dots$

$t = 4.01, 4.70, 5.88$ (weeks) A2

[3 marks]

recognises that $h_{A}' (t)$ and $h_{B}' (t)$ are required (M1)

attempts to solve $h_{A}' (t) = h_{B}' (t)$ for $t$ (M1)

$t = 1.18879 \dots$ and $2.23598 \dots$ OR $4.33038 \dots$ and $5.37758 \dots$ OR $7.47197 \dots$ and $8.51917 \dots$ (A1)

Note: Award full marks for $t = \frac{4 π}{3} - 3, \frac{5 π}{3} - 3, (\frac{7 π}{3} - 3, \frac{8 π}{3} - 3 \frac{10 π}{3} - 3, \frac{11 π}{3} - 3)$ .

Award subsequent marks for correct use of these exact values.

$1.18879 \dots < t < 2.23598 \dots$ OR $4.33038 \dots < t < 5.37758 \dots$ OR

$7.47197 \dots < t < 8.51917 \dots$ (A1)

attempts to calculate the total amount of time (M1)

$3 (2.2359 \dots - 1.1887 \dots) (= 3 ((\frac{5 π}{3} - 3) - (\frac{4 π}{3} - 3)))$

$= 3.14 (= π)$ (weeks) A1

[6 marks]

Examiners report

Many students did not change their calculators back to radian mode. This meant they had no chance of correctly answering parts (c) and (d), since even if follow through was given, there were not enough intersections on the graphs.

Most managed part (a) and some attempted to equate the functions in part b) but few recognised that 'rate of growth' was the derivatives of the given functions, and of those who did, most were unable to find them.

Almost all the candidates who did solve part (c) gave the answer $3 \times 1.05 = 3.15$ , when working with more significant figures would have given them 3.14. They lost the last mark.

a.i.

[N/A]

a.ii.

[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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18M.2.SL.TZ1.S_10a:
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18M.2.SL.TZ1.S_10b.i:
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17M.2.SL.TZ2.S_4c:
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19M.2.SL.TZ1.S_8b:
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19M.2.AHL.TZ1.H_10c:
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Show that $\frac{a}{b} = {\text{tan}}\,\theta$ .
17M.2.SL.TZ1.S_8a.i:
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EXM.3.AHL.TZ0.5c.ii:
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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:
$C$ .
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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:
Given that $p\left( t \right)$ can be written as $p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d$ where $a$ , $b$ , $c$ , $d$ > 0, use your graph to find the values of $a$ , $b$ , $c$ and $d$ .
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 $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(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:
Hence prove your conjectures in part (e).
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 $r$ .
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Find the minimum height of the ball above the ground.
EXN.1.SL.TZ0.9d:
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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:
The equation $g\left( x \right) = 12$ has two solutions where $\pi$ ≤ $x$ ≤ ${\frac{{4\pi }}{3}}$ . Find both solutions.
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).
20N.2.SL.TZ0.S_8a.ii:
Calculate the number of seconds it takes for the water wheel to complete one rotation.
20N.2.SL.TZ0.S_8a.i:
Find the height of point $A$ above the ground.
19M.2.AHL.TZ1.H_10f:
With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.
17M.2.SL.TZ2.S_4b:
Find the value of $q$ .
19M.2.SL.TZ1.S_8a:
The range of $f$ is $k$ ≤ $f\left( x \right)$ ≤ $m$ . Find $k$ and $m$ .
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:
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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 $r$ .
17M.2.SL.TZ1.S_8c:
There are two high tides on 12 December 2017. At what time does the second high tide occur?
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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Markscheme

Examiners report

Syllabus sections

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