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Date	November 2018	Marks available	3	Reference code	18N.2.SL.TZ0.S_10
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_10	Adapted from	N/A

Question

All lengths in this question are in metres.

Consider the function $f\left( x \right) = \sqrt {\frac{{4 - {x^2}}}{8}}$ , for −2 ≤ $x$ ≤ 2. In the following diagram, the shaded region is enclosed by the graph of $f$ and the $x$ -axis.

A container can be modelled by rotating this region by 360˚ about the $x$ -axis.

Water can flow in and out of the container.

The volume of water in the container is given by the function $g\left( t \right)$ , for 0 ≤ $t$ ≤ 4 , where $t$ is measured in hours and $g\left( t \right)$ is measured in m³. The rate of change of the volume of water in the container is given by $g'\left( t \right) = 0.9 - 2.5\,{\text{cos}}\left( {0.4{t^2}} \right)$ .

The volume of water in the container is increasing only when $p$ < $t$ < $q$ .

Find the volume of the container.

[3]

a.

Find the value of $p$ and of $q$ .

[3]

b.i.

During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .

[3]

b.ii.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

[5]

c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to substitute correct limits or the function into formula involving ${f^2}$ (M1)

eg $\pi \int_{ - 2}^2 {{y^2}\,{\text{d}}y}$ , $\pi {\int {\left( {\sqrt {\frac{{4 - {x^2}}}{8}} } \right)} ^2}{\text{d}}x$

4.18879

volume = 4.19, $\frac{4}{3}\pi$ (exact) (m³) A2 N3

Note: If candidates have their GDC incorrectly set in degrees, award M marks where appropriate, but no A marks may be awarded. Answers from degrees are p = 13.1243 and q = 26.9768 in (b)(i) and 12.3130 or 28.3505 in (b)(ii).

[3 marks]

a.

recognizing the volume increases when ${g'}$ is positive (M1)

eg $g'\left( t \right)$ > 0, sketch of graph of ${g'}$ indicating correct interval

1.73387, 3.56393

$p$ = 1.73, $p$ = 3.56 A1A1 N3

[3 marks]

b.i.

valid approach to find change in volume (M1)

eg $g\left( q \right) - g\left( p \right)$ , $\int_p^q {g'\left( t \right){\text{d}}t}$

3.74541

total amount = 3.75 (m³) A2 N3

[3 marks]

b.ii.

Note: There may be slight differences in the final answer, depending on which values candidates carry through from previous parts. Accept answers that are consistent with correct working.

recognizing when the volume of water is a maximum (M1)

eg maximum when $t = q$ , $\int_0^q {g'\left( t \right){\text{d}}t}$

valid approach to find maximum volume of water (M1)

eg $2.3 + \int_0^q {g'\left( t \right){\text{d}}t}$ , $2.3 + \int_0^p {g'\left( t \right){\text{d}}t} + 3.74541$ , 3.85745

correct expression for the difference between volume of container and maximum value (A1)

eg $4.18879 - \left( {2.3 + \int_0^q {g'\left( t \right){\text{d}}t} } \right)$ , 4.19 − 3.85745

0.331334

0.331 (m³) A2 N3

[5 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques

Show 80 related questions

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18N.2.AHL.TZ0.H_10d:
Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.
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State the hypotheses for this t-test.
22M.1.SL.TZ2.7a:
Show that the test score of $25$ would not be considered an outlier.
18M.1.AHL.TZ2.H_3b.i:
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18N.2.AHL.TZ0.H_10a.ii:
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18N.2.AHL.TZ0.H_10b.i:
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18M.1.SL.TZ1.S_2a:
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17M.1.SL.TZ1.S_6a.i:
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17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
EXM.2.SL.TZ0.5f:
Find the p-value for this t-test.
EXM.2.SL.TZ0.5c.ii:
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19M.1.SL.TZ2.S_9d:
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Given that $M$ is parallel to $L$ , find the $x$ -coordinate of P.
18M.1.AHL.TZ2.H_3b.ii:
Find P(X > μ).
SPM.1.SL.TZ0.3b:
Find the value of $k$ .
EXM.2.SL.TZ0.5g:
State the conclusion of this test, in context, giving a reason.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
19M.2.SL.TZ1.S_9d.iii:
Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x = 8$ and the tangent line to $f$ at $x = 2$ .
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
18N.2.AHL.TZ0.H_10a.i:
Using this distribution model, find ${\text{P}}\left( {X < 60} \right)$ .
18M.1.SL.TZ1.S_2b:
One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.
18N.2.AHL.TZ0.H_10e:
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19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18N.2.AHL.TZ0.H_10c:
Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18N.1.SL.TZ0.S_10a:
Find the coordinates of P.
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

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

(iii) Find the value of $a$ .
EXM.2.SL.TZ0.5a:
State the name for this type of sampling technique.
SPM.1.SL.TZ0.3a:
State whether the data is discrete or continuous.
17M.1.SL.TZ1.S_6b:
Determine the concavity of the graph of $f$ when $4 < x < 5$ and justify your answer.
17M.1.SL.TZ1.S_9b:
Find the value of $a$ .
16N.2.AHL.TZ0.H_11d:
(i) Hence show that $X$ has two modes ${m_1}$ and ${m_2}$ .

(ii) State the values of ${m_1}$ and ${m_2}$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.S_6a.ii:
Find the equation of the normal to the curve of $f$ at P.
19M.1.SL.TZ2.S_9b:
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17M.1.SL.TZ1.S_9a:
Find the value of $p$ .
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
18N.1.SL.TZ0.S_10c:
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Find the value of $a$ .
16N.2.AHL.TZ0.H_11b:
Find the values of the constants $a$ and $b$ .
16N.2.AHL.TZ0.H_11a:
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18N.2.SL.TZ0.S_10c:
When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.
19M.1.SL.TZ2.S_9c:
Find the derivative of $f$ .
EXN.1.SL.TZ0.1b:
The Principal selects the students for the sample by asking those who took part in a previous survey if they would like to take part in another. She takes the first of those who reply positively, up to the maximum needed for the sample.

State which two of the sampling methods listed below best describe the method used.

Stratified Quota Convenience Systematic Simple random
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
17M.2.SL.TZ2.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
EXM.1.SL.TZ0.5b:
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17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
SPM.1.SL.TZ0.3c:
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Identify the sampling technique used in the survey.
18N.2.SL.TZ0.S_10a:
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19M.2.SL.TZ1.S_3a:
Find $f'\left( x \right)$ .
21M.2.SL.TZ1.1d:
Determine whether Jason is correct. Support your reasoning.
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Identify the type of sampling used by the restaurant manager.
16N.2.AHL.TZ0.H_11e:
Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.
EXM.1.SL.TZ0.5a:
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17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19M.2.SL.TZ1.S_3b:
The graph of $f$ has a horizontal tangent line at $x = 0$ and at $x = a$ . Find $a$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
21M.2.SL.TZ1.1a:
State which of the two sampling methods, systematic or quota, Jason has used.
18M.1.AHL.TZ2.H_3a:
Find the value of p.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and 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.3.AHL.TZ1.2b:
Show that $11$ employees are selected for the sample from the national department.
21M.2.SL.TZ2.1a:
State the sampling method being used.
16N.2.AHL.TZ0.H_11c:
Deduce that $\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}$ for $n > 3$ .
21M.3.AHL.TZ2.1b:
Juliet classifies response $K$ as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.
EXN.1.SL.TZ0.1a:
Calculate the number of grade $12$ students who should be in the sample.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
18M.1.SL.TZ1.S_7:
Consider f(x), g(x) and h(x), for x∈ $\mathbb{R}$ where h(x) = $\left( {f \circ g} \right)$ (x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

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