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Date	May 2017	Marks available	2	Reference code	17M.2.SL.TZ2.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Write down	Question number	S_8	Adapted from	N/A

Question

Let $f(x) = - 0.5{x^4} + 3{x^2} + 2x$ . The following diagram shows part of the graph of $f$ .

M17/5/MATME/SP2/ENG/TZ2/08

There are $x$ -intercepts at $x = 0$ and at $x = p$ . There is a maximum at A where $x = a$ , and a point of inflexion at B where $x = b$ .

Find the value of $p$ .

[2]

Write down the coordinates of A.

[2]

b.i.

Write down the rate of change of $f$ at A.

[1]

b.ii.

Find the coordinates of B.

[4]

c.i.

Find the the rate of change of $f$ at B.

[3]

c.ii.

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.

[3]

Markscheme

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

evidence of valid approach (M1)

eg $\,\,\,\,\,$ $f(x) = 0,{\text{ }}y = 0$

2.73205

$p = 2.73$ A1 N2

[2 marks]

1.87938, 8.11721

$(1.88,{\text{ }}8.12)$ A2 N2

[2 marks]

b.i.

rate of change is 0 (do not accept decimals) A1 N1

[1 marks]

b.ii.

METHOD 1 (using GDC)

valid approach M1

eg $\,\,\,\,\,$ $f’’ = 0$ , max/min on $f’,{\text{ }}x = - 1$

sketch of either $f^{'}$ or $f^{″}$ , with max/min or root (respectively) (A1)

$x = 1$ A1 N1

Substituting their $x$ value into $f$ (M1)

eg $\,\,\,\,\,$ $f(1)$

$y = 4.5$ A1 N1

METHOD 2 (analytical)

$f’’ = - 6{x^2} + 6$ A1

setting $f’’ = 0$ (M1)

$x = 1$ A1 N1

substituting their $x$ value into $f$ (M1)

eg $\,\,\,\,\,$ $f(1)$

$y = 4.5$ A1 N1

[4 marks]

c.i.

recognizing rate of change is $f^{'}$ (M1)

eg $\,\,\,\,\,$ $y’,{\text{ }}f’(1)$

rate of change is 6 A1 N2

[3 marks]

c.ii.

attempt to substitute either limits or the function into formula (M1)

involving ${f^2}$ (accept absence of $\pi$ and/or ${\text{d}}x$ )

eg $\,\,\,\,\,$ $\pi \int {{{( - 0.5{x^4} + 3{x^2} + 2x)}^2}{\text{d}}x,{\text{ }}\int_1^{1.88} {{f^2}} }$

128.890

${\text{volume}} = 129$ A2 N3

[3 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

Syllabus sections

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

Show 29 related questions

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EXM.2.SL.TZ0.1c:
State the name for this type of sampling technique.
EXM.2.SL.TZ0.1a:
State the name for this type of sampling technique.
EXM.2.SL.TZ0.1d.i:
Calculate Pearson’s product moment correlation coefficient, $r$ .
18M.1.SL.TZ1.S_2a:
Find the value of the interquartile range.
19M.2.SL.TZ1.S_3b:
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17M.2.SL.TZ1.S_1b.i:
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17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
EXM.2.SL.TZ0.1b.i:
Show that 3 students will be selected from grade 12.
EXM.2.SL.TZ0.1d.iii:
Explain why the value of $r$ makes it appropriate to find the equation of a regression line.
EXM.2.SL.TZ0.1b.ii:
Calculate the number of students in each grade in the sample.
EXM.1.SL.TZ0.2b:
Hence, show that a data set with only 5 numbers in it cannot have any outliers.
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
EXM.2.SL.TZ0.1d.ii:
Interpret the meaning of the value of $r$ in the context of the principal’s concerns.
EXM.1.SL.TZ0.2a:
Recalling definitions, such as the Lower Quartile is the $\frac{{n + 1}}{4}th$ piece of data with the data placed in order, find an expression for the Interquartile Range.
17M.2.SL.TZ2.S_8a:
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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.
17M.2.SL.TZ1.S_1a.ii:
Find the value of the range.
22M.2.SL.TZ2.5b:
Verify that the measurement of $0.46$ seconds is not an outlier.
17M.2.SL.TZ1.S_1a.i:
Write down the mode.
21M.1.SL.TZ1.4b:
Hence, find the minimum possible value of $L$ .
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.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
21M.1.SL.TZ2.7e.i:
Explain why this sampling method might not provide an accurate representation of the amount of time all of the students in the school spend doing homework.
21M.1.SL.TZ2.7e.ii:
Suggest a more appropriate sampling method.
EXM.2.SL.TZ0.1e:
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By finding the equation of an appropriate regression line, estimate the time Jasmine spent on social media the previous day.
21M.1.SL.TZ1.4a:
Find the minimum possible value of $U$ .
22M.1.SL.TZ1.3a:
Find the largest value of $c$ that would not be considered an outlier.
17M.2.SL.TZ1.S_1b.ii:
Find the variance.

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