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Date	November Example questions	Marks available	4	Reference code	EXN.1.AHL.TZ0.11
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Sketch	Question number	11	Adapted from	N/A

Question

A function $f$ is defined by $f (x) = \frac{3}{x^{2} + 2}, x \in ℝ$ .

The region $R$ is bounded by the curve $y = f (x)$ , the $x$ -axis and the lines $x = 0$ and $x = \sqrt{6}$ . Let $A$ be the area of $R$ .

The line $x = k$ divides $R$ into two regions of equal area.

Let $m$ be the gradient of a tangent to the curve $y = f (x)$ .

Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

[4]

Show that $A = \frac{\sqrt{2} π}{2}$ .

[4]

Find the value of $k$ .

[4]

Show that $m = - \frac{6 x}{{(x^{2} + 2)}^{2}}$ .

[2]

Show that the maximum value of $m$ is $\frac{27}{32} \sqrt{\frac{2}{3}}$ .

[7]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

a curve symmetrical about the $y$ -axis with correct concavity that has a local maximum point on the positive $y$ -axis A1

a curve clearly showing that $y \to 0$ as $x \to \pm \infty$ A1

$(0, \frac{3}{2})$ A1

horizontal asymptote $y = 0$ ( $x$ -axis) A1

[4 marks]

attempts to find $\int \frac{3}{x^{2} + 2} d x$ (M1)

$= [\frac{3}{\sqrt{2}} \arctan \frac{x}{\sqrt{2}}]$ A1

Note: Award M1A0 for obtaining $[k \arctan \frac{x}{\sqrt{2}}]$ where $k \neq \frac{3}{\sqrt{2}}$ .

Note: Condone the absence of or use of incorrect limits to this stage.

$= \frac{3}{\sqrt{2}} (\arctan \sqrt{3} - \arctan 0)$ (M1)

$= \frac{3}{\sqrt{2}} \times \frac{π}{3} (= \frac{π}{\sqrt{2}})$ A1

$A = \frac{\sqrt{2} π}{2}$ AG

[4 marks]

METHOD 1

EITHER

$\int_{0}^{k} \frac{3}{x^{2} + 2} d x = \frac{\sqrt{2} π}{4}$

$\frac{3}{\sqrt{2}} \arctan \frac{k}{\sqrt{2}} = \frac{\sqrt{2} π}{4}$ (M1)

$\int_{k}^{\sqrt{6}} \frac{3}{x^{2} + 2} d x = \frac{\sqrt{2} π}{4}$

$\frac{3}{\sqrt{2}} (\arctan \sqrt{3} - \arctan \frac{k}{\sqrt{2}}) = \frac{\sqrt{2} π}{4}$ (M1)

$\arctan \sqrt{3} - \arctan \frac{k}{\sqrt{2}} = \frac{π}{6}$

THEN

$\arctan \frac{k}{\sqrt{2}} = \frac{π}{6}$ A1

$\frac{k}{\sqrt{2}} = \tan \frac{π}{6} (= \frac{1}{\sqrt{3}})$ A1

$k = \frac{\sqrt{6}}{3} (= \sqrt{\frac{2}{3}})$ A1

METHOD 2

$\int_{0}^{k} \frac{3}{x^{2} + 2} d x = \int_{k}^{\sqrt{6}} \frac{3}{x^{2} + 2} d x$

$\frac{3}{\sqrt{2}} \arctan \frac{k}{\sqrt{2}} = \frac{3}{\sqrt{2}} (\arctan \sqrt{3} - \arctan \frac{k}{\sqrt{2}})$ (M1)

$\arctan \frac{k}{\sqrt{2}} = \frac{π}{6}$ A1

$\frac{k}{\sqrt{2}} = \tan \frac{π}{6} (= \frac{1}{\sqrt{3}})$ A1

$k = \frac{\sqrt{6}}{3} (= \sqrt{\frac{2}{3}})$ A1

[4 marks]

attempts to find $\frac{d}{d x} (\frac{3}{x^{2} + 2})$ (M1)

$= (3) (- 1) (2 x) {(x^{2} + 2)}^{- 2}$ A1

so $m = - \frac{6 x}{{(x^{2} + 2)}^{2}}$ AG

[2 marks]

attempts product rule or quotient rule differentiation M1

EITHER

$\frac{d m}{d x} = (- 6 x) (- 2) (2 x) {(x^{2} + 2)}^{- 3} + {(x^{2} + 2)}^{- 2} (- 6)$ A1

$\frac{d m}{d x} = \frac{{(x^{2} + 2)}^{2} (- 6) - (- 6 x) (2) (2 x) (x^{2} + 2)}{{(x^{2} + 2)}^{4}}$ A1

Note: Award A0 if the denominator is incorrect. Subsequent marks can be awarded.

THEN

attempts to express their $\frac{d m}{d x}$ as a rational fraction with a factorized numerator M1

$\frac{d m}{d x} = \frac{6 (x^{2} + 2) (3 x^{2} - 2)}{{(x^{2} + 2)}^{4}} (= \frac{6 (3 x^{2} - 2)}{{(x^{2} + 2)}^{3}})$

attempts to solve their $\frac{d m}{d x} = 0$ for $x$ M1

$x = \pm \sqrt{\frac{2}{3}}$ A1

from the curve, the maximum value of $m$ occurs at $x = - \sqrt{\frac{2}{3}}$ R1

(the minimum value of $m$ occurs at $x = \sqrt{\frac{2}{3}}$ )

Note: Award R1 for any equivalent valid reasoning.

maximum value of $m$ is $- \frac{6 (- \sqrt{\frac{2}{3}})}{{({(- \sqrt{\frac{2}{3}})}^{2} + 2)}^{2}}$ A1

leading to a maximum value of $\frac{27}{32} \sqrt{\frac{2}{3}}$ AG

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology

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22M.1.SL.TZ2.4a.ii:
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22M.1.SL.TZ2.4a.i:
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22M.1.SL.TZ2.4b:
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21M.2.SL.TZ2.2a:
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Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
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Find the value of $k$ , giving your answer correct to four significant figures.
EXN.2.SL.TZ0.9f:
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Find the value of $x$ for which $f' (x) = 0$ .
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Find the height of the water at $12 : 00$ .
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Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.
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Find the coordinates of the bridges relative to the centre of Orangeton.
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Write down the coordinates of the local minimum point.
19M.2.SL.TZ1.T_4e:
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State whether the straight road will ever cross the river. Justify your answer.
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Write down the $y$ -intercept of the graph.
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Show that $a = 8$ .
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State the domain of $P$ .
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Find $f'(x)$ .
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The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
21N.2.SL.TZ0.7a.i:
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21N.2.SL.TZ0.7b.ii:
Find the value of $x$ so that $T$ is a minimum.
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Find the new value of $x$ so that the total cost $C$ to travel from $A$ to $B$ via $D$ is a minimum.
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.
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Determine the value of $m$ .
21N.2.AHL.TZ0.7b:
Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .
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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Find the function, $P\left( x \right)$ , that would define this footpath on the map.
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Find $f(2)$ .
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Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .

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