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Date	May 2021	Marks available	3	Reference code	21M.1.AHL.TZ2.12
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Verify	Question number	12	Adapted from	N/A

Question

The following diagram shows the graph of $y = arctan (2 x + 1) + \frac{π}{4}$ for $x \in ℝ$ , with asymptotes at $y = - \frac{π}{4}$ and $y = \frac{3 π}{4}$ .

Describe a sequence of transformations that transforms the graph of $y = arctan x$ to the graph of $y = arctan (2 x + 1) + \frac{π}{4}$ for $x \in ℝ$ .

[3]

Show that $arctan p + arctan q \equiv arctan (\frac{p + q}{1 - p q})$ where $p, q > 0$ and $p q < 1$ .

[4]

Verify that $arctan (2 x + 1) = arctan (\frac{x}{x + 1}) + \frac{π}{4}$ for $x \in ℝ, x > 0$ .

[3]

Using mathematical induction and the result from part (b), prove that $Σ_{r = 1}^{n} arctan (\frac{1}{2 r^{2}}) = arctan (\frac{n}{n + 1})$ for $n \in ℤ^{+}$ .

[9]

Markscheme

EITHER
horizontal stretch/scaling with scale factor $\frac{1}{2}$

Note: Do not allow ‘shrink’ or ‘compression’

followed by a horizontal translation/shift $\frac{1}{2}$ units to the left A2

Note: Do not allow ‘move’

horizontal translation/shift $1$ unit to the left

followed by horizontal stretch/scaling with scale factor $\frac{1}{2}$ A2

THEN

vertical translation/shift up by $\frac{π}{4}$ (or translation through $(\begin{matrix} 0 \\ \frac{π}{4} \end{matrix})$ A1
(may be seen anywhere)

[3 marks]

let $α = arctan p$ and $β = arctan q$ M1

$p = tan α$ and $q = tan β$ (A1)

$\tan (α + β) = \frac{p + q}{1 - p q}$ A1

$α + β = arctan (\frac{p + q}{1 - p q})$ A1

so $arctan p + arctan q \equiv arctan (\frac{p + q}{1 - p q})$ where $p, q > 0$ and $p q < 1$ . AG

[4 marks]

METHOD 1

$\frac{π}{4} = arctan 1$ (or equivalent) A1

$arctan (\frac{x}{x + 1}) + arctan 1 = arctan (\frac{\frac{x}{x + 1} + 1}{1 - \frac{x}{x + 1} (1)})$ A1

$= arctan (\frac{\frac{x + x + 1}{x + 1}}{\frac{x + 1 - x}{x + 1}})$ A1

$= arctan (2 x + 1)$ AG

METHOD 2

$\tan \frac{π}{4} = 1$ (or equivalent) A1

Consider $arctan (2 x + 1) - arctan (\frac{x}{x + 1}) = \frac{π}{4}$

$\tan (arctan (2 x + 1) - arctan (\frac{x}{x + 1}))$

$= arctan (\frac{2 x +1 - \frac{x}{x + 1}}{1 + \frac{x (2 x + 1)}{x + 1}})$ A1

$= arctan (\frac{(2 x +1) (x + 1) - x}{x + 1 + x (2 x + 1)})$ A1

$= arctan 1$ AG

METHOD 3

$tan (arctan (2 x + 1)) = \tan (arctan (\frac{x}{x + 1}) + \frac{π}{4})$

$\tan \frac{π}{4} = 1$ (or equivalent) A1

$LHS = 2 x + 1$ A1

$RHS = \frac{\frac{x}{x + 1} + 1}{1 - \frac{x}{x + 1}} (= 2 x + 1)$ A1

[3 marks]

let $P (n)$ be the proposition that $Σ_{r = 1}^{n} arctan (\frac{1}{2 r^{2}}) = arctan (\frac{n}{n + 1})$ for $n \in ℤ^{+}$

consider $P (1)$

when $n = 1, Σ_{r = 1}^{1} arctan (\frac{1}{2 r^{2}}) = arctan (\frac{1}{2}) = RHS$ and so $P (1)$ is true R1

assume $P (k)$ is true, ie. $Σ_{r = 1}^{k} arctan (\frac{1}{2 r^{2}}) = arctan (\frac{k}{k + 1}) (k \in ℤ^{+})$ M1

Note: Award M0 for statements such as “let $n = k$ ”.
Note: Subsequent marks after this M1 are independent of this mark and can be awarded.

consider $P (k + 1)$ :

$Σ_{r = 1}^{k + 1} arctan (\frac{1}{2 r^{2}}) = Σ_{r = 1}^{k} arctan (\frac{1}{2 r^{2}}) + arctan (\frac{1}{2 {(k + 1)}^{2}})$ (M1)

$= arctan (\frac{k}{k + 1}) + arctan (\frac{1}{2 {(k + 1)}^{2}})$ A1

$= arctan (\frac{\frac{k}{k + 1} + \frac{1}{2 {(k + 1)}^{2}}}{1 - (\frac{k}{k + 1}) (\frac{1}{2 {(k + 1)}^{2}})})$ M1

$= arctan (\frac{(k + 1) (2 k^{2} + 2 k + 1)}{2 {(k + 1)}^{3} - k})$ A1

Note: Award A1 for correct numerator, with $(k + 1)$ factored. Denominator does not need to be simplified

$= arctan (\frac{(k + 1) (2 k^{2} + 2 k + 1)}{2 k^{3} + 6 k^{2} + 5 k + 2})$ A1

Note: Award A1 for denominator correctly expanded. Numerator does not need to be simplified. These two A marks may be awarded in any order

$= arctan (\frac{(k + 1) (2 k^{2} + 2 k + 1)}{(k + 2) (2 k^{2} + 2 k + 1)}) = arctan (\frac{k + 1}{k + 2})$ A1

Note: The word ‘arctan’ must be present to be able to award the last three A marks

$P (k + 1)$ is true whenever $P (k)$ is true and $P (1)$ is true, so

$P (n)$ is true for for $n \in ℤ^{+}$ R1

Note: Award the final R1 mark provided at least four of the previous marks have been awarded.
Note: To award the final R1, the truth of $P (k)$ must be mentioned. ‘ $P (k)$ implies $P (k + 1)$ ’ is insufficient to award the mark.

[9 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.9—Reciprocal trig ratios and their pythagorean identities. Inverse circular functions

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18N.1.AHL.TZ0.H_3b:
Write down the least value of $a$ such that $g$ has an inverse.
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
SPM.3.AHL.TZ0.2b.ii:
local minimum points;
18N.1.AHL.TZ0.H_10b:
Hence, show that $\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c$ , $c \in \mathbb{R}$ .
SPM.2.AHL.TZ0.12b:
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SPM.3.AHL.TZ0.2d.ii:
local minimum points.
17M.1.AHL.TZ1.H_3:
Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$ .
18N.1.AHL.TZ0.H_10a:
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SPM.3.AHL.TZ0.2c:
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22M.1.AHL.TZ2.7:
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22M.1.AHL.TZ2.11c:
Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .
22M.2.AHL.TZ2.7b:
Using l’Hôpital’s rule, show algebraically that the value of the limit is $- \frac{1}{4}$ .
SPM.2.AHL.TZ0.12d:
Using the results from parts (b) and (c) find the exact value of ${\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}$ .

Give your answer in the form $a + b\sqrt 3$ where $a$ , $b \in \mathbb{Z}$ .
21M.1.AHL.TZ1.6:
It is given that $cosec θ = \frac{3}{2}$ , where $\frac{π}{2} < θ < \frac{3 π}{2}$ . Find the exact value of $cot θ$ .
SPM.3.AHL.TZ0.2g:
Use an appropriate trigonometric identity to show that ${f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)$ .
SPM.3.AHL.TZ0.2e:
Solve the equation ${f_n}^\prime (x) = 0$ and hence show that the stationary points on the graph of $y = {f_n}(x)$ occur at $x = {\text{cos}}\frac{{k\pi }}{n}$ where $k \in {\mathbb{Z}^ + }$ and 0 < $k$ < $n$ .
SPM.3.AHL.TZ0.2f:
Use an appropriate trigonometric identity to show that ${f_2}(x) = 2{x^2} - 1$ .
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
18N.1.AHL.TZ0.H_3c.i:
For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .
SPM.3.AHL.TZ0.2a:
On the same set of axes, sketch the graphs of $y = {f_1}(x)$ and $y = {f_3}(x)$ for −1 ≤ $x$ ≤ 1.
21M.2.AHL.TZ2.12f:
Sketch the graph of $y = g^{- 1} (x)$ , clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.
21M.2.AHL.TZ2.12a:
Show that $f$ is an even function.
18N.1.AHL.TZ0.H_10d:
Find the area enclosed by the curve and the $x$ -axis between B and D, as shaded on the diagram.
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
21M.2.AHL.TZ2.12d:
Find an expression for $g^{- 1} (x)$ , justifying your answer.
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21M.1.AHL.TZ2.12b:
Show that $arctan p + arctan q \equiv arctan (\frac{p + q}{1 - p q})$ where $p, q > 0$ and $p q < 1$ .
21M.2.AHL.TZ2.12c.ii:
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22M.2.AHL.TZ2.7a:
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18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
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local maximum points;
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21M.2.AHL.TZ2.12c.i:
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18N.1.AHL.TZ0.H_10c:
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21M.2.AHL.TZ2.12e:
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SPM.3.AHL.TZ0.2h.ii:
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local maximum points;
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21N.2.AHL.TZ0.7b:
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