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Date	May 2021	Marks available	3	Reference code	21M.3.AHL.TZ2.1
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 2
Command term	Show that	Question number	1	Adapted from	N/A

Question

This question asks you to explore the behaviour and some key features of the function $f_{n} (x) = x^{n} (a - x)^{n}$ , where $a \in ℝ^{+}$ and $n \in ℤ^{+}$ .

In parts (a) and (b), only consider the case where $a = 2$ .

Consider $f_{1} (x) = x (2 - x)$ .

Consider $f_{n} (x) = x^{n} {(2 - x)}^{n}$ , where $n \in ℤ^{+}, n > 1$ .

Now consider $f_{n} (x) = x^{n} {(a - x)}^{n}$ where $a \in ℝ^{+}$ and $n \in ℤ^{+}, n > 1$ .

By using the result from part (f) and considering the sign of ${f_{n}}^{'} (- 1)$ , show that the point $(0, 0)$ on the graph of $y = f_{n} (x)$ is

Sketch the graph of $y = f_{1} (x)$ , stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

[3]

Use your graphic display calculator to explore the graph of $y = f_{n} (x)$ for

• the odd values $n = 3$ and $n = 5$ ;

• the even values $n = 2$ and $n = 4$ .

Hence, copy and complete the following table.

[6]

Show that ${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ .

[5]

State the three solutions to the equation ${f_{n}}^{'} (x) = 0$ .

[2]

Show that the point $(\frac{a}{2}, f_{n} (\frac{a}{2}))$ on the graph of $y = f_{n} (x)$ is always above the horizontal axis.

[3]

Hence, or otherwise, show that ${f_{n}}^{'} (\frac{a}{4}) > 0$ , for $n \in ℤ^{+}$ .

[2]

a local minimum point for even values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .

[3]

g.i.

a point of inflexion with zero gradient for odd values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .

[2]

g.ii.

Consider the graph of $y = x^{n} {(a - x)}^{n} - k$ , where $n \in ℤ^{+}$ , $a \in ℝ^{+}$ and $k \in ℝ$ .

State the conditions on $n$ and $k$ such that the equation $x^{n} {(a - x)}^{n} = k$ has four solutions for $x$ .

[5]

Markscheme

inverted parabola extended below the $x$ -axis A1

$x$ -axis intercept values $x = 0, 2$ A1

Note: Accept a graph passing through the origin as an indication of $x = 0$ .

local maximum at $(1, 1)$ A1

Note: Coordinates must be stated to gain the final A1.
Do not accept decimal approximations.

[3 marks]

A1A1A1A1A1A1

Note: Award A1 for each correct value.

For a table not sufficiently or clearly labelled, assume that their values are in the same order as the table in the question paper and award marks accordingly.

[6 marks]

METHOD 1

attempts to use the product rule (M1)

${f_{n}}^{'} (x) = - n x^{n} {(a - x)}^{n - 1} + n x^{n - 1} {(a - x)}^{n}$ A1A1

Note: Award A1 for a correct $u \frac{d v}{d x}$ and A1 for a correct $v \frac{d u}{d x}$ .

EITHER

attempts to factorise ${f_{n}}^{'} (x)$ (involving at least one of $n x^{n - 1}$ or ${(a - x)}^{n - 1}$ ) (M1)

$= n x^{n - 1} {(a - x)}^{n - 1} ((a - x) - x)$ A1

attempts to express ${f_{n}}^{'} (x)$ as the difference of two products with each product containing at least one of $n x^{n - 1}$ or ${(a - x)}^{n - 1}$ (M1)

$= (- x) (n x^{n - 1}) {(a - x)}^{n - 1} + (a - x) (n x^{n - 1}) {(a - x)}^{n - 1}$ A1

THEN

${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ AG

Note: Award the final (M1)A1 for obtaining any of the following forms:

${f_{n}}^{'} (x) = n x^{n} {(a - x)}^{n} (\frac{a - x - x}{x (a - x)}); {f_{n}}^{'} (x) = \frac{n x^{n} {(a - x)}^{n}}{x (a - x)} (a - x - x);$

${f_{n}}^{'} (x) = n x^{n - 1} ({(a - x)}^{n} - x {(a - x)}^{n - 1});$

${f_{n}}^{'} (x) = {(a - x)}^{n - 1} (n x^{n - 1} {(a - x)}^{n} - n x^{n})$

METHOD 2

$f_{n} (x) = {(x (a - x))}^{n}$ (M1)

$= {(a x - x^{2})}^{n}$ A1

attempts to use the chain rule (M1)

${f_{n}}^{'} (x) = n (a - 2 x) {(a x - x^{2})}^{n - 1}$ A1A1

Note: Award A1 for $n (a - 2 x)$ and A1 for ${(a x - x^{2})}^{n - 1}$ .

${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ AG

[5 marks]

$x = 0, x = \frac{a}{2}, x = a$ A2

Note: Award A1 for either two correct solutions or for obtaining $x = 0, x = - a, x = - \frac{a}{2}$
Award A0 otherwise.

[2 marks]

attempts to find an expression for $f_{n} (\frac{a}{2})$ (M1)

$f_{n} (\frac{a}{2}) = {(\frac{a}{2})}^{n} {(a - \frac{a}{2})}^{n}$

$= {(\frac{a}{2})}^{n} {(\frac{a}{2})}^{n} (= {(\frac{a}{2})}^{2 n}), (= {({(\frac{a}{2})}^{n})}^{2})$ A1

EITHER

since $a \in ℝ^{+}, {(\frac{a}{2})}^{2 n} > 0$ (for $n \in ℤ^{+}, n > 1$ and so $f_{n} (\frac{a}{2}) > 0$ ) R1

Note: Accept any logically equivalent conditions/statements on $a$ and $n$ .
Award R0 if any conditions/statements specified involving $a$ , $n$ or both are incorrect.

(since $a \in ℝ^{+}$ ), $\frac{a}{2}$ raised to an even power ( $2 n$ ) (or equivalent reasoning) is always positive (and so $f_{n} (\frac{a}{2}) > 0$ ) R1

Note: The condition $a \in ℝ^{+}$ is given in the question. Hence some candidates will assume $a \in ℝ^{+}$ and not state it. In these instances, award R1 for a convincing argument.
Accept any logically equivalent conditions/statements on on $a$ and $n$ .
Award R0 if any conditions/statements specified involving $a$ , $n$ or both are incorrect.

THEN

so $(\frac{a}{2}, f_{n} (\frac{a}{2}))$ is always above the horizontal axis AG

Note: Do not award (M1)A0R1.

[3 marks]

METHOD 1

${f_{n}}^{'} (\frac{a}{4}) = n {(\frac{a}{4})}^{n - 1} (a - \frac{a}{2}) {(a - \frac{a}{4})}^{n - 1} (= n {(\frac{a}{4})}^{n - 1} (\frac{a}{2}) {(\frac{3 a}{4})}^{n - 1})$ A1

EITHER

$n {(\frac{a}{4})}^{n - 1} (\frac{a}{2}) {(\frac{3 a}{4})}^{n - 1} > 0$ as $a \in ℝ^{+}$ and $n \in ℤ^{+}$ R1

$n {(\frac{a}{4})}^{n - 1}, (a - \frac{a}{2})$ and ${(a - \frac{a}{4})}^{n - 1}$ are all $> 0$ R1

Note: Do not award A0R1.
Accept equivalent reasoning on correct alternative expressions for ${f_{n}}^{'} (\frac{a}{4})$ and accept any logically equivalent conditions/statements on $a$ and $n$ .

Exceptions to the above are condone $n > 1$ and condone $n > 0$ .

An alternative form for ${f_{n}}^{'} (\frac{a}{4})$ is $(2 n) {(3)}^{n - 1} {(\frac{a}{4})}^{2 n - 1}$ .

THEN

hence ${f_{n}}^{'} (\frac{a}{4}) > 0$ AG

METHOD 2

$f_{n} (0) = 0$ and $f_{n} (\frac{a}{2}) > 0$ A1

(since $f_{n}$ is continuous and there are no stationary points between $x = 0$ and $x = \frac{a}{2}$ )

the gradient (of the curve) must be positive between $x = 0$ and $x = \frac{a}{2}$ R1

Note: Do not award A0R1.

hence ${f_{n}}^{'} (\frac{a}{4}) > 0$ AG

[2 marks]

${f_{n}}^{'} (- 1) = n {(- 1)}^{n - 1} (a + 2) {(a + 1)}^{n - 1}$

for $n$ even:

$n {(- 1)}^{n - 1} (= - n) < 0$ (and $(a + 2), {(a + 1)}^{n - 1}$ are both $> 0$ ) R1

${f_{n}}^{'} (- 1) < 0$ A1

${f_{n}}^{'} (0) = 0$ and ${f_{n}}^{'} (\frac{a}{4}) > 0$ (seen anywhere) A1

Note: Candidates can give arguments based on the sign of ${(- 1)}^{n - 1}$ to obtain the R mark.
For example, award R1 for the following:
If $n$ is even, then $n - 1$ is odd and hence ${(- 1)}^{n - 1} < 0 (= - 1)$ .
Do not award R0A1.
The second A1 is independent of the other two marks.
The A marks can be awarded for correct descriptions expressed in words.
Candidates can state $(0, 0)$ as a point of zero gradient from part (d) or show, state or explain (words or diagram) that ${f_{n}}^{'} (0) = 0$ . The last A mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.
The last A1 can be awarded for use of a specific case (e.g. $n = 2$ ).

hence $(0, 0)$ is a local minimum point AG

[3 marks]

g.i.

for $n$ odd:

$n {(- 1)}^{n - 1} (= n) < 0$ , (and $(a + 2), {(a + 1)}^{n - 1}$ are both $> 0$ ) so ${f_{n}}^{'} (- 1) > 0$ R1

Note: Candidates can give arguments based on the sign of ${(- 1)}^{n - 1}$ to obtain the R mark.
For example, award R1 for the following:
If $n$ is odd, then $n - 1$ is even and hence ${(- 1)}^{n - 1} > 0 (= 1)$ .

${f_{n}}^{'} (0) = 0$ and ${f_{n}}^{'} (\frac{a}{4}) > 0$ (seen anywhere) A1

Note: The A1 is independent of the R1.
Candidates can state $(0, 0)$ as a point of zero gradient from part (d) or show, state or explain (words or diagram) that ${f_{n}}^{'} (0) = 0$ . The last A mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.
The last A1 can be awarded for use of a specific case (e.g. $n = 3$ ).

hence $(0, 0)$ is a point of inflexion with zero gradient AG

[2 marks]

g.ii.

considers the parity of $n$ (M1)

Note: Award M1 for stating at least one specific even value of $n$ .

$n$ must be even (for four solutions) A1

Note: The above 2 marks are independent of the 3 marks below.

$0 < k < {(\frac{a}{2})}^{2 n}$ A1A1A1

Note: Award A1 for the correct lower endpoint, A1 for the correct upper endpoint and A1 for strict inequality signs.

The third A1 (strict inequality signs) can only be awarded if A1A1 has been awarded.
For example, award A1A1A0 for $0 \leq k \leq {(\frac{a}{2})}^{2 n}$ . Award A1A0A0 for $k > 0$ .

Award A1A0A0 for $0 < k < f_{n} (\frac{a}{2})$ .

[5 marks]

Examiners report

[N/A]

g.i.

[N/A]

g.ii.

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules

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21N.1.SL.TZ0.5a:
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19N.1.AHL.TZ0.H_10a.i:
Find $f'\left( x \right)$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
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17N.1.SL.TZ0.S_7:
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local minimum points.
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Find the maximum speed of P.
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
19M.2.SL.TZ1.S_4c:
Hence find the values of $x$ for which the graph of $f$ is concave-down.
19M.2.SL.TZ2.S_5a:
Find the population of fish at $t$ = 10.
SPM.3.AHL.TZ0.2c:
On a new set of axes, sketch the graphs of $y = {f_2}(x)$ and $y = {f_4}(x)$ for −1 ≤ $x$ ≤ 1.
19N.1.AHL.TZ0.H_10a.ii:
Show that, if $f'\left( x \right) = 0$ , then $x = 2 - \sqrt 3$ .
22M.3.AHL.TZ1.2d.i:
Show that $g' (x) = 2 (x - r) (x - a) + x^{2} - 2 a x + a^{2} + b^{2}$ .
22M.1.AHL.TZ1.12c.ii:
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22M.1.AHL.TZ2.6b:
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Find the value of $a$ and the value of $b$ .
22M.2.SL.TZ1.5c:
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22M.2.SL.TZ2.6b:
Find the times when the particle’s acceleration is $- 1.9 {m s}^{- 2}$ .
22M.2.SL.TZ2.6c:
Find the particle’s acceleration when its speed is at its greatest.
22M.2.SL.TZ2.8c:
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22M.2.AHL.TZ2.7b:
Using l’Hôpital’s rule, show algebraically that the value of the limit is $- \frac{1}{4}$ .
20N.2.SL.TZ0.S_10a:
Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .
21M.1.SL.TZ1.5c:
Hence, show that $k = e + \frac{e^{2}}{4}$ .
21M.1.SL.TZ1.8a:
Show that $\frac{d y}{d x} = \frac{1 - 4 \ln x}{x^{5}}$ .
21M.1.SL.TZ1.5a:
Find $f' (x)$ .
21M.1.SL.TZ1.5b:
Show that $h = \frac{e + 6}{2}$ .
21M.1.SL.TZ1.8b:
The graph of $f$ has a horizontal tangent at point $P$ . Find the coordinates of $P$ .
17M.2.AHL.TZ1.H_8b:
Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .
21M.2.SL.TZ1.9b:
Find the exact coordinates of $Q$ .
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 ${f_2}(x) = 2{x^2} - 1$ .
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
20N.2.SL.TZ0.S_10b:
Find the least value of $n$ .
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.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
21M.3.AHL.TZ2.1c:
Show that ${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ .
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.
SPM.1.SL.TZ0.9d:
The region R is enclosed by the graph of $f$ , the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of $k$ .
19M.2.SL.TZ2.S_5b:
Find the rate at which the population of fish is increasing at $t$ = 10.
SPM.1.SL.TZ0.9b:
Find the x-coordinate of P.
18M.2.SL.TZ2.S_9c:
Write down the number of times that the acceleration of P is 0 m s⁻² .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
20N.2.AHL.TZ0.H_8b:
At the point on the curve where $x = 4$ , it is given that $\frac{d y}{d t} = - 0.1 {m s}^{- 1}$

Find the value of $\frac{d x}{d t}$ at this exact same instant.
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
EXN.1.AHL.TZ0.11d:
Show that $m = - \frac{6 x}{{(x^{2} + 2)}^{2}}$ .
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
SPM.1.SL.TZ0.9c:
Show that the x-coordinate of Q is $\frac{{\text{1}}}{5}{{\text{e}}^{\frac{3}{2}}}$ .
21M.1.SL.TZ2.5b:
Given that the gradient of $L$ is $\frac{1}{3}$ , find the $x$ -coordinate of $B$ .
20N.2.SL.TZ0.S_10c:
Find $\int \frac{6 x}{x^{2} + 4} d x$ .
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
22M.1.SL.TZ1.5:
Consider the curve with equation $y = (2 x - 1) e^{k x}$ , where $x \in ℝ$ and $k \in ℚ$ .

The tangent to the curve at the point where $x = 1$ is parallel to the line $y = 5 e^{k} x$ .

Find the value of $k$ .
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.7a:
Show that a finite limit only exists for $k = \frac{π}{4}$ .
22M.3.AHL.TZ2.1f.i:
Find the equation of the tangent to $C$ at $P$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2b.ii:
By substituting $Y = \frac{d y}{d t}$ , show that $Y = B e^{2 t}$ where $B$ is a constant.
21N.3.AHL.TZ0.2c.iii:
Let the two values found in part (c)(ii) be $λ_{1}$ and $λ_{2}$ .

Verify that $y = F e^{λ_{1} t} + G e^{λ_{2} t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.
17M.2.AHL.TZ1.H_8a:
Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
21N.1.SL.TZ0.5c:
Find $h (4)$ .
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
16N.2.AHL.TZ0.H_6:
An earth satellite moves in a path that can be described by the curve $72.5{x^2} + 71.5{y^2} = 1$ where $x = x(t)$ and $y = y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}$ when $x = 3.2 \times {10^{ - 3}}$ , find the possible values of $\frac{{{\text{d}}y}}{{{\text{d}}t}}$ .

Give your answers in standard form.
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
SPM.3.AHL.TZ0.2d.i:
local maximum points;
21M.2.AHL.TZ1.11b:
Find an expression for $f' (x)$ .
SPM.1.SL.TZ0.9a:
Show that $f'\left( x \right) = \frac{{1 - {\text{ln}}\,5x}}{{k{x^2}}}$ .
18M.1.AHL.TZ1.H_7b:
Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
21M.2.AHL.TZ2.12c.i:
Show that $f' (x) = \frac{2 x}{\sqrt{x^{2}} (x^{2} + 1)}$ for $x \in ℝ, x \neq 0$ .
21M.3.AHL.TZ2.1d:
State the three solutions to the equation ${f_{n}}^{'} (x) = 0$ .
SPM.3.AHL.TZ0.2h.ii:
Hence express ${f_3}(x)$ as a cubic polynomial.
20N.2.SL.TZ0.S_10d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ . The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .
SPM.3.AHL.TZ0.2h.i:
Hence show that ${f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)$ , $n \in {\mathbb{Z}^ + }$ .
EXN.1.SL.TZ0.9e:
Find the rate of change of the ball’s height above the ground when $t = \frac{1}{3}$ . Give your answer in the form $p π \sqrt{q} {ms}^{- 1}$ where $p \in ℚ$ and $q \in ℤ^{+}$ .
21M.2.SL.TZ1.9c:
Show that the equation of $L$ is $y = - x + 2 \ln 45 + 2$ .
20N.2.AHL.TZ0.H_8a:
Find an expression for $\frac{d y}{d x}$ .
18M.1.AHL.TZ2.H_6b:
Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .
19M.2.SL.TZ1.S_4b:
Find the $x$ -coordinates of the points of inflexion of the graph of $f$ .
SPM.3.AHL.TZ0.2b.i:
local maximum points;
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
17M.2.SL.TZ1.S_10c:
Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y = x$ . There is a point ${\text{P}}(a,{\text{ }}b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where $0.111 < a < 3.31$ .
17M.2.SL.TZ1.S_10a.ii:
Write down the value of $h$ ;
17M.2.SL.TZ1.S_10b.ii:
Hence, find the area of the region enclosed by the graphs of $h$ and ${h^{ - 1}}$ .
21N.1.SL.TZ0.5d:
Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .
21N.3.AHL.TZ0.2a.i:
By solving the differential equation $\frac{d y}{d t} = y$ , show that $y = A e^{t}$ where $A$ is a constant.
21N.3.AHL.TZ0.2a.ii:
Show that $\frac{d x}{d t} - x = - A e^{t}$ .
21N.3.AHL.TZ0.2a.iii:
Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$ .
21N.3.AHL.TZ0.2b.i:
By differentiating $\frac{d y}{d t} = - x + y$ with respect to $t$ , show that $\frac{d^{2} y}{d t^{2}} = 2 \frac{d y}{d t}$ .
21N.3.AHL.TZ0.2b.iii:
Hence find $y$ as a function of $t$ .
21N.3.AHL.TZ0.2b.iv:
Hence show that $x = - \frac{B}{2} e^{2 t} + C$ , where $C$ is a constant.
21N.3.AHL.TZ0.2c.i:
Show that $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .
21N.3.AHL.TZ0.2c.ii:
Find the two values for $λ$ that satisfy $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .

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