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HL Paper 2

Consider the function $f (x) = \frac{x^{2} - x - 12}{2 x - 15}, x \in ℝ, x \neq \frac{15}{2}$ .

Find the coordinates where the graph of $f$ crosses the

$x$ -axis.

[2]

a.i.

$y$ -axis.

[1]

a.ii.

Write down the equation of the vertical asymptote of the graph of $f$ .

[1]

The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .

[4]

Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.

[3]

Express $\frac{1}{f (x)}$ in partial fractions.

[3]

e.i.

Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.

[4]

e.ii.

Prove the identity ${(p + q)}^{3} - 3 p q (p + q) \equiv p^{3} + q^{3}$ .

[2]

The equation $2 x^{2} - 5 x + 1 = 0$ has two real roots, $α$ and $β$ .

Consider the equation $x^{2} + m x + n = 0$ , where $m, n \in ℤ$ and which has roots $\frac{1}{α^{3}}$ and $\frac{1}{β^{3}}$ .
Without solving $2 x^{2} - 5 x + 1 = 0$ , determine the values of $m$ and $n$ .

[6]

The function $f$ is defined by $f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}$ , 0 < $x$ < 3.

Draw a set of axes showing $x$ and $y$ values between −3 and 3. On these axes

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .

[4]

sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.

[4]

c.i.

sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.

[4]

c.ii.

Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .

[3]

A function $f$ is defined by $f (x) = \frac{k e^{\frac{x}{2}}}{1 + e^{x}}$ , where $x \in ℝ, x \geq 0$ and $k \in ℝ^{+}$ .

The region enclosed by the graph of $y = f (x)$ , the $x$ -axis, the $y$ -axis and the line $x = \ln 16$ is rotated $360 °$ about the $x$ -axis to form a solid of revolution.

Pedro wants to make a small bowl with a volume of $300 {cm}^{3}$ based on the result from part (a). Pedro’s design is shown in the following diagrams.

The vertical height of the bowl, $BO$ , is measured along the $x$ -axis. The radius of the bowl’s top is $OA$ and the radius of the bowl’s base is $BC$ . All lengths are measured in $cm$ .

For design purposes, Pedro investigates how the cross-sectional radius of the bowl changes.

Show that the volume of the solid formed is $\frac{15 k^{2} π}{34}$ cubic units.

[6]

Find the value of $k$ that satisfies the requirements of Pedro’s design.

[2]

Find $OA$ .

[2]

c.i.

Find $BC$ .

[2]

c.ii.

By sketching the graph of a suitable derivative of $f$ , find where the cross-sectional radius of the bowl is decreasing most rapidly.

[4]

d.i.

State the cross-sectional radius of the bowl at this point.

[2]

d.ii.

A continuous random variable $X$ has a probability density function given by

$f (x) = \{\begin{matrix} arccos x & 0 \leq x \leq 1 \\ 0 & otherwise \end{matrix}$

The median of this distribution is $m$ .

Determine the value of $m$ .

[2]

Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .

[4]

Consider the function $f (x) = \sqrt{x^{2} - 1}$ , where $1 \leq x \leq 2$ .

The curve $y = f (x)$ is rotated $2 π$ about the $y$ -axis to form a solid of revolution that is used to model a water container.

At $t = 0$ , the container is empty. Water is then added to the container at a constant rate of $0.4 m^{3} s^{- 1}$ .

Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.

[2]

Show that the inverse function of $f$ is given by $f^{- 1} (x) = \sqrt{x^{2} + 1}$ .

[3]

b.i.

State the domain and range of $f^{- 1}$ .

[2]

b.ii.

Show that the volume, $V m^{3}$ , of water in the container when it is filled to a height of $h$ metres is given by $V = π (\frac{1}{3} h^{3} + h)$ .

[3]

c.i.

Hence, determine the maximum volume of the container.

[2]

c.ii.

Find the time it takes to fill the container to its maximum volume.

[2]

Find the rate of change of the height of the water when the container is filled to half its maximum volume.

[6]

The voltage $v$ in a circuit is given by the equation

$v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)$ , $t \geqslant 0$ where $t$ is measured in seconds.

The current $i$ in this circuit is given by the equation

$i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)$ .

The power $p$ in this circuit is given by $p\left( t \right) = v\left( t \right) \times i\left( t \right)$ .

The average power ${p_{av}}$ in this circuit from $t = 0$ to $t = T$ is given by the equation

${p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t}$ , where $T > 0$ .

Write down the maximum and minimum value of $v$ .

[2]

Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .

[2]

Sketch the graph of $y = p\left( t \right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

[3]

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.

[3]

Find ${p_{av}}$ (0.007).

[2]

With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.

[2]

Given that $p\left( t \right)$ can be written as $p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d$ where $a$ , $b$ , $c$ , $d$ > 0, use your graph to find the values of $a$ , $b$ , $c$ and $d$ .

[6]

Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = $p$ , OQ = $q$ and OR = $r$ , such that 0 < $p$ < $q$ < $r$ < 10.

Let ${\theta _p}$ be the angle APO, ${\theta _q}$ be the angle AQO and ${\theta _r}$ be the angle ARO.

Consider the case when ${\theta _p} = {\theta _q} + {\theta _r}$ and QR = 1.

Find an expression for ${\theta _p}$ in terms of $p$ .

[3]

Show that $p = \frac{{{q^2} + q - 1}}{{2q + 1}}$ .

[6]

By sketching the graph of $p$ as a function of $q$ , determine the range of values of $p$ for which there are possible values of $q$ .

[4]

Show that ${\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}$ .

[1]

Verify that $x = {\text{tan}}\,\theta$ and $x = - \,{\text{cot}}\,\theta$ satisfy the equation ${x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0$ .

[7]

Hence, or otherwise, show that the exact value of ${\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3$ .

[5]

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 $a$ , $b \in \mathbb{Z}$ .

[6]

A function $f$ is defined by $f (x) = arcsin (\frac{x^{2} - 1}{x^{2} + 1}), x \in ℝ$ .

A function $g$ is defined by $g (x) = arcsin (\frac{x^{2} - 1}{x^{2} + 1}), x \in ℝ, x \geq 0$ .

Show that $f$ is an even function.

[1]

By considering limits, show that the graph of $y = f (x)$ has a horizontal asymptote and state its equation.

[2]

Show that $f' (x) = \frac{2 x}{\sqrt{x^{2}} (x^{2} + 1)}$ for $x \in ℝ, x \neq 0$ .

[6]

c.i.

By using the expression for $f' (x)$ and the result $\sqrt{x^{2}} = |x|$ , show that $f$ is decreasing for $x < 0$ .

[3]

c.ii.

Find an expression for $g^{- 1} (x)$ , justifying your answer.

[5]

State the domain of $g^{- 1}$ .

[1]

Sketch the graph of $y = g^{- 1} (x)$ , clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

[3]

The height of water, in metres, in Dungeness harbour is modelled by the function $H (t) = a \sin (b (t - c)) + d$ , where $t$ is the number of hours after midnight, and $a, b, c$ and $d$ are constants, where $a > 0, b > 0$ and $c > 0$ .

The following graph shows the height of the water for $13$ hours, starting at midnight.

The first high tide occurs at $04 : 30$ and the next high tide occurs $12$ hours later. Throughout the day, the height of the water fluctuates between $2.2 m$ and $6.8 m$ .

All heights are given correct to one decimal place.

Show that $b = \frac{π}{6}$ .

[1]

Find the value of $a$ .

[2]

Find the value of $d$ .

[2]

Find the smallest possible value of $c$ .

[3]

Find the height of the water at $12 : 00$ .

[2]

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

[3]

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.

[2]

It is given that $f(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4$ where $a$ and $b$ are positive integers.

Given that ${x^2} - 1$ is a factor of $f(x)$ find the value of $a$ and the value of $b$ .

[4]

Factorize $f(x)$ into a product of linear factors.

[3]

Using your graph state the range of values of $c$ for which $f(x) = c$ has exactly two distinct real roots.

[3]

Sketch the graphs $y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x$ and $y = 1 + {\text{cos}}\,x$ on the following axes for 0 < $x$ ≤ 9.

[2]

Hence solve ${\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0$ in the range 0 < $x$ ≤ 9.

[4]

Consider $f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)$

The function $f$ is defined by $f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D$

The function $g$ is defined by $g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[$ .

Find the largest possible domain $D$ for $f$ to be a function.

[2]

Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

[3]

Explain why $f$ is an even function.

[1]

Explain why the inverse function ${f^{ - 1}}$ does not exist.

[1]

Find the inverse function ${g^{ - 1}}$ and state its domain.

[4]

Find $g'(x)$ .

[3]

Hence, show that there are no solutions to $g'(x) = 0$ ;

[2]

g.i.

Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .

[2]

g.ii.

The function $f$ is defined by $f (x) = \frac{3 x + 2}{4 x^{2} - 1}$ , for $x \in ℝ$ , $x \neq p$ , $x \neq q$ .

The graph of $y = f (x)$ has exactly one point of inflexion.

The function $g$ is defined by $g (x) = \frac{4 x^{2} - 1}{3 x + 2}$ , for $x \in ℝ, x \neq - \frac{2}{3}$ .

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

[2]

Find an expression for $f' (x)$ .

[3]

Find the $x$ -coordinate of the point of inflexion.

[2]

Sketch the graph of $y = f (x)$ for $- 3 \leq x \leq 3$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

[5]

Find the equations of all the asymptotes on the graph of $y = g (x)$ .

[4]

By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .

[4]

Consider the function $f$ defined by $f(x) = 3x\arccos (x)$ where $- 1 \leqslant x \leqslant 1$ .

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

[3]

State the range of $f$ .

[2]

Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .

[4]

The population, $P$ , of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation

$\frac{d P}{d t} = k P (1 - \frac{P}{N})$

where $t$ is the time measured in years and $k, N$ are positive constants.

The constant $N$ represents the maximum population of this species of marsupial that the island can sustain indefinitely.

Let $P_{0}$ be the initial population of marsupials.

In the context of the population model, interpret the meaning of $\frac{d P}{d t}$ .

[1]

Show that $\frac{d^{2} P}{d t^{2}} = k^{2} P (1 - \frac{P}{N}) (1 - \frac{2 P}{N})$ .

[4]

Hence show that the population of marsupials will increase at its maximum rate when $P = \frac{N}{2}$ . Justify your answer.

[5]

Hence determine the maximum value of $\frac{d P}{d t}$ in terms of $k$ and $N$ .

[2]

By solving the logistic differential equation, show that its solution can be expressed in the form

$k t = \ln \frac{P}{P_{0}} (\frac{N - P_{0}}{N - P})$ .

[7]

After $10$ years, the population of marsupials is $3 P_{0}$ . It is known that $N = 4 P_{0}$ .

Find the value of $k$ for this population model.

[2]

Consider the function $f (x) = 2^{x} - \frac{1}{2^{x}}, x \in ℝ$ .

The function $g$ is given by $g (x) = \frac{x - 1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x \neq - 1, x \neq 3$ .

Show that $f$ is an odd function.

[2]

Solve the inequality $f (x) \geq g (x)$ .

[4]

A scientist conducted a nine-week experiment on two plants, $A$ and $B$ , of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant $A$ was given fertilizer regularly, while Plant $B$ was not.

The scientist found that the height of Plant $A, h_{A} cm$ , at time $t$ weeks can be modelled by the function $h_{A} (t) = \sin (2 t + 6) + 9 t + 27$ , where $0 \leq t \leq 9$ .

The scientist found that the height of Plant $B, h_{B} cm$ , at time $t$ weeks can be modelled by the function $h_{B} (t) = 8 t + 32$ , where $0 \leq t \leq 9$ .

Use the scientist’s models to find the initial height of

Plant $B$ .

[1]

a.i.

Plant $A$ correct to three significant figures.

[2]

a.ii.

Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .

[3]

For $t > 6$ , prove that Plant $A$ was always taller than Plant $B$ .

[3]

For $0 \leq t \leq 9$ , find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$ .

[6]

Consider the expression $f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{cot}}\left( {\frac{\pi }{4} - x} \right)$ .

The expression $f\left( x \right)$ can be written as $g\left( t \right)$ where $t = {\text{tan}}\,x$ .

Let $\alpha$ , β be the roots of $g\left( t \right) = k$ , where 0 < $k$ < 1.

Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .

[2]

a.i.

With reference to your graph, explain why $f$ is a function on the given domain.

[1]

a.ii.

Explain why $f$ has no inverse on the given domain.

[1]

a.iii.

Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .

[1]

a.iv.

Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .

[3]

Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

[3]

Find $\alpha$ and β in terms of $k$ .

[5]

d.i.

Show that $\alpha$ + β < −2.

[2]

d.ii.

Let $P\left( x \right) = 2{x^4} - 15{x^3} + a{x^2} + bx + c$ , where $a$ , $b$ , $c \in \mathbb{R}$

Given that $\left( {x - 5} \right)$ is a factor of $P\left( x \right)$ , find a relationship between $a$ , $b$ and $c$ .

[2]

Given that ${\left( {x - 5} \right)^2}$ is a factor of $P\left( x \right)$ , write down the value of $P'\left( 5 \right)$ .

[1]

Given that ${\left( {x - 5} \right)^2}$ is a factor of $P\left( x \right)$ , and that $a = 2$ , find the values of $b$ and $c$ .

[3]

Consider the equation ${x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0$ , where $m$ , $n$ , $p$ , $q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as ${\text{lo}}{{\text{g}}_2}\,a$ , ${\text{lo}}{{\text{g}}_2}\,b$ and ${\text{lo}}{{\text{g}}_2}\,c$ .

The equation also has two imaginary roots, one of which is $d{\text{i}}$ where $d \in \mathbb{R}$ .

The values $a$ , $b$ , and $c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8{d^2}$ , find the other two real roots.

[9]

The following diagram shows the graph of $y = f\left( x \right)$ , $- 3 \leqslant x \leqslant 5$ .

Find the value of $\left( {f \circ f} \right)\left( 1 \right)$ .

[2]

Given that ${f^{ - 1}}\left( a \right) = 3$ , determine the value of $a$ .

[2]

Given that $g\left( x \right) = 2f\left( {x - 1} \right)$ , find the domain and range of $g$ .

[2]

Find the set of values of $k$ that satisfy the inequality ${k^2} - k - 12 < 0$ .

[2]

The triangle ABC is shown in the following diagram. Given that $\cos B < \frac{1}{4}$ , find the range of possible values for AB.

M17/5/MATHL/HP2/ENG/TZ2/04.b

[4]

Consider the function $f\left( x \right) = \frac{{ax + 1}}{{bx + c}}$ , $x \ne - \frac{c}{b}$ , where $a$ , $b$ , $c \in \mathbb{Z}$ .

The following graph shows the curve $y = {\left( {f\left( x \right)} \right)^2}$ . It has asymptotes at $x = p$ and $y = q$ and meets the $x$ -axis at A.

On the following axes, sketch the two possible graphs of $y = f\left( x \right)$ giving the equations of any asymptotes in terms of $p$ and $q$ .

[4]

Given that $p = \frac{4}{3}$ , $q = \frac{4}{9}$ and A has coordinates $\left( { - \frac{1}{2},\,\,0} \right)$ , determine the possible sets of values for $a$ , $b$ and $c$ .

[4]

The function $f$ has a derivative given by $f' (x) = \frac{1}{x (k - x)}, x \in ℝ, x \neq o, x \neq k$ where $k$ is a positive constant.

Consider $P$ , the population of a colony of ants, which has an initial value of $1200$ .

The rate of change of the population can be modelled by the differential equation $\frac{d P}{d t} = \frac{P (k - P)}{5 k}$ , where $t$ is the time measured in days, $t \geq 0$ , and $k$ is the upper bound for the population.

At $t = 10$ the population of the colony has doubled in size from its initial value.

The expression for $f' (x)$ can be written in the form $\frac{a}{x} + \frac{b}{k - x}$ , where $a, b \in ℝ$ . Find $a$ and $b$ in terms of $k$ .

[3]

Hence, find an expression for $f (x)$ .

[3]

By solving the differential equation, show that $P = \frac{1200 k}{(k - 1200) e^{- \frac{t}{5}} + 1200}$ .

[8]

Find the value of $k$ , giving your answer correct to four significant figures.

[3]

Find the value of $t$ when the rate of change of the population is at its maximum.

[3]

Two airplanes, $A$ and $B$ , have position vectors with respect to an origin $O$ given respectively by

$r_{A} = (\begin{matrix} 19 \\ - 1 \\ 1 \end{matrix}) + t (\begin{matrix} - 6 \\ 2 \\ 4 \end{matrix})$

$r_{B} = (\begin{matrix} 1 \\ 0 \\ 12 \end{matrix}) + t (\begin{matrix} 4 \\ 2 \\ - 2 \end{matrix})$

where $t$ represents the time in minutes and $0 \leq t \leq 2.5$ .

Entries in each column vector give the displacement east of $O$ , the displacement north of $O$ and the distance above sea level, all measured in kilometres.

The two airplanes’ lines of flight cross at point $P$ .

Find the three-figure bearing on which airplane $B$ is travelling.

[2]

Show that airplane $A$ travels at a greater speed than airplane $B$ .

[2]

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

[4]

Find the coordinates of $P$ .

[5]

d.i.

Determine the length of time between the first airplane arriving at $P$ and the second airplane arriving at $P$ .

[2]

d.ii.

Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .

[5]

The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.

Find the probability that Lucca eats at least one banana in a particular day.

[2]

Find the expected number of weeks in the year in which Lucca eats no bananas.

[4]

A continuous random variable $X$ has the probability density function $f$ given by

$f (x) = \{\begin{cases} \frac{x}{\sqrt{{(x^{2} + k)}^{3}}} 0 \leq x \leq 4 \\ 0 otherwise \end{cases}$

where $k \in ℝ^{+}$ .

Show that $\sqrt{16 + k} - \sqrt{k} = \sqrt{k} \sqrt{16 + k}$ .

[5]

Find the value of $k$ .

[2]

The polynomial ${x^4} + p{x^3} + q{x^2} + rx + 6$ is exactly divisible by each of $\left( {x - 1} \right)$ , $\left( {x - 2} \right)$ and $\left( {x - 3} \right)$ .

Find the values of $p$ , $q$ and $r$ .

The function $f$ is defined by $f\left( x \right) = {\text{sec}}\,x + 2$ , $0 \leqslant x < \frac{\pi }{2}$ .

Write down the range of $f$ .

[1]

Find $f'\left( x \right)$ , stating its domain.

[4]

Consider the graphs of $y = \frac{{{x^2}}}{{x - 3}}$ and $y = m\left( {x + 3} \right)$ , $m \in \mathbb{R}$ .

Find the set of values for $m$ such that the two graphs have no intersection points.

Consider the equation $k x^{2} - (k + 3) x + 2 k + 9 = 0$ , where $k \in ℝ$ .

Write down an expression for the product of the roots, in terms of $k$ .

[1]

Hence or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root.

[3]

Consider the function $f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi$ .

Consider the region bounded by the curve $y = f(x)$ , the $x$ -axis and the lines $x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}$ .

Show that the $x$ -coordinate of the minimum point on the curve $y = f(x)$ satisfies the equation $\tan x = 2x$ .

[5]

a.i.

Determine the values of $x$ for which $f(x)$ is a decreasing function.

[2]

a.ii.

Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.

[3]

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y = - x$ .

[4]

This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.

[3]