\documentclass[10pt]{article}

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\usepackage{amsmath,amssymb}
\usepackage{array}
\usepackage[table]{xcolor}
\usepackage{enumitem}
\usepackage{fancyhdr}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
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\usepackage{titlesec}

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\lhead{\footnotesize\color{ibblue}\bfseries Quadratics \& Trigonometry Review}
\rhead{\footnotesize\color{accent}IBDP Prep $\cdot$ Calc}
\cfoot{\footnotesize\thepage}
\rfoot{\footnotesize J. Murphy}

% ---- Compact lists ----
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% ---- Question heading macro ----
\newcommand{\Q}[2]{%
  \vspace{6pt}%
  {\color{ibblue}\bfseries\large Question #1}\hfill{\color{accent}\itshape\small[#2 marks]}\par
  {\color{ibblue}\rule{\linewidth}{0.8pt}}\vspace{2pt}\par}

\newcommand{\m}[1]{\hfill{\itshape\small[#1]}}

\begin{document}

\begin{center}
  {\color{ibblue}\Large\bfseries Quadratics \& Trigonometry Review}\\[2pt]
  {\color{accent}\footnotesize 9 questions $\cdot$ 54 marks $\cdot$ Calculator allowed throughout}
\end{center}
\vspace{2pt}
\small

%% ===================== Q1 =====================
\Q{1}{6}

The depth of water, $D$ metres, at a harbour entrance is monitored over time and modelled by the function
\[
  D(t) = 4.8 - 1.6\cos\!\left(30^{\circ} \times t\right)
\]
where $t$ is the time in hours measured from midnight.

\begin{enumerate}[label=(\alph*)]
  \item Find the minimum and maximum depths of the water at the harbour entrance.\m{3}
  \item Find the first time after $t = 10$ hours at which the depth of water reaches $5.6\,\mathrm{m}$.\m{3}
\end{enumerate}

\vspace{8pt}

%% ===================== Q2 =====================
\Q{2}{6}

The total cost, $C$, in euros (EUR), of hiring a car from EcoCars can be modelled by the function
\[
  C = 45d + 120,
\]
where $d$ is the number of days of hire.

\begin{enumerate}[label=(\alph*)]
  \item Calculate the total cost of hiring a car from EcoCars for 5 days.\m{1}
  \item On the grid below, sketch the graph of $C = 45d + 120$, for $d \ge 0$.\m{2}
\end{enumerate}

\vspace{4pt}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
  width=9cm, height=6cm,
  axis lines=left,
  xmin=0, xmax=7.5, ymin=0, ymax=560,
  xtick={1,2,3,4,5,6,7}, ytick={100,200,300,400,500},
  minor xtick={0,1,...,7}, minor ytick={0,50,...,550},
  tick label style={font=\scriptsize},
  xlabel={$d$ (days)}, ylabel={$C$ (EUR)},
  xlabel style={right}, ylabel style={above},
  grid=both, grid style={line width=0.3pt, draw=gray!30},
  major grid style={line width=0.5pt, draw=gray!50},
  clip=false]
\end{axis}
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*),start=3]
  \item Calculate the number of days, $d$, when the total cost from EcoCars reaches 435\,EUR.\m{1}
\end{enumerate}

PremierDrive, a rival company, charges an insurance fee of 40\,EUR less than EcoCars; however, their daily rate is 10\,EUR higher.

\begin{enumerate}[label=(\alph*),start=4]
  \item Find the least number of days for which hiring from EcoCars is cheaper than hiring from PremierDrive.\m{2}
\end{enumerate}

\newpage

%% ===================== Q3 =====================
\Q{3}{6}

The diagram below shows the graph of a quadratic function $f(x) = 3x^2 + bx + c$.

\vspace{4pt}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
  width=10cm, height=7cm,
  axis lines=middle,
  xmin=-3.5, xmax=6.5, ymin=-42, ymax=15,
  xtick={-3,-2,-1,0,1,2,3,4,5,6}, ytick={-40,-35,-30,-25,-20,-15,-10,-5,0,5,10},
  tick label style={font=\scriptsize},
  xlabel={$x$}, ylabel={$y$},
  xlabel style={right}, ylabel style={above},
  clip=false]
  \addplot[ibblue,thick,smooth,domain=-2.6:5.6,samples=100]
        {3*x^2 - 9*x - 30};
  \addplot[accent,only marks,mark=*,mark size=1.5pt] coordinates {(-2,0) (5,0) (0,-30)};
  \node[above right,font=\scriptsize,accent] at (axis cs:-2,0) {$(-2,0)$};
  \node[above right,font=\scriptsize,accent] at (axis cs:5,0) {$(5,0)$};
  \node[left,font=\scriptsize,accent] at (axis cs:0,-30) {$(0,c)$};
\end{axis}
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*)]
  \item Write down the value of $c$.\m{1}
  \item Find the value of $b$ and write down $f(x)$.\m{3}
  \item Calculate the coordinates of the vertex of the graph of $f$.\m{2}
\end{enumerate}

\vspace{8pt}

%% ===================== Q4 =====================
\Q{4}{6}

The temperature, $T$, in degrees Celsius, at a weather station in northern Canada over a 24-hour period is modelled by
\[
  T(t) = a\cos(bt) + d
\]
where $t$ is the time in hours measured from midnight, $a$ and $d$ are constants, and $b$ is measured in degrees.

The graph of $T$ versus $t$ is shown below.

\vspace{4pt}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
  width=10cm, height=5.5cm,
  axis lines=left,
  xmin=0, xmax=25, ymin=-18, ymax=10,
  xtick={0,4,8,12,16,20,24},
  ytick={-15,-12,-9,-6,-3,0,3,6},
  tick label style={font=\scriptsize},
  xlabel={$t$ (hours)}, ylabel={$T$ (${}^{\circ}$C)},
  xlabel style={right}, ylabel style={above},
  grid=both, grid style={line width=0.3pt, draw=gray!30},
  major grid style={line width=0.5pt, draw=gray!50},
  clip=false]
  \addplot[ibblue,thick,smooth,domain=0:24,samples=200]
        {11*cos(15*x) - 4};
\end{axis}
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*)]
  \item Find the value of:
  \begin{enumerate}[label=(\roman*)]
    \item $a$;
    \item $d$.\m{2}
  \end{enumerate}
  \item Find the value of $b$.\m{2}
  \item Determine the interval of time during which the temperature is decreasing from $3\,^{\circ}\mathrm{C}$ to $-9\,^{\circ}\mathrm{C}$.\m{2}
\end{enumerate}

\newpage

%% ===================== Q5 =====================
\Q{5}{6}

The height, $H$ metres, of a gondola on a large observation wheel above the ground is modelled by
\[
  H(t) = 35\sin\!\left(60^{\circ} \times t\right) + 40,
\]
where $t$ is the elapsed time in seconds since the wheel began rotating at full speed.

\begin{enumerate}[label=(\alph*)]
  \item Write down the minimum height of the gondola above the ground.\m{2}
  \item Find the height of the gondola above the ground after 7 seconds.\m{2}
  \item Find the time it takes the gondola to complete one full revolution.\m{2}
\end{enumerate}

\vspace{8pt}

%% ===================== Q6 =====================
\Q{6}{6}

The total cost of a mobile phone data plan from TalkDirect, $C$ EUR, can be modelled by
\[
  C = 25m + 15,
\]
where $m$ is the number of months of the plan.

\begin{enumerate}[label=(\alph*)]
  \item Calculate the total cost of a TalkDirect plan over 4 months.\m{1}
  \item On the grid below, sketch the graph of $C = 25m + 15$, for $m \ge 0$.\m{2}
\end{enumerate}

\vspace{4pt}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
  width=9cm, height=6cm,
  axis lines=left,
  xmin=0, xmax=8.5, ymin=0, ymax=230,
  xtick={1,2,3,4,5,6,7,8}, ytick={50,100,150,200},
  minor xtick={0,1,...,8}, minor ytick={0,25,...,225},
  tick label style={font=\scriptsize},
  xlabel={$m$ (months)}, ylabel={$C$ (EUR)},
  xlabel style={right}, ylabel style={above},
  grid=both, grid style={line width=0.3pt, draw=gray!30},
  major grid style={line width=0.5pt, draw=gray!50},
  clip=false]
\end{axis}
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*),start=3]
  \item Calculate the number of months, $m$, when the total TalkDirect cost reaches 165\,EUR.\m{1}
\end{enumerate}

DataPlus, a rival provider, charges a connection fee 35\,EUR higher than TalkDirect; however, their monthly fee is lower at 18\,EUR per month.

\begin{enumerate}[label=(\alph*),start=4]
  \item Find the least number of months for which DataPlus is the cheaper option.\m{2}
\end{enumerate}

\newpage

%% ===================== Q7 =====================
\Q{7}{6}

The Singapore Flyer is one of the world's largest observation wheels. The height, $h$ metres, of a passenger above the ground after $t$ minutes can be modelled by
\[
  h(t) = 60\cos\!\!\left(\frac{\pi}{15}(t - 15)\right) + 65.
\]

\begin{enumerate}[label=(\alph*)]
  \item Find the maximum height reached by a passenger on the Singapore Flyer.\m{1}
  \item Find the height above the ground of a passenger 12 minutes after the ride begins.\m{1}
  \item Find the time, in minutes, for the Singapore Flyer to complete one rotation.\m{2}
  \item Given that passengers complete exactly one rotation, calculate for how long they are more than 95 metres above the ground.\m{2}
\end{enumerate}

\vspace{8pt}

%% ===================== Q8 =====================
\Q{8}{6}

A ball is launched horizontally from a platform. The height of the ball above the ground, $y$ metres, is modelled by the quadratic curve
\[
  y = 6 + 4x - 0.4x^2,
\]
where $x$ represents the horizontal distance, in metres, from the base of the platform. The platform stands against a wall along the $y$-axis. The ball starts at point A, reaches its maximum height at point B, and lands on the ground at point C, as shown in the diagram below.

\vspace{4pt}
\begin{center}
\begin{tikzpicture}[scale=0.85]
  \draw[-{Stealth[length=2mm]}] (-0.2,0) -- (12.5,0) node[right] {$x$};
  \draw[-{Stealth[length=2mm]}] (0,-0.3) -- (0,5.2) node[above] {$y$};
  \node[below left,font=\small] at (0,0) {O};
  \draw[ibblue,thick,smooth,domain=0:11.33,samples=120]
        plot (\x, {(6 + 4*\x - 0.4*\x*\x)*0.3});
  % Points
  \fill[accent] (0, 1.8) circle (2.2pt) node[left,font=\small] {A};
  \fill[accent] (5, 4.8) circle (2.2pt) node[above,font=\small] {B};
  \fill[accent] (11.33, 0) circle (2.2pt) node[below right,font=\small] {C};
\end{tikzpicture}
\end{center}

\begin{enumerate}[label=(\alph*)]
  \item Write down the height, in metres, from which the ball was launched.\m{1}
  \item Calculate the maximum height above the ground reached by the ball.\m{3}
  \item Find the horizontal distance from the base of the platform to the point where the ball hits the ground.\m{2}
\end{enumerate}

\newpage

%% ===================== Q9 =====================
\Q{9}{6}

The graph of a quadratic function has a $y$-intercept at $\mathrm{A}(0,\,18)$ and \textbf{one} of its $x$-intercepts is at $\mathrm{B}(3,\,0)$.

The $x$-coordinate of the vertex of the graph is $6$.

The equation of the quadratic function is in the form $y = ax^2 + bx + c$.

\begin{enumerate}[label=(\alph*)]
  \item Write down the value of $c$.\m{1}
  \item Find the value of $a$ and the value of $b$.\m{4}
  \item Write down the coordinates of the second $x$-intercept of the function.\m{1}
\end{enumerate}

\vspace{30pt}

\begin{center}
{\color{ibblue!60}\rule{0.8\linewidth}{0.4pt}}\\[4pt]
{\footnotesize\color{ibblue!60}\itshape End of paper — 9 questions, 54 marks}
\end{center}

\end{document}
