SPS SPS SM Statistics 2022 February — Question 6 9 marks

Exam BoardSPS
ModuleSPS SM Statistics (SPS SM Statistics)
Year2022
SessionFebruary
Marks9
TopicHarmonic Form
TypeApplied context modeling
DifficultyStandard +0.3 This is a standard harmonic form question with straightforward application. Part (a) uses the routine R cos(x-α) transformation with standard techniques. Parts (b) and (c) involve direct substitution and basic differentiation of the resulting expression. While it requires multiple steps and careful arithmetic, it follows a well-established template with no novel problem-solving required, making it slightly easier than average.
Spec1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07i Differentiate x^n: for rational n and sums
6. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Express \(3 \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\) where
    • \(\quad R\) and \(\alpha\) are constants
    • \(R > 0\)
    • \(0 < \alpha < \frac { \pi } { 2 }\)
    Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a rabbit hole on a particular day is modelled by the equation $$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
    Using the equation of the model and your answer to part (a)
    1. deduce the minimum value of \(\theta\) during this day,
    2. find the time of day when this minimum value occurs, giving your answer to the nearest minute.
  3. Find the rate of temperature increase in the rabbit hole at midday.
    [0pt]
    [0pt]
    [0pt]
    [0pt] [TURN OVER FOR QUESTION 7]
6.

\section*{In this question you must show all stages of your working.}
\section*{Solutions relying entirely on calculator technology are not acceptable.}
\begin{enumerate}[label=(\alph*)]
\item Express $3 \cos x + \sin x$ in the form $R \cos ( x - \alpha )$ where

\begin{itemize}
  \item $\quad R$ and $\alpha$ are constants
  \item $R > 0$
  \item $0 < \alpha < \frac { \pi } { 2 }$
\end{itemize}

Give the exact value of $R$ and the value of $\alpha$ in radians to 3 decimal places.

The temperature, $\theta ^ { \circ } \mathrm { C }$, inside a rabbit hole on a particular day is modelled by the equation

$$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$

where $t$ is the number of hours after midnight.\\
Using the equation of the model and your answer to part (a)
\item \begin{enumerate}[label=(\roman*)]
\item deduce the minimum value of $\theta$ during this day,
\item find the time of day when this minimum value occurs, giving your answer to the nearest minute.
\end{enumerate}\item Find the rate of temperature increase in the rabbit hole at midday.\\[0pt]
\\[0pt]
\\[0pt]
\\[0pt]
[TURN OVER FOR QUESTION 7]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q6 [9]}}
This paper (14 questions)
View full paper
Q1 7 Q2 4 Q3 7 Q4 8 Q5 7 Q6 9 Q7 6 Q8 11 Q9 5 Q10 6 Q11 4 Q12 6 Q13 8 Q14 12