6.

A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point $A$.

The toy is initially at the point with displacement 3 metres from $A$. Its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds is defined by

$$v = 0.06 \left( 2 + t - t ^ { 2 } \right)$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for the displacement, $r$ metres, of the toy from $A$ at time $t$ seconds.
\item In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

At time $t = 2$ seconds, the toy launches a ball which travels directly upwards with initial speed $3.43 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Find the time taken for the ball to reach its highest point.

Name:

\section*{U8th AS LEVEL Single Mathematics Assessment \\
 Statistics \\
 18 ${ } ^ { \text {th } }$ January 2021 }
Instructions

\begin{itemize}
  \item Answer all the questions
  \item Write your answer to each question in the space provided under each question. The question number(s) must be clearly shown.
  \item Use black or blue ink. Pencil may be used for graphs and diagrams only.
  \item You should clearly write your name at the top of this page and on any additional sheets that you use. There are blank pages at the end of the paper which you can use if needed.
  \item You are permitted to use a scientific or graphical calculator in this paper.
  \item Final answers should be given to a degree of accuracy appropriate to the context.
\end{itemize}

Information

\begin{itemize}
  \item The total mark for this paper is $\mathbf { 2 7 }$ marks.
  \item The marks for each question are shown in brackets.
  \item You are reminded of the need for clear presentation in your answers.
  \item You should allow approximately 30 minutes for this section of the test
\end{itemize}

\section*{Formulae}
\section*{AS Level Mathematics A (H230)}
\section*{Binomial series}
$$( a + b ) ^ { n } = a ^ { n } + { } ^ { n } \mathrm { C } _ { 1 } a ^ { n - 1 } b + { } ^ { n } \mathrm { C } _ { 2 } a ^ { n - 2 } b ^ { 2 } + \ldots + { } ^ { n } \mathrm { C } _ { r } a ^ { n - r } b ^ { r } + \ldots + b ^ { n } \quad ( n \in \mathbb { N } ) ,$$

where ${ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }$

\section*{Differentiation from first principles}
$$\mathrm { f } ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$$

\section*{Standard deviation}
$$\sqrt { \frac { \sum ( x - \bar { x } ) ^ { 2 } } { n } } = \sqrt { \frac { \sum x ^ { 2 } } { n } - \bar { x } ^ { 2 } } \text { or } \sqrt { \frac { \sum f ( x - \bar { x } ) ^ { 2 } } { \sum f } } = \sqrt { \frac { \sum f x ^ { 2 } } { \sum f } - \bar { x } ^ { 2 } }$$

\section*{The binomial distribution}
If $X \sim \mathrm {~B} ( n , p )$ then $P ( X = x ) = \binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }$, mean of $X$ is $n p$, variance of $X$ is $n p ( 1 - p )$

\section*{Kinematics}
$v = u + a t$\\
$s = u t + \frac { 1 } { 2 } a t ^ { 2 }$\\
$s = \frac { 1 } { 2 } ( u + v ) t$\\
$v ^ { 2 } = u ^ { 2 } + 2 a s$\\
$s = v t - \frac { 1 } { 2 } a t ^ { 2 }$\\
1.

The table below shows the probability distribution for a discrete random variable $X$.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 0 & 1 & 2 & 3 & 4 or more \\
\hline
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$ & 0.35 & 0.25 & $k$ & 0.14 & 0.1 \\
\hline
\end{tabular}
\end{center}

Find the value of $k$.\\
2. Given that $\sum x = 364 , \sum x ^ { 2 } = 19412 , n = 10$, find $\sigma$, the standard deviation of $X$.\\
3.

Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart.

Nicola throws eight practice darts.\\
(a) Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times.\\
(b) Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions.
\item State two assumptions that are necessary for the distribution you have used in part (a) to be valid.\\
4.

Kevin is the Principal of a college.\\
He wishes to investigate types of transport used by students to travel to college.\\
There are 3200 students in the college and Kevin decides to survey 60 of them.\\
Describe how he could obtain a simple random sample of size 60 from the 3200 students.\\
5.

Jennie is a piano teacher who teaches nine pupils.\\
She records how many hours per week they practice the piano along with their most recent practical exam score.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Student & Practice (hours per week) & Practical exam score (out of 100) \\
\hline
Donovan & 50 & 64 \\
\hline
Vazquez & 6 & 71 \\
\hline
Higgins & 3 & 55 \\
\hline
Begum & 2.5 & 47 \\
\hline
Collins & 1 & 80 \\
\hline
Coldbridge & 4 & 61 \\
\hline
Nedbalek & 4.5 & 65 \\
\hline
Carter & 8 & 83 \\
\hline
White & 11 & 92 \\
\hline
\end{tabular}
\end{center}

She plots a scatter diagram of this data, as shown below.\\
\includegraphics[max width=\textwidth, alt={}, center]{1a7b75e9-eab4-4264-ab15-5292c504fb4d-09_880_1550_1361_246}\\
(a) Identify two possible outliers by name, giving a possible explanation for the position on the scatter diagram of each outlier.\\[0pt]
[4 marks]\\
(b) Jennie discards the two outliers.\\
(b) (i) Describe the correlation shown by the scatter diagram for the remaining points.

\section*{6.}
A factory buys $10 \%$ of its components from supplier $A , 30 \%$ from supplier $B$ and the rest from supplier $C$. It is known that $6 \%$ of the components it buys are faulty.

Of the components bought from supplier $A , 9 \%$ are faulty and of the components bought from supplier $B , 3 \%$ are faulty.\\
(a) Find the percentage of components bought from supplier $C$ that are faulty.

A component is selected at random.\\
(b) Explain why the event "the component was bought from supplier $B$ " is not statistically independent from the event "the component is faulty".
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM 2021 Q6 [7]}}