4.
\section*{In this question you must show detailed reasoning.}
A biased four-sided spinner has edges numbered \(1,2,3,4\). When the spinner is spun, the probability that it will land on the edge numbered \(X\) is given by
\(P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 ,
0 & \text { otherwise } . \end{cases}\)
- Draw a table showing the probability distribution of \(X\).
The spinner is spun three times and the value of \(X\) is noted each time.
- Find the probability that the third value of \(X\) is greater than the sum of the first two values of \(X\).
Name:
\section*{U8th AS LEVEL Single Mathematics Assessment
Mechanics }
22 \({ ^ { \text {nd } }\) February 2021}
Instructions
- Answer all the questions
- Write your answer to each question in the space provided under each question. The question number(s) must be clearly shown.
- Use black or blue ink. Pencil may be used for graphs and diagrams only.
- 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.
- You are permitted to use a scientific or graphical calculator in this paper.
- Final answers should be given to a degree of accuracy appropriate to the context.
Information
- The total mark for this paper is \(\mathbf { 2 5 }\) marks.
- The marks for each question are shown in brackets.
- You are reminded of the need for clear presentation in your answers.
- You should allow approximately 30 minutes for this section of the test
\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}
\(v = u + a t\)
\(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\)
\(s = \frac { 1 } { 2 } ( u + v ) t\)
\(v ^ { 2 } = u ^ { 2 } + 2 a s\)
\(s = v t - \frac { 1 } { 2 } a t ^ { 2 }\)
\section*{1.}
A particle is in equilibrium under the action of the following three forces:
\(( 2 p \mathbf { i } - 4 \mathbf { j } ) \mathrm { N } , ( - 3 q \mathbf { i } + 5 p \mathbf { j } ) \mathrm { N }\) and \(( - 13 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\).
Find the values of p and q .
\section*{2.}
A crane lifts a car vertically. The car is inside a crate which is raised by the crane by means of a strong cable. The cable can withstand a maximum tension of 9500 N without breaking. The crate has a mass of 55 kg and the car has a mass of 830 kg . - Find the maximum acceleration with which the crate and car can be raised.
- Show on a clearly labelled diagram the forces acting on the crate while it is in motion.
- Determine the magnitude of the reaction force between the crate and the car when they are ascending with maximum acceleration.
\section*{3.}
A particle \(P\) is moving in a straight line. At time \(t\) seconds \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) where \(v = ( 2 t + 1 ) ( 3 - t )\).
- Find the deceleration of \(P\) when \(t = 4\).
- State the positive value of \(t\) for which \(P\) is instantaneously at rest.
- Find the total distance that \(P\) travels between times \(t = 0\) and \(t = 4\).
\section*{4.}
A car starts from rest at a set of traffic lights and moves along a straight road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A motorcycle, travelling parallel to the car with constant speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passes the same traffic lights exactly 1.5 seconds after the car starts to move. The time after the car starts to move is denoted by \(t\) seconds.
- Determine the two values of \(t\) at which the car and motorcycle are the same distance from the traffic lights.
These two values of \(t\) are denoted by \(t _ { 1 }\) and \(t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
- Describe the relative positions of the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\).
- Determine the maximum distance between the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\).