8.
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts.
Of the shirts with defects, the proportion of each type of defect is as shown in the table below.
| Type of defect | Colour | Fabric | Sewing | Sizing |
| Probability | 0.25 | 0.30 | 0.40 | 0.05 |
Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect.
She wishes to test the hypotheses
$$\begin{aligned}
& \mathrm { H } _ { 0 } : p = 0.3
& \mathrm { H } _ { 1 } : p < 0.3
\end{aligned}$$
She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
- Using a \(5 \%\) level of significance, find the critical region for \(x\).
- In her sample she finds 13 shirts with a fabric defect.
Complete the test stating her conclusion in context.
Instructions
- Answer all the questions
- Write your answer to each question on file paper 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 each page.
- 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.
- At the end you must upload your solutions to the mechanics questions to the google classroom of your mechanics teacher before you leave the examination google Meet.
Information
- The total mark for this paper is \(\mathbf { 6 1 }\) 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 65 minutes for this section of the test
1.
A vehicle is driven at a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) along a straight horizontal road. Only one of the statements below is correct.
Identify the correct statement.
Tick \(( \checkmark )\) one box.
The vehicle is accelerating □
The vehicle's driving force exceeds the total force resisting its motion □
The resultant force acting on the vehicle is zero □
The resultant force acting on the vehicle is dependent on its mass □
2.
A number of forces act on a particle such that the resultant force is \(\binom { 6 } { - 3 } \mathrm {~N}\)
One of the forces acting on the particle is \(\binom { 8 } { - 5 } \mathrm {~N}\)
Calculate the total of the other forces acting on the particle.
Circle your answer.
[0pt]
[1 mark]
$$\binom { 2 } { - 2 } \mathrm {~N} \quad \binom { 14 } { - 8 } \mathrm {~N} \quad \binom { - 2 } { 2 } \mathrm {~N} \quad \binom { - 14 } { 8 } \mathrm {~N}$$
3.
A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
A brick \(P\) of mass \(m\) is placed on the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\)
Brick \(P\) is in equilibrium and on the point of sliding down the plane.
Brick \(P\) is modelled as a particle.
Using the model,
(a) find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on brick \(P\)
(b) show that \(\mu = \frac { 3 } { 4 }\)
For parts (c) and (d), you are not required to do any further calculations.
Brick \(P\) is now removed from the plane and a much heavier brick \(Q\) is placed on the plane.
The coefficient of friction between \(Q\) and the plane is also \(\frac { 3 } { 4 }\)
(c) Explain briefly why brick \(Q\) will remain at rest on the plane.
Brick \(Q\) is now projected with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane.
Brick \(Q\) is modelled as a particle.
Using the model,
(d) describe the motion of brick \(Q\), giving a reason for your answer.
4.
A particle \(P\) moves with acceleration \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)
At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
(a) Find the velocity of \(P\) at time \(t = 2\) seconds.
At time \(t = 0 , P\) passes through the origin \(O\).
At time \(t = T\) seconds, where \(T > 0\), the particle \(P\) passes through the point \(A\).
The position vector of \(A\) is \(( \lambda \mathbf { i } - 4.5 \mathbf { j } ) \mathrm { m }\) relative to \(O\), where \(\lambda\) is a constant.
(b) Find the value of \(T\).
(c) Hence find the value of \(\lambda\)
5. - At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) is given by
$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$
At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
(a) Find the velocity of \(P\) when \(t = 4\)
(b) Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to \(\mathbf { i }\) - At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by
$$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$
Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64d3256a-9007-4e8a-86d4-8375c006a4ce-16_529_993_374_529}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A small ball is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
The point \(O\) is 25 m vertically above the point \(N\) which is on horizontal ground.
The ball is projected at an angle of \(45 ^ { \circ }\) above the horizontal.
The ball hits the ground at a point \(A\), where \(A N = 100 \mathrm {~m}\), as shown in Figure 2 .
The motion of the ball is modelled as that of a particle moving freely under gravity.
Using this initial model,
(a) show that \(U = 28\)
(b) find the greatest height of the ball above the horizontal ground \(N A\).
In a refinement to the model of the motion of the ball from \(O\) to \(A\), the effect of air resistance is included.
This refined model is used to find a new value of \(U\).
(c) How would this new value of \(U\) compare with 28 , the value given in part (a)?
(d) State one further refinement to the model that would make the model more realistic.
7.
Block \(A\), of mass 0.2 kg , lies at rest on a rough plane.
The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac { 7 } { 24 }\)
A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope.
The other end of this string is attached to particle \(B\), of mass 2 kg , which is held at rest so that the string is taut, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{64d3256a-9007-4e8a-86d4-8375c006a4ce-17_424_1070_815_486}
(a) \(\quad B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac { 543 } { 625 } \mathrm {~g} \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
Show that the coefficient of friction between \(A\) and the surface of the inclined plane is 0.17
(b) In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
When \(A\) reaches a speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) the string breaks.
(b) (i) Find the distance travelled by \(A\) after the string breaks until first coming to rest.
(b) (ii) State an assumption that could affect the validity of your answer to part (b)(i).
\section*{8.}
A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an acute angle \(\theta\) above the horizontal.
The ball needs to first land at a point at least \(d\) metres away from \(P\).
You may assume the ball may be modelled as a particle and that air resistance may be ignored.
Show that
$$\sin 2 \theta \geq \frac { d g } { u ^ { 2 } }$$
[6 marks]