Questions — Edexcel (10514 questions)

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Edexcel PURE 2024 October Q3
Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-06_638_643_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$
  1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
  2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$
Edexcel PURE 2024 October Q4
Moderate -0.3
  1. The number of bacteria on a surface is being monitored.
The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).
  1. Find the initial number of bacteria on the surface.
  2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
  3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.
Edexcel PURE 2024 October Q5
Challenging +1.2
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Show that \(\sin 3 x\) can be written in the form $$P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be found.
  2. Hence or otherwise, solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$2 \sin 3 \theta = 5 \sin 2 \theta$$ giving your answers, in degrees, to one decimal place as appropriate.
Edexcel PURE 2024 October Q6
Standard +0.3
  1. The functions f and g are defined by
$$\begin{array} { l l } \mathrm { f } ( x ) = 6 - \frac { 21 } { 2 x + 3 } & x \geqslant 0 \\ \mathrm {~g} ( x ) = x ^ { 2 } + 5 & x \in \mathbb { R } \end{array}$$
  1. Find \(\mathrm { gf } ( 2 )\)
  2. Find \(f ^ { - 1 }\)
  3. Solve the equation $$\operatorname { gg } ( x ) = 126$$
Edexcel PURE 2024 October Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-20_554_559_264_753} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$
  1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
  2. Find the coordinates of \(P\).
  3. Hence find the range of the function g defined by $$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
  4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$
Edexcel PURE 2024 October Q8
Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-24_472_595_246_735} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The heart rate of a horse is being monitored.
The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}
  1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
  2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
  3. Find the value of \(T\), giving your answer to 3 decimal places.
    (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
  4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
    1. find, to 4 decimal places, the value of \(t _ { 2 }\)
    2. find, to 4 decimal places, the value of \(M\)
Edexcel PURE 2024 October Q9
Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-28_753_1111_248_477} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
  2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
  3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
  4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
    (Solutions based entirely on calculator technology are not acceptable.)
Edexcel PURE 2024 October Q1
Moderate -0.3
  1. Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
  2. Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)
Edexcel PURE 2024 October Q2
Standard +0.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
The curve \(C _ { 1 }\) has equation $$y = x ^ { 4 } + 10 x ^ { 2 } + 8 \quad x \in \mathbb { R }$$ The curve \(C _ { 2 }\) has equation $$y = 2 x ^ { 2 } - 7 \quad x \in \mathbb { R }$$ Use algebra to prove by contradiction that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
Edexcel PURE 2024 October Q3
Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-06_549_750_251_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$
  1. Show that $$\frac { d y } { d x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leqslant x \leqslant q$$ where \(q\) is a constant to be found.
Edexcel PURE 2024 October Q4
Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-10_634_638_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).
  1. Verify that \(P\) lies on \(C\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
  3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.
Edexcel PURE 2024 October Q5
Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-14_569_616_242_785} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a container in the shape of a hollow, inverted, right circular cone.
The height of the container is 30 cm and the radius is 12 cm , as shown in Figure 3.
The container is initially empty when water starts flowing into it.
When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\)
  1. Show that $$V = \frac { 4 \pi h ^ { 3 } } { 75 }$$ [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ] Given that water flows into the container at a constant rate of \(2 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
  2. find, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), the rate at which \(h\) is changing, exactly 1.5 minutes after water starts flowing into the container.
Edexcel PURE 2024 October Q6
Challenging +1.2
  1. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that
$$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + c$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
Edexcel PURE 2024 October Q7
Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-18_510_680_251_696} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$
  1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.
Edexcel PURE 2024 October Q8
Standard +0.3
  1. Relative to a fixed origin \(O\)
  • the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
  • the point \(B\) has coordinates \(( - 6,4 , - 1 )\)
The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
  2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
  3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
  4. find the exact length of \(A C\), giving your answer as a surd.
Edexcel PURE 2024 October Q9
Challenging +1.2
  1. Express \(\frac { 1 } { x ( 2 x - 1 ) }\) in partial fractions. The height above ground, \(h\) metres, of a carriage on a fairground ride is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 1 } { 50 } h ( 2 h - 1 ) \cos \left( \frac { t } { 10 } \right)$$ where \(t\) seconds is the time after the start of the ride.
    Given that, at the start of the ride, the carriage is 2.5 m above ground,
  2. solve the differential equation to show that, according to the model, $$h = \frac { 5 } { 10 - 8 \mathrm { e } ^ { k \sin \left( \frac { t } { 10 } \right) } }$$ where \(k\) is a constant to be found.
  3. Hence find, according to the model, the time taken for the carriage to reach its maximum height above ground for the 3rd time.
    Give your answer to the nearest second.
    (Solutions relying entirely on calculator technology are not acceptable.)
Edexcel PURE 2024 October Q10
Standard +0.8
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-30_563_602_255_735} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.
  1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
  2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.
Edexcel M2 2024 October Q1
Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) is moving with velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = 3 ( t + 2 ) ^ { 2 } \mathbf { i } + 5 t ( t + 2 ) \mathbf { j }$$ Position vectors are given relative to the fixed point \(O\) At time \(t = 0 , P\) is at the point with position vector \(( - 30 \mathbf { i } - 45 \mathbf { j } ) \mathrm { m }\).
  1. Find the position vector of \(P\) when \(t = 3\)
  2. Find the magnitude of the acceleration of \(P\) when \(t = 3\) At time \(T\) seconds, \(P\) is moving in the direction of the vector \(2 \mathbf { i } + \mathbf { j }\)
  3. Find the value of \(T\)
Edexcel M2 2024 October Q2
Standard +0.3
  1. A particle \(Q\) of mass 3 kg is moving on a smooth horizontal surface.
Particle \(Q\) is moving with velocity \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives a horizontal impulse of magnitude \(3 \sqrt { 82 } \mathrm { Ns }\). Immediately after receiving the impulse, the velocity of \(Q\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(x\) and \(y\) are positive constants. The kinetic energy gained by \(Q\) as a result of receiving the impulse is 138 J .
Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(Q\) immediately after receiving the impulse.
Edexcel M2 2024 October Q3
Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-06_275_1143_303_461} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A van of mass 900 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 240 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 15 kW .
  1. Find the acceleration of the van at the instant when the speed of the van is \(12 \mathrm {~ms} ^ { - 1 }\) At the instant when the speed of the van is \(14 \mathrm {~ms} ^ { - 1 }\), the trailer is passing the point \(A\) on the slope and the towbar breaks. The trailer continues to move up the slope until it comes to rest at the point \(B\).
    The resistance to the motion of the trailer from non-gravitational forces is still modelled as a constant force of magnitude 240 N .
  2. Use the work-energy principle to find the distance \(A B\).
Edexcel M2 2024 October Q4
Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_301_871_319_598} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C D\) shown in Figure 2 is in the shape of an isosceles trapezium.
  • \(B C\) is parallel to \(A D\) and angle \(B A D\) is equal to angle \(A D C\)
  • \(B C = 5 a\) and \(A D = 7 a\)
  • the perpendicular distance between \(B C\) and \(A D\) is \(3 a\)
  • the distance of the centre of mass of \(A B C D\) from \(A D\) is \(d\)
    1. Show that \(d = \frac { 17 } { 12 } a\)
The uniform lamina \(P Q R S\) is a rectangle with \(P Q = 5 a\) and \(Q R = 9 a\).
The lamina \(A B C D\) in Figure 2 is used to cut a hole in \(P Q R S\) to form the template shown shaded in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-10_364_876_1567_593} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
The template is freely suspended from \(P\) and hangs in equilibrium with \(P S\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
  • Find the value of \(\theta\)
    •
    Edexcel M2 2024 October Q5
    Standard +0.3
    1. The fixed points \(X\) and \(Y\) lie on horizontal ground.
    At time \(t = 0\), a particle \(P\) is projected from \(X\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at angle \(\theta\) to the ground. Particle \(P\) moves freely under gravity and first hits the ground at \(Y\).
    1. Show that \(X Y = \frac { u ^ { 2 } \sin 2 \theta } { g }\) The points \(A\) and \(B\) lie on horizontal ground. A vertical pole \(C D\) has length 5 m .
      The end \(C\) is fixed to the ground between \(A\) and \(B\), where \(A C = 12 \mathrm {~m}\).
      At time \(t = 0\), a particle \(Q\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the ground.
      Particle \(Q\) moves freely under gravity, passes over the pole and first hits the ground at \(B\), as shown in Figure 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-14_335_1179_1032_443} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure}
    2. Find the distance \(C B\).
    3. Find the height of \(Q\) above \(D\) at the instant when \(Q\) passes over the pole.
    Edexcel M2 2024 October Q6
    Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e78f951-041d-4227-aa4b-e67a6ab5b4cd-18_419_1307_315_379} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A uniform beam \(A B\), of weight \(5 W\) and length \(12 a\), rests with end \(A\) on rough horizontal ground.
    A package of weight \(W\) is attached to the beam at \(B\).
    The beam rests in equilibrium on a smooth horizontal peg at \(C\), with \(A C = 9 a\), as shown in Figure 5.
    The beam is inclined at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 12 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg. The package is modelled as a particle. The normal reaction between the beam and the peg at \(C\) has magnitude \(k W\) Using the model,
    1. show that \(k = \frac { 56 } { 13 }\) The coefficient of friction between \(A\) and the ground is \(\mu\) Given that the beam is resting in limiting equilibrium,
    2. find the value of \(\mu\)
    Edexcel M2 2024 October Q7
    Standard +0.3
    1. A particle \(P\) has mass \(5 m\) and a particle \(Q\) has mass \(2 m\).
    The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
    Particle \(P\) collides directly with particle \(Q\).
    Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
    The direction of motion of \(Q\) is reversed as a result of the collision.
    The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Find the set of values of \(e\) for which the direction of motion of \(P\) is unchanged as a result of the collision. In the collision, \(Q\) receives an impulse of magnitude \(\frac { 60 } { 7 } m u\)
    2. Show that \(e = \frac { 1 } { 5 }\) After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds and there is a second collision between \(P\) and \(Q\).
      The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
    3. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the second collision between \(P\) and \(Q\).
    Edexcel S2 2024 October Q1
    Standard +0.3
    During an annual beach-clean, the people doing the clean are asked to conduct a litter survey.
    At a particular beach-clean, litter was found at a rate of 4 items per square metre.
    1. Find the probability that, in a randomly selected area of 2 square metres on this beach, exactly 5 items of litter were found. Of the litter found on the beach, 30\% of the items were face masks.
    2. Find the probability that, in a randomly selected area of 5 square metres on this beach, more than 4 face masks were found.
    3. Using a suitable approximation, find the probability that, in a randomly selected area of 20 square metres on this beach, less than 60 items of litter were found that were not face masks.