Questions CP1 (59 questions)

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Edexcel CP1 2023 June Q1
5 marks Moderate -0.3
  1. The cubic equation
$$x ^ { 3 } - 7 x ^ { 2 } - 12 x + 6 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine a cubic equation whose roots are ( \(\alpha + 2\) ), \(( \beta + 2 )\) and \(( \gamma + 2 )\), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.
Edexcel CP1 2023 June Q2
6 marks Standard +0.3
  1. (a) Write \(x ^ { 2 } + 4 x - 5\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are integers.
    (b) Hence use a standard integral from the formula book to find
$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form \(A \ln B\) where \(A\) and \(B\) are constants in simplest form.
Edexcel CP1 2023 June Q3
10 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
$$z _ { 1 } = - 4 + 4 i$$
  1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
  2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
    1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
    2. \(\left( z _ { 2 } \right) ^ { 4 }\)
  3. Show on a single Argand diagram
    1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
    2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)
Edexcel CP1 2023 June Q4
5 marks Standard +0.3
  1. Prove by induction that for \(n \in \mathbb { N }\)
$$\left( \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & - 2 n \\ 0 & 1 \end{array} \right)$$
Edexcel CP1 2023 June Q5
12 marks Challenging +1.8
5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)
  1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
  2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
  3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
    Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
  4. determine the value of \(a\) and the value of \(b\).
Edexcel CP1 2023 June Q6
12 marks Challenging +1.2
6
2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)\\ (c) Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.\\ Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,\\ (d) determine the value of \(a\) and the value of \(b\).
  1. Water is flowing into and out of a large tank.
Initially the tank contains 10 litres of water.\\ The rate of flow of the water is modelled so that
  • there are \(V\) litres of water in the tank at time \(t\) minutes after the water begins to flow
  • water enters the tank at a rate of \(\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)\) litres per minute
  • water leaves the tank at a rate proportional to the volume of water remaining in the tank
Given that when \(t = 0\) the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to\\ (a) show that the volume of water in the tank at time \(t\) satisfies $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$ (b) Determine \(\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)\) Hence, by solving the differential equation from part (a),
(c) determine an equation for the volume of water in the tank at time \(t\). Give your answer in simplest form as \(V = \mathrm { f } ( t )\) After 10 minutes, the volume of water in the tank was 8 litres.
(d) Evaluate the model in light of this information.
Edexcel CP1 2023 June Q7
12 marks Standard +0.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    1. Explain why, for \(n \in \mathbb { N }\)
    $$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function \(\mathrm { f } ( r )\).
  2. Use the standard summation formulae to show that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$
  3. Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$
Edexcel CP1 2023 June Q10
Challenging +1.2
10
6
2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)\\ (c) Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.\\ Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,\\ (d) determine the value of \(a\) and the value of \(b\).
  1. Water is flowing into and out of a large tank.
Initially the tank contains 10 litres of water.\\ The rate of flow of the water is modelled so that
  • there are \(V\) litres of water in the tank at time \(t\) minutes after the water begins to flow
  • water enters the tank at a rate of \(\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)\) litres per minute
  • water leaves the tank at a rate proportional to the volume of water remaining in the tank
Given that when \(t = 0\) the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to\\ (a) show that the volume of water in the tank at time \(t\) satisfies $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$ (b) Determine \(\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)\) Hence, by solving the differential equation from part (a),\\ (c) determine an equation for the volume of water in the tank at time \(t\). Give your answer in simplest form as \(V = \mathrm { f } ( t )\) After 10 minutes, the volume of water in the tank was 8 litres.\\ (d) Evaluate the model in light of this information.
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.\\ (a) Explain why, for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function \(\mathrm { f } ( r )\).\\ (b) Use the standard summation formulae to show that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$ (c) Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$
  1. A colony of small mammals is being studied.
In the study, the mammals are divided into 3 categories
\(N\) (newborns)0 to less than 1 month old
\(J\) (juveniles)1 to 3 months old
\(B\) (breeders)over 3 months old
(a) State one limitation of the model regarding the division into these categories. A model for the population of the colony is given by the matrix equation $$\left( \begin{array} { l } N _ { n + 1 }
J _ { n + 1 }
B _ { n + 1 } \end{array} \right) = \left( \begin{array} { c c c } 0 & 0 & 2
a & b & 0
0 & 0.48 & 0.96 \end{array} \right) \left( \begin{array} { l } N _ { n }
J _ { n }
B _ { n } \end{array} \right)$$ where \(a\) and \(b\) are constants, and \(N _ { n } , J _ { n }\) and \(B _ { n }\) are the respective numbers of the mammals in each category \(n\) months after the start of the study. At the start of the study the colony has breeders only, with no newborns or juveniles.\\ According to the model, after 2 months the number of newborns is 48 and the number of juveniles is 40\\ (b) (i) Determine the number of mammals in the colony at the start of the study.\\ (ii) Show that \(a = 0.8\)\\ (c) Determine, in terms of \(b\), $$\left( \begin{array} { c c c } 0 & 0 & 2
0.8 & b & 0
0 & 0.48 & 0.96 \end{array} \right) ^ { - 1 }$$ Given that the model predicts approximately 1015 mammals in total at the start of a particular month, and approximately 596 newborns, 464 juveniles and 437 breeders at the start of the next month,
(d) determine the value of \(b\), giving your answer to 2 decimal places. It is decided to monitor the number of newborn males and females as a part of the study. Assuming that \(42 \%\) of newborns are male,
(e) refine the matrix equation for the model to reflect this information, giving a reason for your answer.
(There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)
Edexcel CP1 2024 June Q1
9 marks Standard +0.3
1. $$\mathrm { f } ( z ) = z ^ { 4 } - 6 z ^ { 3 } + a z ^ { 2 } + b z + 145$$ where \(a\) and \(b\) are real constants.
Given that \(2 + 5 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. determine the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  2. Show all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
Edexcel CP1 2024 June Q2
8 marks Moderate -0.5
  1. The roots of the equation
$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are \(\alpha , \beta\) and \(\gamma\) Without solving the equation,
  1. write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
  2. Use the answers to part (a) to determine the value of
    1. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }\)
    2. \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
    3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
Edexcel CP1 2024 June Q3
8 marks Challenging +1.2
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc3e8e46-c60b-4263-9652-d7c2a322cfae-10_563_561_395_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,
  1. determine the value of \(a\).
  2. Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).
Edexcel CP1 2024 June Q4
10 marks Standard +0.8
  1. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.
    1. Show that
    $$z ^ { n } + \frac { 1 } { z ^ { n } } \equiv 2 \cos n \theta$$ where \(n\) is a positive integer.
  2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
  3. Hence, making your reasoning clear, determine all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)
Edexcel CP1 2024 June Q5
9 marks Standard +0.3
  1. A raindrop falls from rest from a cloud. The velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, of the raindrop, \(t\) seconds after the raindrop starts to fall, is modelled by the differential equation
$$( t + 2 ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 v = k ( t + 2 ) - 3 \quad t \geqslant 0$$ where \(k\) is a positive constant.
  1. Solve the differential equation to show that $$v = \frac { k } { 4 } ( t + 2 ) - 1 + \frac { 4 ( 2 - k ) } { ( t + 2 ) ^ { 3 } }$$ Given that \(v = 4\) when \(t = 2\)
  2. determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.
  3. Comment on the validity of the model for very large values of \(t\)
Edexcel CP1 2024 June Q6
6 marks Standard +0.3
  1. Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
Edexcel CP1 2024 June Q7
10 marks Standard +0.3
  1. The line \(l _ { 1 }\) has equation
$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)
  1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
  2. Hence determine a Cartesian equation of \(\Pi\)
  3. Hence determine the value of \(p\) Given that
    • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
    • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
    • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)
Edexcel CP1 2024 June Q8
15 marks Challenging +1.2
  1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.
At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
  3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
  4. determine the value of \(T\), giving your answer to 2 decimal places.
  5. Suggest a limitation of the model.
Edexcel CP1 Specimen Q1
5 marks Standard +0.8
  1. Prove that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$ where \(a\) and \(b\) are constants to be found.
Edexcel CP1 Specimen Q2
6 marks Standard +0.3
  1. Prove by induction that for all positive integers \(n\),
$$f ( n ) = 2 ^ { 3 n + 1 } + 3 \left( 5 ^ { 2 n + 1 } \right)$$ is divisible by 17
Edexcel CP1 Specimen Q3
9 marks Standard +0.3
3. $$\mathrm { f } ( z ) = z ^ { 4 } + a z ^ { 3 } + 6 z ^ { 2 } + b z + 65$$ where \(a\) and \(b\) are real constants.
Given that \(z = 3 + 2 \mathbf { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\), show the roots of \(\mathrm { f } ( z ) = 0\) on a single Argand diagram.
Edexcel CP1 Specimen Q4
9 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-08_492_1063_214_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 4 + \cos 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 9 } { 2 }\) The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are rational numbers to be found.
Edexcel CP1 Specimen Q5
10 marks Standard +0.8
  1. A pond initially contains 1000 litres of unpolluted water.
The pond is leaking at a constant rate of 20 litres per day.
It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water. It is assumed that the pollutant instantly dissolves throughout the pond upon entry.
Given that there are \(x\) grams of the pollutant in the pond after \(t\) days,
  1. show that the situation can be modelled by the differential equation, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 50 - \frac { 4 x } { 200 + t }$$
  2. Hence find the number of grams of pollutant in the pond after 8 days.
  3. Explain how the model could be refined.
Edexcel CP1 Specimen Q6
9 marks Standard +0.3
6. $$\mathrm { f } ( x ) = \frac { x + 2 } { x ^ { 2 } + 9 }$$
  1. Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \ln \left( x ^ { 2 } + 9 \right) + B \arctan \left( \frac { x } { 3 } \right) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
  2. Hence show that the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\) is $$\frac { 1 } { 6 } \ln 2 + \frac { 1 } { 18 } \pi$$
  3. Use the answer to part (b) to find the mean value, over the interval \([ 0,3 ]\), of $$\mathrm { f } ( x ) + \ln k$$ where \(k\) is a positive constant, giving your answer in the form \(p + \frac { 1 } { 6 } \ln q\), where \(p\) and \(q\) are constants and \(q\) is in terms of \(k\).
Edexcel CP1 Specimen Q7
8 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b36bdc3-a68d-4982-bf23-f780773df5cc-14_259_327_214_868} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the image of a gold pendant which has height 2 cm . The pendant is modelled by a solid of revolution of a curve \(C\) about the \(y\)-axis. The curve \(C\) has parametric equations $$x = \cos \theta + \frac { 1 } { 2 } \sin 2 \theta , \quad y = - ( 1 + \sin \theta ) \quad 0 \leqslant \theta \leqslant 2 \pi$$
  1. Show that a Cartesian equation of the curve \(C\) is $$x ^ { 2 } = - \left( y ^ { 4 } + 2 y ^ { 3 } \right)$$
  2. Hence, using the model, find, in \(\mathrm { cm } ^ { 3 }\), the volume of the pendant.
Edexcel CP1 Specimen Q8
7 marks Challenging +1.8
  1. The line \(l _ { 1 }\) has equation \(\frac { x - 2 } { 4 } = \frac { y - 4 } { - 2 } = \frac { z + 6 } { 1 }\)
The plane \(\Pi\) has equation \(x - 2 y + z = 6\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi\).
Find a vector equation of the line \(l _ { 2 }\)
Edexcel CP1 Specimen Q9
12 marks Standard +0.3
  1. A company plans to build a new fairground ride. The ride will consist of a capsule that will hold the passengers and the capsule will be attached to a tall tower. The capsule is to be released from rest from a point half way up the tower and then made to oscillate in a vertical line.
The vertical displacement, \(x\) metres, of the top of the capsule below its initial position at time \(t\) seconds is modelled by the differential equation, $$m \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 200 \cos t , \quad t \geqslant 0$$ where \(m\) is the mass of the capsule including its passengers, in thousands of kilograms.
The maximum permissible weight for the capsule, including its passengers, is 30000 N .
Taking the value of \(g\) to be \(10 \mathrm {~ms} ^ { - 2 }\) and assuming the capsule is at its maximum permissible weight,
    1. explain why the value of \(m\) is 3
    2. show that a particular solution to the differential equation is $$x = 40 \sin t - 20 \cos t$$
    3. hence find the general solution of the differential equation.
  2. Using the model, find, to the nearest metre, the vertical distance of the top of the capsule from its initial position, 9 seconds after it is released.