Questions — Edexcel CP1 (59 questions)

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Edexcel CP1 2019 June Q1
9 marks Standard +0.3
1. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are real constants.
Given that \(- 1 + 2 \mathrm { i }\) and \(3 - \mathrm { i }\) are two roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
  2. find the values of \(a , b , c\) and \(d\).
Edexcel CP1 2019 June Q2
7 marks Challenging +1.2
  1. Show that
$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.
Edexcel CP1 2019 June Q3
10 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f5761f9-15d0-499a-992a-c98539f2785c-10_508_874_244_609} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not to scale Figure 1 shows the design for a table top in the shape of a rectangle \(A B C D\). The length of the table, \(A B\), is 1.2 m . The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood. The perimeter of the glass is modelled by the curve with polar equation $$r = 0.4 + a \cos 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) is a constant.
  1. Show that \(a = 0.2\) Hence, given that \(A D = 60 \mathrm {~cm}\),
  2. find the area of the wooden part of the table top, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.
Edexcel CP1 2019 June Q4
5 marks Challenging +1.2
  1. Prove that, for \(n \in \mathbb { Z } , n \geqslant 0\)
$$\sum _ { r = 0 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { ( n + a ) ( n + b ) } { c ( n + 2 ) ( n + 3 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel CP1 2019 June Q5
13 marks Standard +0.8
  1. A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.
Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt. It is assumed that the salt water mixes instantly with the contents of the tank upon entry.
At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute. Given that there are \(S\) grams of salt in the tank after \(t\) minutes,
  1. show that the situation can be modelled by the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } = 3 - \frac { 2 S } { 100 + t }$$
  2. Hence find the number of grams of salt in the tank after 10 minutes. When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.
  3. Find, to the nearest minute, when the valve would need to be closed.
  4. Evaluate the model.
Edexcel CP1 2019 June Q6
6 marks Standard +0.3
  1. Prove by induction that for all positive integers \(n\)
$$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
(6)
Edexcel CP1 2019 June Q7
7 marks Standard +0.3
  1. The line \(l _ { 1 }\) has equation
$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
  2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
  3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)
Edexcel CP1 2019 June Q8
18 marks Challenging +1.2
  1. A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish.
    At time \(t\) years, the number of white-clawed crayfish, \(w\), and the number of signal crayfish, \(s\), are modelled by the differential equations
$$\begin{aligned} & \frac { \mathrm { d } w } { \mathrm {~d} t } = \frac { 5 } { 2 } ( w - s ) \\ & \frac { \mathrm { d } s } { \mathrm {~d} t } = \frac { 2 } { 5 } w - 90 \mathrm { e } ^ { - t } \end{aligned}$$
  1. Show that $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} w } { \mathrm {~d} t } + 2 w = 450 \mathrm { e } ^ { - t }$$
  2. Find a general solution for the number of white-clawed crayfish at time \(t\) years.
  3. Find a general solution for the number of signal crayfish at time \(t\) years. The model predicts that, at time \(T\) years, the population of white-clawed crayfish will have died out. Given that \(w = 65\) and \(s = 85\) when \(t = 0\)
  4. find the value of \(T\), giving your answer to 3 decimal places.
  5. Suggest a limitation of the model.
Edexcel CP1 2020 June Q1
10 marks Standard +0.3
1. $$f ( z ) = 3 z ^ { 3 } + p z ^ { 2 } + 57 z + q$$ where \(p\) and \(q\) are real constants.
Given that \(3 - 2 \sqrt { 2 } \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
  2. find the value of \(p\) and the value of \(q\).
Edexcel CP1 2020 June Q2
7 marks Standard +0.8
  1. Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x\) is an improper integral.
  2. Prove that $$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.
Edexcel CP1 2020 June Q3
9 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7458ec3b-1be1-4b46-893c-c7470d622e6e-08_549_908_246_790} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations $$\begin{array} { l l } C _ { 1 } : r = ( 1 + \sin \theta ) & 0 \leqslant \theta < 2 \pi \\ C _ { 2 } : r = 3 ( 1 - \sin \theta ) & 0 \leqslant \theta < 2 \pi \end{array}$$ The region \(R\) lies inside \(C _ { 1 }\) and outside \(C _ { 2 }\) and is shown shaded in Figure 1.
Show that the area of \(R\) is $$p \sqrt { 3 } - q \pi$$ where \(p\) and \(q\) are integers to be determined.
Edexcel CP1 2020 June Q4
9 marks Standard +0.3
  1. The plane \(\Pi _ { 1 }\) has equation
$$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
  2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r . } ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
  3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
Edexcel CP1 2020 June Q5
17 marks Challenging +1.2
  1. Two compounds, \(X\) and \(Y\), are involved in a chemical reaction. The amounts in grams of these compounds, \(t\) minutes after the reaction starts, are \(x\) and \(y\) respectively and are modelled by the differential equations
$$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = - 5 x + 10 y - 30 \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = - 2 x + 3 y - 4 \end{aligned}$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 50$$
  2. Find, according to the model, a general solution for the amount in grams of compound \(X\) present at time \(t\) minutes.
  3. Find, according to the model, a general solution for the amount in grams of compound \(Y\) present at time \(t\) minutes. Given that \(x = 2\) and \(y = 5\) when \(t = 0\)
  4. find
    1. the particular solution for \(x\),
    2. the particular solution for \(y\). A scientist thinks that the chemical reaction will have stopped after 8 minutes.
  5. Explain whether this is supported by the model.
Edexcel CP1 2020 June Q6
12 marks Standard +0.3
  1. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r + 2 ) = n ( n + 2 ) ( n + 3 )$$
  2. Prove by induction that for all positive odd integers \(n\) $$f ( n ) = 4 ^ { n } + 5 ^ { n } + 6 ^ { n }$$ is divisible by 15
Edexcel CP1 2020 June Q7
11 marks Standard +0.3
  1. A sample of bacteria in a sealed container is being studied.
The number of bacteria, \(P\), in thousands, is modelled by the differential equation $$( 1 + t ) \frac { \mathrm { d } P } { \mathrm {~d} t } + P = t ^ { \frac { 1 } { 2 } } ( 1 + t )$$ where \(t\) is the time in hours after the start of the study.
Initially, there are exactly 5000 bacteria in the container.
  1. Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.
  2. Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.
  3. State a limitation of the model.
Edexcel CP1 2022 June Q1
6 marks Moderate -0.3
1. $$\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } + 52 \quad \text { where } a \text { is a real constant }$$ Given that \(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. write down the other complex root.
  2. Hence
    1. solve completely \(\mathrm { f } ( \mathrm { z } ) = 0\)
    2. determine the value of \(a\)
  3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
Edexcel CP1 2022 June Q2
4 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Determine the values of \(x\) for which $$64 \cosh ^ { 4 } x - 64 \cosh ^ { 2 } x - 9 = 0$$ Give your answers in the form \(q \ln 2\) where \(q\) is rational and in simplest form.
Edexcel CP1 2022 June Q3
6 marks Standard +0.3
  1. Determine the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \mathrm { e } ^ { 2 x } \cos ^ { 2 } x$$ giving your answer in the form \(y = \mathrm { f } ( x )\) Given that \(y = 3\) when \(x = 0\)
  2. determine the smallest positive value of \(x\) for which \(y = 0\)
Edexcel CP1 2022 June Q4
7 marks Challenging +1.2
  1. Use the method of differences to prove that for \(n > 2\) $$\sum _ { r = 2 } ^ { n } \ln \left( \frac { r + 1 } { r - 1 } \right) \equiv \ln \left( \frac { n ( n + 1 ) } { 2 } \right)$$ (4)
  2. Hence find the exact value of $$\sum _ { r = 51 } ^ { 100 } \ln \left( \frac { r + 1 } { r - 1 } \right) ^ { 35 }$$ Give your answer in the form \(a \ln \left( \frac { b } { c } \right)\) where \(a , b\) and \(c\) are integers to
    be determined.
Edexcel CP1 2022 June Q5
6 marks Standard +0.3
5. $$\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 3 \\ 2 & 3 & 0 \\ 4 & a & 2 \end{array} \right) \quad \text { where } a \text { is a constant }$$
  1. Show that \(\mathbf { M }\) is non-singular for all values of \(a\).
  2. Determine, in terms of \(a , \mathbf { M } ^ { - 1 }\)
Edexcel CP1 2022 June Q6
7 marks Standard +0.8
  1. Express as partial fractions $$\frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) }$$
  2. Hence, show that $$\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x + 6 } { ( x + 1 ) \left( x ^ { 2 } + 4 \right) } d x = \ln ( a \sqrt { 2 } ) + b \pi$$ where \(a\) and \(b\) are constants to be determined.
Edexcel CP1 2022 June Q7
7 marks Standard +0.3
Given that \(z = a + b \mathrm { i }\) is a complex number where \(a\) and \(b\) are real constants,
  1. show that \(z z ^ { * }\) is a real number. Given that
    • \(z z ^ { * } = 18\)
    • \(\frac { z } { z ^ { * } } = \frac { 7 } { 9 } + \frac { 4 \sqrt { 2 } } { 9 } \mathrm { i }\)
    • determine the possible complex numbers \(z\)
Edexcel CP1 2022 June Q8
12 marks Challenging +1.2
  1. Given $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad n \in \mathbb { N }$$ show that $$32 \cos ^ { 6 } \theta \equiv \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-22_218_357_653_331} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-22_307_824_621_897} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 1 shows a solid paperweight with a flat base.
    Figure 2 shows the curve with equation $$y = H \cos ^ { 3 } \left( \frac { x } { 4 } \right) \quad - 4 \leqslant x \leqslant 4$$ where \(H\) is a positive constant and \(x\) is in radians.
    The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = - 4\), the line with equation \(x = 4\) and the \(x\)-axis. The paperweight is modelled by the solid of revolution formed when \(R\) is rotated \(\mathbf { 1 8 0 } ^ { \circ }\) about the \(x\)-axis. Given that the maximum height of the paperweight is 2 cm ,
  2. write down the value of \(H\).
  3. Using algebraic integration and the result in part (a), determine, in \(\mathrm { cm } ^ { 3 }\), the volume of the paperweight, according to the model. Give your answer to 2 decimal places.
    [0pt] [Solutions based entirely on calculator technology are not acceptable.]
  4. State a limitation of the model.
Edexcel CP1 2022 June Q9
6 marks Standard +0.8
  1. (a) Explain why \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is an improper integral.
    (b) Show that \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is divergent.
  2. $$4 \sinh x = p \cosh x \quad \text { where } p \text { is a real constant }$$ Given that this equation has real solutions, determine the range of possible values for \(p\)
Edexcel CP1 2022 June Q10
14 marks Challenging +1.2
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f237de57-ed6d-4bea-8bb0-1b4e5b66d7da-28_428_301_246_881} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The motion of a pendulum, shown in Figure 3, is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ where \(\theta\) is the angle, in radians, that the pendulum makes with the downward vertical, \(t\) seconds after it begins to move. \begin{enumerate}[label=(\alph*)] \item
  1. Show that a particular solution of the differential equation is $$\theta = \frac { 1 } { 12 } t \sin 3 t$$
  2. Hence, find the general solution of the differential equation. Initially, the pendulum
    Given that, 10 seconds after it begins to move, the pendulum makes an angle of \(\alpha\) radians with the downward vertical,
  3. determine, according to the model, the value of \(\alpha\) to 3 significant figures. Given that the true value of \(\alpha\) is 0.62
  4. evaluate the model. The differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ models the motion of the pendulum as moving with forced harmonic motion.
  5. Refine the differential equation so that the motion of the pendulum is simple harmonic motion.