Questions CP1 (58 questions)

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Edexcel CP1 2019 June Q1
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
  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
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
  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
  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
  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
  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
  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
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
  1. (a) Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x\) is an improper integral.
    (b) 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
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
  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
  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
  1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r + 2 ) = n ( n + 2 ) ( n + 3 )$$ (ii) 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
  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 2021 June Q1
  1. The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix
$$\mathbf { M } = \left( \begin{array} { c c } - 4 & - 4 \sqrt { 3 }
4 \sqrt { 3 } & - 4 \end{array} \right)$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\) A square \(S\) has vertices at the points with coordinates ( 0,0 ), ( \(a , - a\) ), ( \(2 a , 0\) ) and ( \(a , a\) ) where \(a\) is a constant. The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
  2. Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
Edexcel CP1 2021 June Q2
  1. (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)
Give each term in simplest form.
(b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ (c) Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
(d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
Edexcel CP1 2021 June Q4
4 \sqrt { 3 } & - 4 \end{array} \right)$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\) A square \(S\) has vertices at the points with coordinates ( 0,0 ), ( \(a , - a\) ), ( \(2 a , 0\) ) and ( \(a , a\) ) where \(a\) is a constant. The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
  2. Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
    1. (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)
    Give each term in simplest form.
  3. Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
  4. Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
  5. Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
    1. The cubic equation
    $$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha , \beta\) and \(\gamma\)
    The cubic equation $$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$ where \(c\) is a constant, has roots \(( 4 \alpha - 1 ) , ( 4 \beta - 1 )\) and \(( 4 \gamma - 1 )\)
    Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
      1. \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.
    Giving a reason for your answer, explain whether it is possible to evaluate
  6. \(\mathbf { A B }\)
  7. \(\mathbf { A } + \mathbf { B }\)
    (ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
    a & 0 & 0
    b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
    2 & 12 & - 1
    - 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where \(a\), \(b\) and \(\lambda\) are constants,
  8. determine
    • the value of \(\lambda\)
    • the value of \(a\)
    • the value of \(b\)
    • Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1
      a & 0 & 0
      b & a & b \end{array} \right)\)
      (iii) Given that
    $$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
    0 & \sin \theta & \cos \theta
    0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.
Edexcel CP1 2021 June Q5
    1. Evaluate the improper integral
$$\int _ { 1 } ^ { \infty } 2 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) The air temperature, \(\theta ^ { \circ } \mathrm { C }\), on a particular day in London is modelled by the equation $$\theta = 8 - 5 \sin \left( \frac { \pi } { 12 } t \right) - \cos \left( \frac { \pi } { 6 } t \right) \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the number of hours after midnight.
  1. Use calculus to show that the mean air temperature on this day is \(8 ^ { \circ } \mathrm { C }\), according to the model. Given that the actual mean air temperature recorded on this day was higher than \(8 ^ { \circ } \mathrm { C }\),
  2. explain how the model could be refined.
Edexcel CP1 2021 June Q6
  1. A tourist decides to do a bungee jump from a bridge over a river.
One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist.
The tourist jumps off the bridge.
At time \(t\) seconds after the tourist reaches their lowest point, their vertical displacement is \(x\) metres above a fixed point 30 metres vertically above the river. When \(t = 0\)
  • \(x = - 20\)
  • the velocity of the tourist is \(0 \mathrm {~ms} ^ { - 1 }\)
  • the acceleration of the tourist is \(13.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation $$5 k \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 17 x = 0 \quad t \geqslant 0$$ where \(k\) is a positive constant.
  1. Determine the value of \(k\)
  2. Determine the particular solution to the differential equation.
  3. Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point.
  4. Give a limitation of the model.
Edexcel CP1 2021 June Q7
  1. The plane \(\Pi\) has equation
$$\mathbf { r } = \left( \begin{array} { l } 3
3
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \mu \left( \begin{array} { l } 2
0
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) is perpendicular to \(\Pi\).
  2. Hence find a Cartesian equation of \(\Pi\). The line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { r } 4
    - 5
    2 \end{array} \right) + t \left( \begin{array} { r } 1
    6
    - 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
    The point \(A\) lies on \(l\).
    Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
  3. determine the possible coordinates of \(A\).
Edexcel CP1 2021 June Q8
  1. Two different colours of paint are being mixed together in a container.
The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint.
The colour of the paint mixture is now altered by
  • adding red paint to the container at a rate of 2 litres per second
  • adding blue paint to the container at a rate of 1 litre per second
  • pumping fully mixed paint from the container at a rate of 3 litres per second.
Let \(r\) litres be the amount of red paint in the container at time \(t\) seconds after the colour of the paint mixture starts to be altered.
  1. Show that the amount of red paint in the container can be modelled by the differential equation $$\frac { \mathrm { dr } } { \mathrm { dt } } = 2 - \frac { \mathrm { r } } { \alpha }$$ where \(\alpha\) is a positive constant to be determined.
  2. By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.
  3. Use this information to evaluate the model, giving a reason for your answer.
Edexcel CP1 2021 June Q9
  1. (a) Use a hyperbolic substitution and calculus to show that
$$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } \left[ x \sqrt { x ^ { 2 } - 1 } + \operatorname { arcosh } x \right] + k$$ where \(k\) is an arbitrary constant. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a62583d5-0109-4bde-ac4e-c331f373d021-32_730_803_616_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 15 } x \operatorname { arcosh } x \quad x \geqslant 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
(b) Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac { 1 } { 15 } [ 17 \ln ( 3 + 2 \sqrt { 2 } ) - 6 \sqrt { 2 } ]$$
Edexcel CP1 2022 June Q1
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
  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.