Questions CP2 (58 questions)

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Edexcel CP2 2019 June Q1
  1. (a) Prove that
$$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$ stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation $$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$
Edexcel CP2 2019 June Q2
  1. The roots of the equation
$$x ^ { 3 } - 2 x ^ { 2 } + 4 x - 5 = 0$$ are \(p , q\) and \(r\).
Without solving the equation, find the value of
  1. \(\frac { 2 } { p } + \frac { 2 } { q } + \frac { 2 } { r }\)
  2. \(( p - 4 ) ( q - 4 ) ( r - 4 )\)
  3. \(p ^ { 3 } + q ^ { 3 } + r ^ { 3 }\)
Edexcel CP2 2019 June Q3
3. $$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$
  1. Using a substitution, that should be stated clearly, show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
  2. Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).
Edexcel CP2 2019 June Q4
  1. The infinite series C and S are defined by
$$\begin{aligned} & \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots
& \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots \end{aligned}$$ Given that the series C and S are both convergent,
  1. show that $$\mathrm { C } + \mathrm { iS } = \frac { 2 \mathrm { e } ^ { \mathrm { i } \theta } } { 2 - \mathrm { e } ^ { 4 \mathrm { i } \theta } }$$
  2. Hence show that $$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$
Edexcel CP2 2019 June Q5
  1. An engineer is investigating the motion of a sprung diving board at a swimming pool.
Let \(E\) be the position of the end of the diving board when it is at rest in its equilibrium position and when there is no diver standing on the diving board.
A diver jumps from the diving board.
The vertical displacement, \(h \mathrm {~cm}\), of the end of the diving board above \(E\) is modelled by the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} h } { \mathrm {~d} t } + 37 h = 0$$ where \(t\) seconds is the time after the diver jumps.
  1. Find a general solution of the differential equation. When \(t = 0\), the end of the diving board is 20 cm below \(E\) and is moving upwards with a speed of \(55 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
  2. Find, according to the model, the maximum vertical displacement of the end of the diving board above \(E\).
  3. Comment on the suitability of the model for large values of \(t\).
Edexcel CP2 2019 June Q6
  1. In an Argand diagram, the points \(A , B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2 \mathrm { i }\).
    1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The points \(D , E\) and \(F\) are the midpoints of the sides of triangle \(A B C\).
  2. Find the exact area of triangle \(D E F\).
Edexcel CP2 2019 June Q7
7. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1
3 & k & 4
3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p
    & 3 x - 6 y + 4 z = 1
    & 3 x + 2 y - z = 0 \end{aligned}$$
    1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1
      & 3 x - 5 y + 4 z = q
      & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.
Edexcel CP2 2019 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e63eca0e-e2e9-4660-855f-0696c680a8f9-24_359_552_274_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e63eca0e-e2e9-4660-855f-0696c680a8f9-24_323_387_283_1238} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the central vertical cross section \(A B C D\) of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain. Using these measurements, the curve \(B D\) is modelled by the equation $$y = \ln ( 3.6 x - k ) \quad 1 \leqslant x \leqslant 1.18$$ as shown in Figure 2.
  1. Find the value of \(k\).
  2. Find the depth of the paddling pool according to this model. The pool is being filled with water from a tap.
  3. Find, in terms of \(h\), the volume of water in the pool when the pool is filled to a depth of \(h \mathrm {~m}\). Given that the pool is being filled at a constant rate of 15 litres every minute,
  4. find, in \(\mathrm { cmh } ^ { - 1 }\), the rate at which the water level is rising in the pool when the depth of the water is 0.2 m .
Edexcel CP2 2020 June Q1
  1. The curve \(C\) has equation
$$y = 31 \sinh x - 2 \sinh 2 x \quad x \in \mathbb { R }$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\).
Edexcel CP2 2020 June Q2
  1. In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
    1. Find the equation of \(C\), giving your answer in the form
    $$| z - a | = b \quad a \in \mathbb { C } , \quad b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 i | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
  2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
Edexcel CP2 2020 June Q3
  1. A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination.
    The concentration of antibodies, \(x\), measured in micrograms ( \(\mu \mathrm { g }\) ) per millilitre ( ml ) of blood, is modelled by the differential equation
$$100 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 60 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26$$ where \(t\) is the number of weeks since the vaccination was given.
  1. Find a general solution of the differential equation. Initially,
    • there are no antibodies in the bloodstream of the patient
    • the concentration of antibodies is estimated to be increasing at \(10 \mu \mathrm {~g} / \mathrm { ml }\) per week
    • Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination.
    A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than \(5 \mu \mathrm {~g} / \mathrm { ml }\).
  2. Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose.
Edexcel CP2 2020 June Q4
  1. (a) Use de Moivre's theorem to prove that
$$\sin 7 \theta = 7 \sin \theta - 56 \sin ^ { 3 } \theta + 112 \sin ^ { 5 } \theta - 64 \sin ^ { 7 } \theta$$ (b) Hence find the distinct roots of the equation $$1 + 7 x - 56 x ^ { 3 } + 112 x ^ { 5 } - 64 x ^ { 7 } = 0$$ giving your answer to 3 decimal places where appropriate.
Edexcel CP2 2020 June Q5
  1. (a)
$$y = \tan ^ { - 1 } x$$ Assuming the derivative of \(\tan x\), prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$$ $$\mathrm { f } ( x ) = x \tan ^ { - 1 } 4 x$$ (b) Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A x ^ { 2 } \tan ^ { - 1 } 4 x + B x + C \tan ^ { - 1 } 4 x + k$$ where \(k\) is an arbitrary constant and \(A , B\) and \(C\) are constants to be determined.
(c) Hence find, in exact form, the mean value of \(\mathrm { f } ( x )\) over the interval \(\left[ 0 , \frac { \sqrt { 3 } } { 4 } \right]\)
Edexcel CP2 2020 June Q6
6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7
1 & 1 & 1
2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1
    x + y + z = p
    2 x + y - z = 2 \end{array}$$
    1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1
      x + y + z = q
      2 x + y - z = 2 \end{array}$$
    2. For this value of \(q\), determine a vector equation for the line of intersection.
Edexcel CP2 2020 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6186130-0027-4670-a6ac-f8a722d2f5fc-24_691_896_255_587} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = 1\), the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) through \(360 ^ { \circ }\) about the \(y\)-axis. The point \(A\) has coordinates ( \(1,0.5\) ) and the point \(B\) has coordinates ( \(0.5,2.5\) ) where the units are centimetres. The curve \(C _ { 1 }\) is modelled by the equation $$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$
  1. Determine the value of \(a\) and the value of \(b\) according to the model. The curve \(C _ { 2 }\) is modelled to be an arc of the circle with centre \(( 0,3 )\).
  2. Use calculus to determine the volume of plastic required to make the chess piece according to the model.
Edexcel CP2 2021 June Q1
  1. Given that
$$\begin{aligned} z _ { 1 } & = 3 \left( \cos \left( \frac { \pi } { 3 } \right) + \mathrm { i } \sin \left( \frac { \pi } { 3 } \right) \right)
z _ { 2 } & = \sqrt { 2 } \left( \cos \left( \frac { \pi } { 12 } \right) - \mathrm { i } \sin \left( \frac { \pi } { 12 } \right) \right) \end{aligned}$$
  1. write down the exact value of
    1. \(\left| Z _ { 1 } Z _ { 2 } \right|\)
    2. \(\arg \left( \mathrm { z } _ { 1 } \mathrm { z } _ { 2 } \right)\) Given that \(w = z _ { 1 } z _ { 2 }\) and that \(\arg \left( w ^ { n } \right) = 0\), where \(n \in \mathbb { Z } ^ { + }\)
  2. determine
    1. the smallest positive value of \(n\)
    2. the corresponding value of \(\left| w ^ { n } \right|\)
Edexcel CP2 2021 June Q2
2. $$A = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.
(5)
Edexcel CP2 2021 June Q3
3. $$f ( x ) = \arcsin x \quad - 1 \leqslant x \leqslant 1$$
  1. Determine the first two non-zero terms, in ascending powers of \(x\), of the Maclaurin series for \(\mathrm { f } ( x )\), giving each coefficient in its simplest form.
  2. Substitute \(x = \frac { 1 } { 2 }\) into the answer to part (a) and hence find an approximate value for \(\pi\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers to be determined.
Edexcel CP2 2021 June Q4
  1. In this question you may assume the results for
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 } \text { and } \sum _ { r = 1 } ^ { n } r$$
  1. Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n ^ { 2 } \left( 2 n ^ { 2 } - 1 \right)$$ The sum of the cubes of 10 consecutive positive odd numbers is 99800
  2. Use the answer to part (a) to determine the smallest of these 10 consecutive positive odd numbers.
Edexcel CP2 2021 June Q5
  1. The curve \(C\) has equation
$$y = \arccos \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2$$
  1. Show that \(C\) has no stationary points. The normal to \(C\), at the point where \(x = 1\), crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\). Given that \(O\) is the origin,
  2. show that the area of the triangle \(O A B\) is \(\frac { 1 } { 54 } \left( p \sqrt { 3 } + q \pi + r \sqrt { 3 } \pi ^ { 2 } \right)\) where \(p\), \(q\) and \(r\) are integers to be determined.
    (5)
Edexcel CP2 2021 June Q6
  1. The curve \(C\) has equation
$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\)
There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
  1. Show that the range of possible values for \(p\) is $$2 < p < 4$$
  2. Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
    The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres.
    Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
    Given that the pond is initially empty,
  3. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
  4. State a limitation of the model.
Edexcel CP2 2021 June Q7
  1. Solutions based entirely on graphical or numerical methods are not acceptable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aaf73eef-4103-48c2-865e-e8288891ae80-20_480_930_299_555} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \operatorname { arsinh } x \quad x \geqslant 0$$ and the straight line with equation \(y = \beta\)
The line and the curve intersect at the point with coordinates \(( \alpha , \beta )\)
Given that \(\beta = \frac { 1 } { 2 } \ln 3\)
  1. show that \(\alpha = \frac { 1 } { \sqrt { 3 } }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \operatorname { arsinh } x\), the \(y\)-axis and the line with equation \(y = \beta\) The region \(R\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
  2. Use calculus to find the exact value of the volume of the solid generated.
Edexcel CP2 2021 June Q8
  1. (i) The point \(P\) is one vertex of a regular pentagon in an Argand diagram.
The centre of the pentagon is at the origin.
Given that \(P\) represents the complex number \(6 + 6 \mathrm { i }\), determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
(ii) (a) On a single Argand diagram, shade the region, \(R\), that satisfies both $$| z - 2 i | \leqslant 2 \quad \text { and } \quad \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi$$ (b) Determine the exact area of \(R\), giving your answer in simplest form.
Edexcel CP2 2021 June Q9
  1. (a) Given that \(| z | < 1\), write down the sum of the infinite series
$$1 + z + z ^ { 2 } + z ^ { 3 } + \ldots$$ (b) Given that \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\),
  1. use the answer to part (a), and de Moivre's theorem or otherwise, to prove that $$\frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$
  2. show that the sum of the infinite series \(1 + z + z ^ { 2 } + z ^ { 3 } + \ldots\) cannot be purely imaginary, giving a reason for your answer.
Edexcel CP2 2022 June Q1
  1. A student was asked to answer the following:
For the complex numbers \(z _ { 1 } = 3 - 3 \mathrm { i }\) and \(z _ { 2 } = \sqrt { 3 } + \mathrm { i }\), find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) The student's attempt is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-02_798_1109_534_338} The student made errors in line 1 and line 3
Correct the error that the student made in
    1. line 1
    2. line 3
  2. Write down the correct value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)