Questions — Edexcel (9671 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel CP2 2020 June Q1
7 marks Standard +0.8
  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
9 marks Standard +0.3
  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
14 marks Standard +0.8
  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
10 marks Challenging +1.2
  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
10 marks Standard +0.8
  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
14 marks Standard +0.8
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
11 marks Standard +0.8
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
5 marks Moderate -0.3
  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
5 marks Standard +0.8
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
6 marks Standard +0.8
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
9 marks Standard +0.8
  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
8 marks Standard +0.8
  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
14 marks
  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
9 marks Challenging +1.2
  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
11 marks Standard +0.8
  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
8 marks Standard +0.8
  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
3 marks Moderate -0.3
  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 \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)
Edexcel CP2 2022 June Q2
8 marks Standard +0.3
  1. In this question you must show all stages of your working.
A college offers only three courses: Construction, Design and Hospitality. Each student enrols on just one of these courses. In 2019, there was a total of 1110 students at this college.
There were 370 more students enrolled on Construction than Hospitality.
In 2020 the number of students enrolled on
  • Construction increased by \(1.25 \%\)
  • Design increased by \(2.5 \%\)
  • Hospitality decreased by \(2 \%\)
In 2020, the total number of students at the college increased by \(0.27 \%\) to 2 significant figures.
    1. Define, for each course, a variable for the number of students enrolled on that course in 2019.
    2. Using your variables from part (a)(i), write down three equations that model this situation.
  2. By forming and solving a matrix equation, determine how many students were enrolled on each of the three courses in 2019.
Edexcel CP2 2022 June Q3
11 marks Standard +0.3
  1. \(\mathbf { M } = \left( \begin{array} { l l } 3 & a \\ 0 & 1 \end{array} \right) \quad\) where \(a\) is a constant
    1. Prove by mathematical induction that, for \(n \in \mathbb { N }\)
    $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { a } { 2 } \left( 3 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$ Triangle \(T\) has vertices \(A , B\) and \(C\).
    Triangle \(T\) is transformed to triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { M } ^ { n }\) where \(n \in \mathbb { N }\) Given that
    • triangle \(T\) has an area of \(5 \mathrm {~cm} ^ { 2 }\)
    • triangle \(T ^ { \prime }\) has an area of \(1215 \mathrm {~cm} ^ { 2 }\)
    • vertex \(A ( 2 , - 2 )\) is transformed to vertex \(A ^ { \prime } ( 123 , - 2 )\)
    • determine
      1. the value of \(n\)
      2. the value of \(a\)
Edexcel CP2 2022 June Q4
6 marks Standard +0.8
  1. (i) Given that
$$z _ { 1 } = 6 \mathrm { e } ^ { \frac { \pi } { 3 } \mathrm { i } } \text { and } z _ { 2 } = 6 \sqrt { 3 } \mathrm { e } ^ { \frac { 5 \pi } { 6 } \mathrm { i } }$$ show that $$z _ { 1 } + z _ { 2 } = 12 \mathrm { e } ^ { \frac { 2 \pi } { 3 } \mathrm { i } }$$ (ii) Given that $$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$ determine the least value of \(| z |\) as \(z\) varies.
Edexcel CP2 2022 June Q5
6 marks Standard +0.3
  1. (a) Given that
$$y = \arcsin x \quad - 1 \leqslant x \leqslant 1$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$$ (b) $$\mathrm { f } ( x ) = \arcsin \left( \mathrm { e } ^ { x } \right) \quad x \leqslant 0$$ Prove that \(\mathrm { f } ( x )\) has no stationary points.
Edexcel CP2 2022 June Q6
10 marks
  1. The cubic equation
$$4 x ^ { 3 } + p x ^ { 2 } - 14 x + q = 0$$ where \(p\) and \(q\) are real positive constants, has roots \(\alpha , \beta\) and \(\gamma\)
Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 16\)
  1. show that \(p = 12\) Given that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { 14 } { 3 }\)
  2. determine the value of \(q\) Without solving the cubic equation,
  3. determine the value of \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
Edexcel CP2 2022 June Q7
10 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-22_678_776_248_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 1 + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 3 }$$ Figure 1 also shows the tangent to \(C\) at the point \(A\).
This tangent is perpendicular to the initial line.
  1. Use differentiation to prove that the polar coordinates of \(A\) are \(\left( 2 , \frac { \pi } { 4 } \right)\) The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the tangent at \(A\) and the initial line.
  2. Use calculus to show that the exact area of \(R\) is \(\frac { 1 } { 2 } ( 1 - \ln 2 )\)
Edexcel CP2 2022 June Q8
13 marks Standard +0.3
  1. Two birds are flying towards their nest, which is in a tree.
Relative to a fixed origin, the flight path of each bird is modelled by a straight line.
In the model, the equation for the flight path of the first bird is $$\mathbf { r } _ { 1 } = \left( \begin{array} { r } - 1 \\ 5 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ a \\ 0 \end{array} \right)$$ and the equation for the flight path of the second bird is $$\mathbf { r } _ { 2 } = \left( \begin{array} { r } 4 \\ - 1 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(a\) is a constant.
In the model, the angle between the birds’ flight paths is \(120 ^ { \circ }\)
  1. Determine the value of \(a\).
  2. Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point. The position of the nest is modelled as being at this common point.
    The tree containing the nest is in a park.
    The ground level of the park is modelled by the plane with equation $$2 x - 3 y + z = 2$$
  3. Hence determine the shortest distance from the nest to the ground level of the park.
  4. By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.
Edexcel CP2 2022 June Q9
8 marks Challenging +1.2
9. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.