Questions — OCR Further Pure Core 2 (129 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
OCR Further Pure Core 2 2017 Specimen Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
OCR Further Pure Core 2 2017 Specimen Q3
4 marks Standard +0.3
3
  1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
  2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.
OCR Further Pure Core 2 2017 Specimen Q4
5 marks Standard +0.3
4 Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.
OCR Further Pure Core 2 2017 Specimen Q5
4 marks Standard +0.8
5 In this question you must show detailed reasoning. Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
OCR Further Pure Core 2 2017 Specimen Q6
8 marks Standard +0.3
6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).
  1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
  2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).
OCR Further Pure Core 2 2017 Specimen Q7
7 marks Challenging +1.2
7
  1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
  2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).
OCR Further Pure Core 2 2021 June Q1
6 marks Moderate -0.8
1
  1. The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347} On the copy of the diagram in the Resource Materials.
    • draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
    • indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
    • You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
    In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Resource Materials. You should join each point to the origin with a straight line.
    1. \(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
    2. \(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)
OCR Further Pure Core 2 2021 June Q2
6 marks Standard +0.8
2 In this question you must show detailed reasoning.
Solve the equation \(2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0\) giving each answer in the form \(\ln ( p + q \sqrt { r } )\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.
  1. State the plane of reflection of \(R\).
  2. Determine the value of \(a\).
  3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
    1. By using Euler's formula show that \(\cosh ( \mathrm { iz } ) = \cos z\).
    2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
      1. Write down the general solution to (\textit{).
      2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where \(\lambda\) is a positive constant.
    4. In the case where \(\lambda = 16\) the door is held open at an angle of 0.9 radians and then released from rest at time \(t = 0\).
      1. Find, in a real form, the general solution of ( \(\dagger\) ).
      2. Find the particular solution of ( \(\dagger\) ).
      3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( \(\dagger\) ) improves the model.
      4. Find the value of \(\lambda\) for which the door is critically damped.
OCR Further Pure Core 2 2021 June Q2
5 marks Moderate -0.3
2 A 2-D transformation \(T\) is a shear which leaves the \(y\)-axis invariant and which transforms the object point \(( 2,1 )\) to the image point \(( 2,9 )\). \(A\) is the matrix which represents the transformation \(T\).
  1. Find A .
  2. By considering the determinant of A , explain why the area of a shape is invariant under T .
OCR Further Pure Core 2 2021 June Q3
11 marks Standard +0.3
3 A particle of mass 2 kg moves along the \(x\)-axis. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\). The particle is subject to two forces.
  • One acts in the positive \(x\)-direction with magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\).
  • One acts in the negative \(x\)-direction with magnitude \(v \mathrm {~N}\).
    1. Show that the motion of the particle can be modelled by the differential equation
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when \(t = 0\).
  • Find \(v\) in terms of \(t\).
  • Find the velocity of the particle when \(t = 2\). When \(t = 2\) the force acting in the positive \(x\)-direction is replaced by a constant force of magnitude \(\frac { 1 } { 2 } \mathrm {~N}\) in the same direction.
  • Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).
  • Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\).
    •
    OCR Further Pure Core 2 2021 June Q4
    8 marks Challenging +1.8
    4 In this question you must show detailed reasoning.
    1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
    2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).
      1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
      2. By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).
    OCR Further Pure Core 2 2021 June Q1
    6 marks Challenging +1.2
    1 In this question you must show detailed reasoning.
    The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
    1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
    2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).
    OCR Further Pure Core 2 2021 June Q2
    6 marks Standard +0.3
    2 In this question you must show detailed reasoning.
    1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
    2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).
    OCR Further Pure Core 2 2021 June Q3
    6 marks Challenging +1.8
    3 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.
    1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
    2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
    3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.
    OCR Further Pure Core 2 2021 June Q4
    9 marks Standard +0.8
    4 In this question you must show detailed reasoning.
    The complex number \(- 4 + i \sqrt { 48 }\) is denoted by \(z\).
    1. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
    2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
    OCR Further Pure Core 2 2021 June Q5
    10 marks Standard +0.3
    5 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
      1. Determine \(f ^ { \prime \prime } ( x )\).
      2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
      3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( \mathrm { x } )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
    2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.
    OCR Further Pure Core 2 2024 June Q1
    5 marks Moderate -0.8
    1. Use the method of differences to show that \(\sum_{r=1}^{n}\left(\frac{1}{r} - \frac{1}{r+1}\right) = 1 - \frac{1}{n+1}\). [1]
    2. Hence determine the following sums.
      1. \(\sum_{r=1}^{90}\frac{1}{r} - \frac{1}{r+1}\) [1]
      2. \(\sum_{r=100}^{\infty}\frac{1}{r} - \frac{1}{r+1}\) [3]
    OCR Further Pure Core 2 2024 June Q2
    6 marks Moderate -0.8
    In this question you must show detailed reasoning.
    1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
    2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]
    OCR Further Pure Core 2 2024 June Q3
    7 marks Standard +0.3
    Matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\).
    1. Find \(2\mathbf{A} - 4\mathbf{B}\). [2]
    2. Write down the matrix \(\mathbf{C}\) such that \(\mathbf{A}\mathbf{C} = 2\mathbf{A}\). [1]
    3. Find the value of \(\det \mathbf{A}\). [1]
    4. In this question you must show detailed reasoning. Use \(\mathbf{A}^{-1}\) to solve the equations \(4x - 3y = 7\) and \(-2x + 2y = 9\). [3]
    OCR Further Pure Core 2 2024 June Q4
    5 marks Challenging +1.2
    In this question you must show detailed reasoning. The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1^2\) to \(779^2\). Determine the value of \(S\). [5]
    OCR Further Pure Core 2 2024 June Q5
    6 marks Challenging +1.2
    Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]
    OCR Further Pure Core 2 2024 June Q6
    11 marks Challenging +1.8
    In polar coordinates, the equation of a curve, \(C\), is \(r = 6\sin(2\theta)\sinh\left(\frac{1}{3}\theta\right)\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.
    1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. [3]
    The incomplete table below shows values of \(r\) for various values of \(\theta\).
    \(\theta\)0\(\frac{1}{12}\pi\)\(\frac{1}{6}\pi\)\(\frac{1}{4}\pi\)\(\frac{1}{3}\pi\)\(\frac{5}{12}\pi\)\(\frac{1}{2}\pi\)
    \(r\)00.2621.851
    1. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). [3]
    The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
    1. Show that \(\phi\) satisfies the equation \(\phi = \frac{3}{4}\ln\left(\frac{6-\tan 2\phi}{6+\tan 2\phi}\right)\) [4]
    2. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to 3 significant figures. [1]
    OCR Further Pure Core 2 2024 June Q7
    10 marks Challenging +1.8
    1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
    A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
    1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]
    OCR Further Pure Core 2 2024 June Q8
    13 marks Standard +0.8
    A children's play centre has two rooms, a room full of bouncy castles and a room full of ball pits. At any given instant, each child in the centre is playing either on the bouncy castles or in the ball pits. Each child can see one room from the other room and can decide to change freely between the two rooms. It is assumed that such changes happen instantaneously. The number of children playing on the bouncy castles at time \(t\) hours, is denoted by \(C\) and the corresponding number of children playing in the ball pits is \(P\). Because the number of children is large for most of the time, \(C\) and \(P\) are modelled as being continuous. When there is a different number of children in each room, some children will move from the room with more children to the room with fewer children. A researcher therefore decides to model \(C\) and \(P\) with the following coupled differential equations. $$\frac{dP}{dt} = \alpha(P-C) + \gamma t$$ $$\frac{dC}{dt} = \alpha(C-P)$$
    1. Explain why \(\alpha\) must be negative. [1]
    After examining data, the researcher chooses \(\alpha = -2\) and \(\gamma = 32\).
    1. Show that \(P\) satisfies the second order differential equation \(\frac{d^2P}{dt^2} + 4\frac{dP}{dt} = 64t + 32\). [2]
      1. Find the complementary function for the differential equation from part (b). [1]
      2. Explain why a particular integral of the form \(P = at + b\) will not work in this situation. [1]
      3. Using a particular integral of the form \(P = at^2 + bt\), find the general solution of the differential equation from part (b). [3]
    At a certain time there are 55 children playing in the ball pits and 24 children per hour are arriving at the ball pits.
    1. Use the model, starting from this time, to estimate the number of children in the ball pits 30 minutes later. [4]
    2. Explain why the model becomes unreliable as \(t\) gets very large. [1]
    OCR Further Pure Core 2 2024 June Q9
    12 marks Challenging +1.2
    In this question, the argument of a complex number is defined as being in the range \([0, 2\pi)\). You are given that \(\omega_k\), where \(k = 0, 1, 2, ..., n-1\), are the \(n\) \(n\)th roots of unity for some integer \(n\), \(n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega_0 = 1\)).
    1. With the help of a diagram explain why \(\omega_k = (\omega_1)^k\) for \(k = 2, ..., n-1\). [3]
    2. Using the identity given in part (a), show that \(\sum_{k=0}^{n-1}\omega_k = 0\). [2]
    3. Show that if \(z\) is a complex number then \(z + z^* = 2\text{Re}(z)\). [1]
    4. Using the results from parts (b) and (c) show that \(\sum_{k=0}^{n-1}\text{Re}(\omega_k) = 0\). [1]
    5. With the help of a diagram explain why \(\text{Re}(\omega_k) = \text{Re}(\omega_{n-k})\) for \(k = 1, 2, ..., n-1\). [1]
    You should now consider the case when \(n = 5\).
      1. Use parts (d) and (e) to deduce that \(\cos\frac{4\pi}{5} = a + b\cos\frac{2\pi}{5}\), for some rational constants \(a\) and \(b\). [2]
      2. Hence determine the exact value of \(\cos\frac{2\pi}{5}\). [2]