Edexcel CP2 (Core Pure 2) 2021 June

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 \(\left| w ^ { n } \right|\)

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)

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.

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.

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)

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.

1. Solutions based entirely on graphical or numerical methods are not acceptable.

\begin{figure}[h] \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.

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.

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.