Edexcel CP2 (Core Pure 2) 2024 June

1. (a) Using the definition of \(\sinh x\) in terms of exponentials, prove that

$$4 \sinh ^ { 3 } x + 3 \sinh x \equiv \sinh 3 x$$ (b) Hence solve the equation $$\sinh 3 x = 19 \sinh x$$ giving your answers as simplified natural logarithms where appropriate.

2. $$f ( x ) = \tanh ^ { - 1 } \left( \frac { 3 - x } { 6 + x } \right) \quad | x | < \frac { 3 } { 2 }$$

1. Show that $$f ^ { \prime } ( x ) = - \frac { 1 } { 2 x + 3 }$$
2. Hence determine \(\mathrm { f } ^ { \prime \prime } ( x )\)
3. Hence show that the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 2 }\), is $$\ln p + q x + r x ^ { 2 }$$ where \(p , q\) and \(r\) are constants to be determined.

1. (a) Explain why

$$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x$$ is an improper integral.
(b) Show that $$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x = k \pi$$ where \(k\) is a constant to be determined.

1. Use the method of differences to show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 4 ) ( r + 6 ) } = \frac { n ( a n + b ) } { 30 ( n + 5 ) ( n + 6 ) }$$ where \(a\) and \(b\) are integers to be determined.

1. The locus \(C\) is given by

$$| z - 4 | = 4$$ The locus \(D\) is given by $$\arg z = \frac { \pi } { 3 }$$

1. Sketch, on the same Argand diagram, the locus \(C\) and the locus \(D\) The set of points \(A\) is defined by $$A = \{ z \in \mathbb { C } : | z - 4 | \leqslant 4 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\}$$
2. Show, by shading on your Argand diagram, the set of points \(A\)
3. Find the area of the region defined by \(A\), giving your answer in the form \(p \pi + q \sqrt { 3 }\) where \(p\) and \(q\) are constants to be determined.

1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation

$$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 4 t + 12$$ where \(P\) is \(x\) metres from the origin \(O\) at time \(t\) seconds, \(t \geqslant 0\)

1. Determine the general solution of the differential equation.
2. Hence determine the particular solution for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2\) when \(t = 0\)
3. 1. Show that, according to the model, the minimum distance between \(O\) and \(P\) is \(( 2 + \ln 2 )\) metres.
   2. Justify that this distance is a minimum. For large values of \(t\) the particle is expected to move with constant speed.
4. Comment on the suitability of the model in light of this information.

1. (a) Determine the roots of the equation

$$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
(b) Show the roots of the equation in part (a) on a single Argand diagram.
(c) Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$ (d) Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)

8. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1
1 & 1 & 1
k & 3 & 6 \end{array} \right) \quad k \neq 0$$

1. Find, in terms of \(k , \mathbf { A } ^ { - 1 }\)
2. Determine, in simplest form in terms of \(k\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 3 x + y - z = 3
   x + y + z = 1
   k x + 3 y + 6 z = 6 \end{array}$$

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section \(A B C D E F A\) of a vase together with measurements that have been taken from the vase. The horizontal cross-section between \(A B\) and \(F C\) is a circle with diameter 4 cm .
The base of the vase \(E D\) is horizontal and the point \(E\) is vertically below \(F\) and the point \(D\) is vertically below \(C\). Using these measurements, the curve \(C D\) is modelled by the parametric equations $$x = a + 3 \sin 2 t \quad y = b \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Determine the value of \(a\) and the value of \(b\) according to the model.
2. Using algebraic integration and showing all your working, determine, according to the model, the volume of the vase, giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\)
3. State a limitation of the model.