Edexcel CP2 (Core Pure 2) 2023 June

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = 2 \sqrt { \sinh \theta + \cosh \theta } \quad 0 \leqslant \theta \leqslant \pi$$ The region \(R\), shown shaded in Figure 1, is bounded by the initial line, the curve and the line with equation \(\theta = \pi\) Use algebraic integration to determine the exact area of \(R\) giving your answer in the form \(p \mathrm { e } ^ { q } - r\) where \(p , q\) and \(r\) are real numbers to be found.

1. (a) Write down the Maclaurin series of \(\mathrm { e } ^ { x }\), in ascending power of \(x\), up to and including the term in \(x ^ { 3 }\)
   (b) Hence, without differentiating, determine the Maclaurin series of

$$\mathrm { e } ^ { \left( \mathrm { e } ^ { x } - 1 \right) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.

3. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5
6 & k \end{array} \right)$$ where \(k\) is a constant.
Given that $$\mathbf { M } ^ { 2 } + 11 \mathbf { M } = a \mathbf { I }$$ where \(a\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = - 9\)
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf { M }\).
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.

1. (a) Sketch the polar curve \(C\), with equation

$$r = 3 + \sqrt { 5 } \cos \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ On your sketch clearly label the pole, the initial line and the value of \(r\) at the point where the curve intersects the initial line. The tangent to \(C\) at the point \(A\), where \(0 < \theta < \frac { \pi } { 2 }\), is parallel to the initial line.
(b) Use calculus to show that at \(A\) $$\cos \theta = \frac { 1 } { \sqrt { 5 } }$$ (c) Hence determine the value of \(r\) at \(A\).

1. The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.

The centre of \(H\) represents the complex number \(\alpha\)

1. Show that \(\alpha = 3 + 7 \mathrm { i }\) Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
2. show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of \(H\) are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
3. 1. Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
   2. Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.

1. Given that

$$y = \mathrm { e } ^ { 2 x } \sinh x$$ prove by induction that for \(n \in \mathbb { N }\) $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = \mathrm { e } ^ { 2 x } \left( \frac { 3 ^ { n } + 1 } { 2 } \sinh x + \frac { 3 ^ { n } - 1 } { 2 } \cosh x \right)$$

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} John picked 100 berries from a plant.
The largest berry picked was approximately 2.8 cm long.
The shape of this berry is modelled by rotating the curve with equation $$16 x ^ { 2 } + 3 y ^ { 2 } - y \cos \left( \frac { 5 } { 2 } y \right) = 6 \quad x \geqslant 0$$ shown in Figure 2, about the \(y\)-axis through \(2 \pi\) radians, where the units are cm .
Given that the \(y\) intercepts of the curve are - 1.545 and 1.257 to four significant figures,

1. use algebraic integration to determine, according to the model, the volume of this berry. Given that the 100 berries John picked were then squeezed for juice,
2. use your answer to part (a) to decide whether, in reality, there is likely to be enough juice to fill a \(200 \mathrm {~cm} ^ { 3 }\) cup, giving a reason for your answer.

1. Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane,
   1. describe the possible lines the roots can lie on.
   $$f ( z ) = 8 z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(b , c\) and \(d\) are real constants.
   The roots of \(f ( z )\) are distinct and lie on a straight line in the complex plane.
   Given that one of the roots is \(\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }\)
2. state the other two roots of \(\mathrm { f } ( \mathrm { z } )\) $$g ( z ) = z ^ { 3 } + P z ^ { 2 } + Q z + 12$$ where \(P\) and \(Q\) are real constants, has 3 distinct roots.
   The roots of \(g ( z )\) lie on a different straight line in the complex plane than the roots of \(\mathrm { f } ( \mathrm { z } )\) Given that
   • \(f ( z )\) and \(g ( z )\) have one root in common
   • one of the roots of \(\mathrm { g } ( \mathrm { z } )\) is - 4
   • 1. write down the value of the common root,
     2. determine the value of the other root of \(\mathrm { g } ( \mathrm { z } )\)
   • Hence solve the equation \(\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )\)

1. A patient is treated by administering an antibiotic intravenously at a constant rate for some time.

Initially there is none of the antibiotic in the patient.
At time \(t\) minutes after treatment began

• the concentration of the antibiotic in the blood of the patient is \(x \mathrm { mg } / \mathrm { ml }\)
• the concentration of the antibiotic in the tissue of the patient is \(y \mathrm { mg } / \mathrm { ml }\)

The concentration of antibiotic in the patient is modelled by the equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.025 y - 0.045 x + 2
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.032 x - 0.025 y \end{aligned}$$

1. Show that $$40000 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2800 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 2560$$
2. Determine, according to the model, a general solution for the concentration of the antibiotic in the patient's tissue at time \(t\) minutes after treatment began.
3. Hence determine a particular solution for the concentration of the antibiotic in the tissue at time \(t\) minutes after treatment began. To be effective for the patient the concentration of antibiotic in the tissue must eventually reach a level between \(185 \mathrm { mg } / \mathrm { ml }\) and \(200 \mathrm { mg } / \mathrm { ml }\).
4. Determine whether the rate of administration of the antibiotic is effective for the patient, giving a reason for your answer.