Questions - Edexcel - A-Level Maths

(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.

Edexcel CP2 2023 June Q3

10 marks Standard +0.3

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

Edexcel CP2 2023 June Q4

7 marks Challenging +1.2

(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 $A$.

Edexcel CP2 2023 June Q5

9 marks Challenging +1.2

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 $\alpha$

Show that $\alpha = 3 + 7 \mathrm { i }$ Given that $\beta = \frac { 1 + \mathrm { i } } { 64 }$
show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of $H$ are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
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.

Edexcel CP2 2023 June Q6

6 marks Challenging +1.8

Given that

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

Edexcel CP2 2023 June Q7

8 marks Challenging +1.2

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{59a57888-8aa8-4ed8-b704-ebf3980c0344-20_557_558_408_756} \captionsetup{labelformat=empty} \caption{Figure 2}

\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,

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,
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.

Edexcel CP2 2023 June Q8

11 marks Challenging +1.2

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 $\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }$
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 $\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )$

Edexcel CP2 2023 June Q9

14 marks Challenging +1.2

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 $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.025 y - 0.045 x + 2 \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.032 x - 0.025 y \end{aligned}$$

Show that $$40000 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2800 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 2560$$
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.
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 $200 \mathrm { mg } / \mathrm { ml }$.
Determine whether the rate of administration of the antibiotic is effective for the patient, giving a reason for your answer.

Edexcel CP2 2024 June Q1

7 marks Standard +0.3

(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.

Edexcel CP2 2024 June Q2

8 marks Standard +0.8

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

Show that $$f ^ { \prime } ( x ) = - \frac { 1 } { 2 x + 3 }$$
Hence determine $\mathrm { f } ^ { \prime \prime } ( x )$
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.

Edexcel CP2 2024 June Q3

5 marks Challenging +1.2

(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.

Edexcel CP2 2024 June Q4

6 marks Standard +0.8

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.

Edexcel CP2 2024 June Q5

9 marks Standard +0.8

The locus $C$ is given by

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

Sketch, on the same Argand diagram, the locus $C$ and the locus $D$ The set of points $A$ is defined by $$A = \{ z \in \mathbb { C } : | z - 4 | \leqslant 4 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\}$$
Show, by shading on your Argand diagram, the set of points $A$
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.

Edexcel CP2 2024 June Q6

14 marks Challenging +1.2

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, $t \geqslant 0$

Determine the general solution of the differential equation.
Hence determine the particular solution for which $x = 3$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2$ when $t = 0$
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.
Comment on the suitability of the model in light of this information.

Edexcel CP2 2024 June Q7

9 marks Moderate -0.8

(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 $0 \leqslant \theta < 2 \pi$

Edexcel CP2 2024 June Q8

7 marks Standard +0.3

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

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

Edexcel CP2 2024 June Q9

10 marks Challenging +1.2

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f2d33c3-eb35-4b50-9a4d-54f43c514f49-28_586_560_246_411} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f2d33c3-eb35-4b50-9a4d-54f43c514f49-28_606_542_269_1110} \captionsetup{labelformat=empty} \caption{Figure 2}

\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.

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

Edexcel CP2 Specimen Q1

8 marks Moderate -0.3

The roots of the equation

$$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are $\alpha , \beta$ and $\gamma$
Without solving the equation, find the value of

$\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }$
$( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )$
$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$

Edexcel CP2 Specimen Q2

8 marks Standard +0.3

The plane $\Pi _ { 1 }$ has vector equation

$$\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

Find the perpendicular distance from the point $( 6,2,12 )$ to the plane $\Pi _ { 1 }$ The plane $\Pi _ { 2 }$ has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$ where $\lambda$ and $\mu$ are scalar parameters.
Show that the vector $- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ is perpendicular to $\Pi _ { 2 }$
Show that the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$ is $52 ^ { \circ }$ to the nearest degree.

Edexcel CP2 Specimen Q3

12 marks Standard +0.3

$$\mathbf { M } = \left( \begin{array} { c c c } 2 & a & 4 \\ 1 & - 1 & - 1 \\ - 1 & 2 & - 1 \end{array} \right)$$ where $a$ is a constant.

For which values of $a$ does the matrix $\mathbf { M }$ have an inverse? Given that $\mathbf { M }$ is non-singular,
find $\mathbf { M } ^ { - 1 }$ in terms of $a$
(ii) Prove by induction that for all positive integers $n$, $$\left( \begin{array} { l l } 3 & 0 \\ 6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0 \\ 3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)$$

Edexcel CP2 Specimen Q4

7 marks Standard +0.8

A complex number $z$ has modulus 1 and argument $\theta$.
1. Show that
$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta , \quad n \in \mathbb { Z } ^ { + }$$
Hence, show that $$\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )$$

Edexcel CP2 Specimen Q5

10 marks Challenging +1.2

5. $$y = \sin x \sinh x$$

Show that $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = - 4 y$
Hence find the first three non-zero terms of the Maclaurin series for $y$, giving each coefficient in its simplest form.
Find an expression for the $n$th non-zero term of the Maclaurin series for $y$.

Edexcel CP2 Specimen Q6

13 marks Challenging +1.8

(a) (i) Show on an Argand diagram the locus of points given by the values of $z$ satisfying

$$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $\theta \in [ \alpha , \alpha + \pi ]$, where $\alpha = - \arctan \left( \frac { 4 } { 3 } \right)$,
(ii) show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points $A$ is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$ (b) (i) Show, by shading on your Argand diagram, the set of points $A$.
(ii) Find the exact area of the region defined by $A$, giving your answer in simplest form.

Edexcel CP2 Specimen Q7

17 marks Standard +0.8

At the start of the year 2000, a survey began of the number of foxes and rabbits on an island.

At time $t$ years after the survey began, the number of foxes, $f$, and the number of rabbits, $r$, on the island are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } f } { \mathrm {~d} t } = 0.2 f + 0.1 r \\ & \frac { \mathrm {~d} r } { \mathrm {~d} t } = - 0.2 f + 0.4 r \end{aligned}$$

Show that $\frac { \mathrm { d } ^ { 2 } f } { \mathrm {~d} t ^ { 2 } } - 0.6 \frac { \mathrm {~d} f } { \mathrm {~d} t } + 0.1 f = 0$
Find a general solution for the number of foxes on the island at time $t$ years.
Hence find a general solution for the number of rabbits on the island at time $t$ years. At the start of the year 2000 there were 6 foxes and 20 rabbits on the island.
1. According to this model, in which year are the rabbits predicted to die out?
2. According to this model, how many foxes will be on the island when the rabbits die out?
3. Use your answers to parts (i) and (ii) to comment on the model.

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