Questions - Edexcel - A-Level Maths

Show that $$\frac { \mathrm { d } h } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 6 t - 17 \right) \left( 9 t ^ { 2 } + 4 t - 3 \right) } { 2 \left( 1 + t ^ { 2 } \right) ^ { 2 } }$$ where $t = \tan \left( \frac { x } { 4 } \right)$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_581_1403_1263_331} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Source: ${ } ^ { 1 }$ Data taken on 29th December 2016 from \href{http://www.ukho.gov.uk/easytide/EasyTide}{http://www.ukho.gov.uk/easytide/EasyTide} Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 08:00 on the 3rd January 2017 to the end of the 4th January $2017 { } ^ { 1 }$. The graph of $k \mathrm {~h} ( x )$, where $k$ is a constant and $x$ is the number of hours after 08:00 on 3rd of January, can be used to model the predicted tide heights, in metres, for this period of time.
1. Suggest a value of $k$ that could be used for the graph of $k \mathrm {~h} ( x )$ to form a suitable model.
2. Why may such a model be suitable to predict the times when the tide heights are at their peaks, but not to predict the heights of these peaks?
Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of the highest tide height on the 4th January 2017.

Edexcel FP2 2019 June Q1

5 marks Standard +0.3

A complex number $z = x + \mathrm { i } y$ is represented by the point $P$ in an Argand diagram.

Given that $$| z - 3 | = 4 | z + 1 |$$

show that the locus of $P$ has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
Hence find the maximum value of $| z |$

Edexcel FP2 2019 June Q2

11 marks Challenging +1.2

The matrix $\mathbf { A }$ is given by

$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2 \\ - 2 & 3 & - 1 \\ 2 & - 1 & 3 \end{array} \right)$$

Show that 2 is a repeated eigenvalue of $\mathbf { A }$ and find the other eigenvalue.
Hence find three non-parallel eigenvectors of $\mathbf { A }$.
Find a matrix $\mathbf { P }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P }$ is a diagonal matrix.

Edexcel FP2 2019 June Q3

8 marks Standard +0.3

The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus $k$ times the number of visits received in the month before that, where $k$ is a positive constant.

Given that $V _ { n }$ is the number of visits to the website in month $n$,

write down a general recurrence relation for $V _ { n + 2 }$ in terms of $V _ { n + 1 } , V _ { n }$ and $k$. For a particular website you are given that
- $k = 0.24$
- In month 1 , there were 65 visits to the website.
- In month 2 , there were 71 visits to the website.
- Show that
$$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month $N$.
Find the value of $N$.

Edexcel FP2 2019 June Q4

12 marks Standard +0.3

1. Use Fermat's Little Theorem to find the least positive residue of $6 ^ { 542 }$ modulo 13
2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if
  1. there are no restrictions where they sit in the row,
3. Alan, Enid, Felix and Graham sit together,
4. Brenda sits at one end of the row and Graham sits at the other end of the row,
5. Charles and Devindra do not sit together.

Edexcel FP2 2019 June Q5

8 marks Challenging +1.8

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$

Prove that, for $n \geqslant 2$ $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$

Edexcel FP2 2019 June Q6

12 marks Challenging +1.2

1. A binary operation * is defined on positive real numbers by

$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) The set $G = \{ 1,2,3,4,5,6 \}$ forms a group under the operation of multiplication modulo 7

Show that $G$ is cyclic. The set $H = \{ 1,5,7,11,13,17 \}$ forms a group under the operation of multiplication modulo 18
List all the subgroups of $H$.
Describe an isomorphism between $G$ and $H$.

Edexcel FP2 2019 June Q7

6 marks Challenging +1.2

A transformation from the $z$-plane to the $w$-plane is given by

$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$

Show that the circle $C$ with equation $| z + \mathrm { i } | = 1$ in the $z$-plane is mapped to a circle $D$ in the $w$-plane, giving a Cartesian equation for $D$.
Sketch $C$ and $D$ on Argand diagrams.

Edexcel FP2 2019 June Q8

13 marks Challenging +1.8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4ba4a815-f53d-4de2-810b-b06e145f457b-24_547_629_242_717} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point $A$ is vertically above the point $B$ and the height of the spinning top is 5 cm . The line $C D$ is perpendicular to $A B$ such that $C D$ is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through $2 \pi$ radians about the initial line.

Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where $k$ is a constant to be determined.
Show that, according to the model, the length $C D$ is $\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}$.

Edexcel FP2 2020 June Q1

6 marks Moderate -0.3

A small sports club has 12 adult members and 14 junior members.

The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if

there are no restrictions on the team,
the team must contain 4 adults and 4 juniors,
more than half the team must be adults.

Edexcel FP2 2020 June Q2

9 marks Challenging +1.2

Solve the recurrence system

$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$

Edexcel FP2 2020 June Q3

10 marks Standard +0.3

3. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & - 2 \\ 2 & - 4 & 1 \\ 1 & 2 & 3 \end{array} \right)$$ where $k$ is a constant.

Show that, in terms of $k$, a characteristic equation for $\mathbf { M }$ is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that $\operatorname { det } \mathbf { M } = 5$
1. find the value of $k$
2. use the Cayley-Hamilton theorem to find the inverse of $\mathbf { M }$.

Edexcel FP2 2020 June Q4

10 marks Challenging +1.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_213_684_257_221} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_193_736_258_1053} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross-section with vertical ends and its length is 30 cm . A side profile of the speed bump is shown in Figure 2. The curve $C$ shown in Figure 2 is modelled by the polar equation $$r = 30 \left( 1 - \theta ^ { 2 } \right) \quad 0 \leqslant \theta \leqslant 1$$ The units for $r$ are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in $\mathrm { cm } ^ { 2 }$, the area that is to be painted, according to the model.

Edexcel FP2 2020 June Q5

10 marks Challenging +1.2

A transformation $T$ from the $z$-plane to the $w$-plane is given by

$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation $| z + \mathrm { i } | = 3$ is mapped by $T$ onto the circle $C$.

Show that the equation for $C$ can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
Hence find
1. a Cartesian equation for $C$,
2. the centre and radius of $C$.

Edexcel FP2 2020 June Q6

10 marks

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-20_371_328_255_870} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, $G$.

Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
Write down, in similar two-line notation, the remaining elements of the group $G$.
Explain why each of the following statements is false, making your reasoning clear.
1. $G$ has a subgroup of order 4
2. $G$ is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
\includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
\section*{Diagram 1} \section*{Spare copy of Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
Only use this diagram if you need to redraw your answer to part (d).

Edexcel FP2 2020 June Q7

8 marks Challenging +1.8

7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$

Show that, for $n > 0$ $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
Find $I _ { 2 }$

Edexcel FP2 2020 June Q8

12 marks Challenging +1.8

The four digit number $n = a b c d$ satisfies the following properties:
(1) $n \equiv 3 ( \bmod 7 )$
(2) $n$ is divisible by 9
(3) the first two digits have the same sum as the last two digits
(4) the digit $b$ is smaller than any other digit
(5) the digit $c$ is even
1. Use property (1) to explain why $6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )$
2. Use properties (2), (3) and (4) to show that $a + b = 9$
3. Deduce that $c \equiv 5 ( a - 1 ) ( \bmod 7 )$
4. Hence determine the number $n$, verifying that it is unique. You must make your reasoning clear.

Edexcel FP2 2021 June Q1

4 marks Standard +0.3

In this question you must show detailed reasoning.

Without performing any division, explain why $n = 20210520$ is divisible by 66

Edexcel FP2 2021 June Q2

8 marks Standard +0.3

A binary operation ★ on the set of non-negative integers, $\mathbb { Z } _ { 0 } ^ { + }$, is defined by

$$m \star n = | m - n | \quad m , n \in \mathbb { Z } _ { 0 } ^ { + }$$

Explain why $\mathbb { Z } _ { 0 } ^ { + }$is closed under the operation
Show that 0 is an identity for $\left( \mathbb { Z } _ { 0 } ^ { + } , \star \right)$
Show that all elements of $\mathbb { Z } _ { 0 } ^ { + }$have an inverse under ★
Determine if $\mathbb { Z } _ { 0 } ^ { + }$forms a group under ★, giving clear justification for your answer.

Edexcel FP2 2021 June Q3

8 marks Standard +0.3

(a) Use the Euclidean Algorithm to find integers $a$ and $b$ such that

$$125 a + 87 b = 1$$ (b) Hence write down a multiplicative inverse of 87 modulo 125
(c) Solve the linear congruence $$87 x \equiv 16 ( \bmod 125 )$$

Edexcel FP2 2021 June Q4

7 marks Challenging +1.8

Let $G$ be a group of order $46 ^ { 46 } + 47 ^ { 47 }$

Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of $G$

Edexcel FP2 2021 June Q5

10 marks Standard +0.8

The point $P$ in the complex plane represents a complex number $z$ such that

$$| z + 9 | = 4 | z - 12 i |$$ Given that, as $z$ varies, the locus of $P$ is a circle,

determine the centre and radius of this circle.
Shade on an Argand diagram the region defined by the set $$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$

Edexcel FP2 2021 June Q6

6 marks Challenging +1.2

A recurrence system is defined by

$$\begin{aligned} u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1 \\ u _ { 1 } & = - 3 , u _ { 2 } = 18 \end{aligned}$$ Prove by induction that, for $n \in \mathbb { N }$, $$u _ { n } = ( - 3 ) ^ { n } n !$$

Edexcel FP2 2021 June Q7

15 marks Challenging +1.8

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

You must not use the integration facility on your calculator. $$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$

Show that, for $n > 1$ $$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve shown in Figure 1 is defined by the parametric equations $$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$ This curve is rotated through $2 \pi$ radians about the $x$-axis to form a hollow open shell.
Show that the external surface area of the shell is given by $$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$ Using the results in parts (a) and (b) and making each step of your working clear,
determine the value of the external surface area of the shell, giving your answer to 3 significant figures.

Edexcel FP2 2021 June Q8

17 marks Challenging +1.2

8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5 \\ 0 & 3 & p \\ - 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that $\left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right)$ is an eigenvector for $\mathbf { A }$

1. determine the eigenvalue corresponding to this eigenvector
2. hence show that $p = 2$
3. determine the remaining eigenvalues and corresponding eigenvectors of $\mathbf { A }$
Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } = \mathbf { P D P } ^ { - 1 }$
1. Solve the differential equation $\dot { u } = k u$, where $k$ is a constant. With respect to a fixed origin $O$, the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right)$$ By considering $\left( \begin{array} { c } u \\ v \\ w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x \\ y \\ z \end{array} \right)$ so that $\left( \begin{array} { c } \dot { u } \\ \dot { v } \\ \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right)$
2. determine a general solution for the displacement of the particle.

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