Questions - Edexcel - A-Level Maths

determine $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ in terms of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }$ and $y$ Given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ at $x = 0$
determine a series solution for $y$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, giving each coefficient in its simplest form.

Edexcel F2 2022 June Q6

13 marks Challenging +1.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ff9ff379-78d8-41c0-a177-ec346e359249-20_497_1196_260_520} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The curve shown in Figure 1 has polar equation $$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$ where $a$ is a positive constant.
The tangent to the curve at the point $A$ is parallel to the initial line.

Show that the polar coordinates of $A$ are $\left( 6 a , \frac { \pi } { 3 } \right)$ The point $B$ lies on the curve such that angle $A O B = \frac { \pi } { 6 }$ The finite region $R$, shown shaded in Figure 1, is bounded by the line $A B$ and the curve.
Use calculus to determine the area of the shaded region $R$, giving your answer in the form $a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )$, where $n , p$ and $q$ are integers.

Edexcel F2 2022 June Q7

12 marks Challenging +1.2

(a) Show that the transformation $y = x v$ transforms the equation

$$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 6 } { x } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 6 y } { x ^ { 2 } } + 3 y = x ^ { 2 } \quad x \neq 0$$ into the equation $$3 \frac { \mathrm {~d} ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 3 v = x$$ (b) Hence obtain the general solution of the differential equation (I), giving your answer in the form $y = \mathrm { f } ( x )$

Edexcel F2 2022 June Q8

14 marks Challenging +1.2

(a) Use de Moivre's theorem to show that

$$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ (b) Hence determine the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 5 } = 0$$ giving your answers to 3 decimal places.
(c) Use the identity given in part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta - 6 \sin \theta \right) \mathrm { d } \theta = a \sqrt { 2 } + b$$ where $a$ and $b$ are rational numbers to be determined.

Edexcel F2 2023 June Q1

7 marks Standard +0.8

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

Solutions relying on calculator technology are not acceptable.

Show that, for $r \geqslant 2$ $$\frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = \sqrt { r } - \sqrt { r - 2 }$$
Hence use the method of differences to determine $$\sum _ { r = 2 } ^ { n } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } }$$ giving your answer in simplest form.
Hence show that $$\sum _ { r = 4 } ^ { 50 } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = A + B \sqrt { 2 } + C \sqrt { 3 }$$ where $A$, $B$ and $C$ are integers to be determined.

Edexcel F2 2023 June Q2

10 marks Standard +0.3

The complex number $z _ { 1 }$ is defined as

$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$

Without using your calculator show that $$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
Shade, on a single Argand diagram, the region $R$ defined by $$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$ Given that the complex number $z$ lies in $R$
determine the smallest possible positive value of $\arg z$

Edexcel F2 2023 June Q3

7 marks Challenging +1.2

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

\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where $k$ is a positive constant,

show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where $p , q$ and $r$ are expressions in terms of $k$ to be determined.
Hence, or otherwise, determine the values for $x$ for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$

Edexcel F2 2023 June Q4

11 marks Challenging +1.2

(a) Determine the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 48 x ^ { 2 } - 34$$ Given that $y = 4$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 21$ at $x = 0$ (b) determine the particular solution of the differential equation.
(c) Hence find the value of $y$ at $x = - 2$, giving your answer in the form $p \mathrm { e } ^ { q } + r$ where $p , q$ and $r$ are integers to be determined.

Edexcel F2 2023 June Q5

7 marks Challenging +1.2

The transformation $T$ from the $z$-plane, where $z = x + \mathrm { i } y$, to the $w$-plane, where $w = u + \mathrm { i } v$ is given by

$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the $z$-plane with equation $y = 4 x$ is mapped by $T$ onto the circle $C$ in the $w$-plane.

Show that $C$ has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where $k$ is a constant to be determined.
Hence determine
1. the coordinates of the centre of $C$
2. the radius of $C$

Edexcel F2 2023 June Q6

9 marks Challenging +1.2

Given that $y = \sec x$
1. show that
$$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sec x \tan x \left( p \sec ^ { 2 } x + q \right)$$ where $p$ and $q$ are integers to be determined.
Hence determine the Taylor series expansion about $\frac { \pi } { 3 }$ of sec $x$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$, up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 3 }$, giving each coefficient in simplest form.
Use the answer to part (b) to determine, to four significant figures, an approximate value of $\sec \left( \frac { 7 \pi } { 24 } \right)$

Edexcel F2 2023 June Q7

11 marks Challenging +1.2

(a) Show that the substitution $z = y ^ { - 2 }$ transforms the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y + 4 x ^ { 2 } y ^ { 3 } \ln x = 0 \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { 2 z } { x } = 8 x \ln x \quad x > 0$$ (b) By solving differential equation (II), determine the general solution of differential equation (I), giving your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$

Edexcel F2 2023 June Q8

13 marks Challenging +1.8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{709ed2f1-f81c-4820-ac31-e1f86baae9d7-28_552_759_246_660} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that $C$ meets the initial line at the point $A$, as shown in Figure 1,

write down the polar coordinates of $A$. The line $l _ { 1 }$, also shown in Figure 1, is the tangent to $C$ at the point $B$ and is parallel to the initial line.
Use calculus to determine the polar coordinates of $B$. The line $l _ { 2 }$, also shown in Figure 1, is the tangent to $C$ at $A$ and is perpendicular to the initial line. The region $R$, shown shaded in Figure 1, is bounded by $C , l _ { 1 }$ and $l _ { 2 }$
Use algebraic integration to find the exact area of $R$, giving your answer in the form $p \sqrt { 3 } + q \pi$ where $p$ and $q$ are constants to be determined.

Edexcel F2 2024 June Q1

4 marks Moderate -0.3

The complex number $z = x + i y$ satisfies the equation

$$| z - 3 - 4 i | = | z + 1 + i |$$

Determine an equation for the locus of $z$ giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.
Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.

Edexcel F2 2024 June Q2

7 marks Challenging +1.8

2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$

Show that $$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where $a$, $b$ and $c$ are integers to be determined. Given that $y = 1$ at $x = 2$
determine the Taylor series expansion for $y$ in ascending powers of $( x - 2 )$, up to and including the term in $( x - 2 ) ^ { 3 }$, giving each coefficient in simplest form.

Edexcel F2 2024 June Q3

8 marks Standard +0.3

(a) Express

$$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all positive integer values of $n$, $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where $p$ and $q$ are integers to be determined.
(c) Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$

Edexcel F2 2024 June Q4

9 marks Challenging +1.2

(a) Show that the substitution $y ^ { 2 } = \frac { 1 } { t }$ transforms the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y = x y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } - 2 t = - 2 x$$ (b) Solve differential equation (II) and determine $y ^ { 2 }$ in terms of $x$.

Edexcel F2 2024 June Q5

6 marks Standard +0.8

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

Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the values of $x$ for which $$\frac { x + 1 } { ( x - 3 ) ( x + 2 ) } \leqslant 1 - \frac { 2 } { x - 3 }$$

Edexcel F2 2024 June Q6

7 marks Challenging +1.2

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

$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that $T$ maps the imaginary axis in the $z$-plane to the circle $C$ in the $w$-plane, determine (i) the coordinates of the centre of $C$ (ii) the radius of $C$

Edexcel F2 2024 June Q7

7 marks Challenging +1.2

Given that $y = \mathrm { e } ^ { x } \sin x$
1. show that
$$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = k \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where $k$ is a constant to be determined.
Hence determine the first 5 non-zero terms in the Maclaurin series expansion for $y$, giving each coefficient in simplest form.

Edexcel F2 2024 June Q8

10 marks Challenging +1.3

(a) Given that $t = \ln x$, where $x > 0$, show that

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$ (b) Hence show that the transformation $t = \ln x$, where $x > 0$, transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 y = 1 + 4 \ln x - 2 ( \ln x ) ^ { 2 }$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } - 2 y = 1 + 4 t - 2 t ^ { 2 }$$ (c) Solve differential equation (II) to determine $y$ in terms of $t$.
(d) Hence determine the general solution of differential equation (I).

Edexcel F2 2024 June Q9

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.

Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.