Questions - Edexcel FP2 - A-Level Maths

Edexcel FP2 2024 June Q2

Determine a closed form for the recurrence system

$$\begin{gathered} u _ { 1 } = 4 \quad u _ { 2 } = 6
u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad ( n = 1,2,3 , \ldots ) \end{gathered}$$

Edexcel FP2 2024 June Q3

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
1. Use the Euclidean Algorithm to determine the highest common factor $h$ of 234 and 96
2. Hence determine integers $a$ and $b$ such that
$$234 a + 96 b = h$$
Solve the congruence equation $$96 x \equiv 36 ( \bmod 234 )$$

Edexcel FP2 2024 June Q4

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

determine the eigenvalue corresponding to this eigenvector.
Hence show that $p = 3$
Determine
1. the remaining eigenvalues of $\mathbf { A }$,
2. corresponding eigenvectors for these eigenvalues.
Hence determine a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } = \mathbf { P D P } ^ { \mathrm { T } }$

Edexcel FP2 2024 June Q5

1. A circle $C$ in the complex plane is defined by the locus of points satisfying

$$| z - 3 i | = 2 | z |$$

Determine a Cartesian equation for $C$, giving your answer in simplest form.
On an Argand diagram, shade the region defined by $$\{ z \in \mathbb { C } : | z - 3 \mathrm { i } | > 2 | z | \}$$ (ii) The transformation $T$ from the $z$-plane to the $w$-plane is given by $$w = z ^ { 3 }$$
Describe the geometric effect of $T$. The region $R$ in the $z$-plane is given by $$\left\{ z \in \mathbb { C } : 0 < \arg z < \frac { \pi } { 4 } \right\}$$
On a different Argand diagram, sketch the image of $R$ under $T$.

Edexcel FP2 2024 June Q6

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$I _ { n } = \int \frac { \cos ( n x ) } { \sin x } \mathrm {~d} x \quad n \geqslant 1$$

Show that, for $n \geqslant 1$ $$I _ { n + 2 } = \frac { 2 \cos ( n + 1 ) x } { n + 1 } + I _ { n }$$
Hence determine the exact value of $$\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \frac { \cos ( 5 x ) } { \sin x } d x$$ giving the answer in the form $a + b \ln c$ where $a , b$ and $c$ are rational numbers to be found.

Edexcel FP2 2024 June Q7

The set of matrices $G = \{ \mathbf { I } , \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E } \}$ where

$$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) \quad \mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { l l } 1 & 1
1 & 0 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 1
0 & 1 \end{array} \right) \quad \mathbf { D } = \left( \begin{array} { l l } 1 & 0
1 & 1 \end{array} \right) \quad \mathbf { E } = \left( \begin{array} { l l } 0 & 1
1 & 1 \end{array} \right)$$ with the operation $\otimes _ { 2 }$ of matrix multiplication with entries evaluated modulo 2 , forms a group.

Show that $\mathbf { B }$ is an element of order 3 in $G$.
Determine the orders of the other elements of $G$.
Give a reason why $G$ is not isomorphic to
1. a cyclic group of order 6
2. the group of symmetries of a regular hexagon. The group $H$ of permutations of the numbers 1, 2 and 3 contains the following elements, denoted in two-line notation, $$\begin{array} { l l l } e = \left( \begin{array} { l l l } 1 & 2 & 3
  1 & 2 & 3 \end{array} \right) & a = \left( \begin{array} { l l l } 1 & 2 & 3
  2 & 3 & 1 \end{array} \right) & b = \left( \begin{array} { l l l } 1 & 2 & 3
  3 & 1 & 2 \end{array} \right)
  c = \left( \begin{array} { l l } 1 & 2
  1 & 3
  2 \end{array} \right) & d = \left( \begin{array} { l l l } 1 & 2 & 3
  2 & 1 & 3 \end{array} \right) & f = \left( \begin{array} { l l } 1 & 2
  3 & 2 \end{array} \right) \end{array}$$
Determine an isomorphism between the groups $G$ and $H$.

Edexcel FP2 2024 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c20a4592-74c6-4f58-b63b-984b171b1bfd-28_552_380_264_468} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{c20a4592-74c6-4f58-b63b-984b171b1bfd-28_524_446_274_1151} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows a French horn with a detachable bell section.
The shape of the bell section can be modelled by rotating an exponential curve through $360 ^ { \circ }$ about the $x$-axis, where units are centimetres. The model uses the curve shown in Figure 2, with equation $$y = \frac { 9 } { 2 } e ^ { \frac { 1 } { 9 } x } \quad 0 \leqslant x \leqslant 9$$

Show that, according to this model, the external surface area of the bell section is given by $$K \int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x$$ where $K$ is a real constant to be determined.
Use the substitution $u = e ^ { \frac { 1 } { 9 } x }$ to show that $$\int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x = 9 \int _ { a } ^ { b } \frac { 2 u + u ^ { 3 } } { \sqrt { 4 u ^ { 2 } + u ^ { 4 } } } \mathrm {~d} u + 18 \int _ { a } ^ { b } \frac { 1 } { \sqrt { 4 + u ^ { 2 } } } \mathrm {~d} u$$ where $a$ and $b$ are constants to be determined. Hence, using algebraic integration,
determine, according to the model, the external surface area of the bell section of the horn, giving your answer to 3 significant figures.

Edexcel FP2 Specimen Q1

(i) Use the Euclidean algorithm to find the highest common factor of 602 and 161.

Show each step of the algorithm.
(ii) The digits which can be used in a security code are the numbers $1,2,3,4,5,6,7,8$ and 9. Originally the code used consisted of two distinct odd digits, followed by three distinct even digits. To enable more codes to be generated, a new system is devised. This uses two distinct even digits, followed by any three other distinct digits. No digits are repeated. Find the increase in the number of possible codes which results from using the new system.

Edexcel FP2 Specimen Q2

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

$$w = z ^ { 2 }$$

Show that the line with equation $\operatorname { Im } ( z ) = 1$ in the $z$-plane is mapped to a parabola in the $w$-plane, giving an equation for this parabola.
Sketch the parabola on an Argand diagram.

Edexcel FP2 Specimen Q3

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

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0
1 & 2 & 0
- 1 & 0 & 4 \end{array} \right)$$

Show that 4 is an eigenvalue of $\mathbf { M }$, and find the other two eigenvalues.
For each of the eigenvalues find a corresponding eigenvector.
Find a matrix $\mathbf { P }$ such that $\mathbf { P } ^ { - 1 } \mathbf { M P }$ is a diagonal matrix.

Edexcel FP2 Specimen Q4

1. A group $G$ contains distinct elements $a , b$ and $e$ where $e$ is the identity element and the group operation is multiplication.

Given $a ^ { 2 } b = b a$, prove $a b \neq b a$
(ii) The set $H = \{ 1,2,4,7,8,11,13,14 \}$ forms a group under the operation of multiplication modulo 15

Find the order of each element of $H$.
Find three subgroups of $H$ each of order 4, and describe each of these subgroups. The elements of another group $J$ are the matrices $\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right)
- \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)$
where $k = 1,2,3,4,5,6,7,8$ and the group operation is matrix multiplication.
Determine whether $H$ and $J$ are isomorphic, giving a reason for your answer.

Edexcel FP2 Specimen Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-14_480_588_210_740} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where $a > 1$ and $A$ is a positive constant. The curve begins and ends on the $x$-axis, as shown in Figure 1.

Show that the length of this curve is $k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)$, stating the value of the constant $k$. The length of the curved cross-section of the miniature arch is required to be 2 m long.
Find the height of the arch, according to this model, giving your answer to 2 significant figures.
Find also the width of the base of the arch giving your answer to 2 significant figures.
Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
(You do not need to consider the arc length of your curve)

Edexcel FP2 Specimen Q6

A curve has equation

$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$

Show that the curve is a circle with equation $x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0$
Sketch the curve on an Argand diagram. The line $l$ has equation $a z ^ { * } + a ^ { * } z = 0$, where $a \in \mathbb { C }$ and $z \in \mathbb { C }$
Given that the line $l$ is a tangent to the curve and that $\arg a = \theta$
find the possible values of $\tan \theta$

Edexcel FP2 Specimen Q7

7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

Prove that, for $n \geqslant 2$, $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of $\pi \mathrm { m }$.
Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.

Edexcel FP2 Specimen Q8

A staircase has $n$ steps. A tourist moves from the bottom (step zero) to the top (step $n$ ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.

If $u _ { n }$ is the number of ways that the tourist can climb up a staircase with $n$ steps,

explain why $u _ { n }$ satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$

Edexcel FP2 2008 June Q2

Find, in terms of $a$, the polar coordinates of the points where the curve $C _ { 1 }$ meets the circle $C _ { 2 }$.(4) \end{enumerate} The regions enclosed by the curves $C _ { 1 }$ and $C _ { 2 }$ overlap and this common region $R$ is shaded in the figure.
Find, in terms of $a$, an exact expression for the area of the
\includegraphics[max width=\textwidth, alt={}, center]{863ef52d-ae75-450c-9eab-8102804868f5-1_523_707_1262_1255}
region $R$.(8)
In a single diagram, copy the two curves in the diagram above and also sketch the curve $C _ { 3 }$ with polar equation $r = 2 a \cos \theta , 0 \leq \theta < 2 \pi$ Show clearly the coordinates of the points of intersection of $\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ with the initial line, $\theta = 0$.(3)(Total 15 marks)

Edexcel FP2 Q1

(a) Express $\frac { 1 } { r ( r + 2 ) }$ in partial fractions.
(b) Hence show that $\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 2 ) } = \frac { n ( 3 n + 5 ) } { ( n + 1 ) ( n + 2 ) }$.
Solve the equation

$$z ^ { 3 } = 4 \sqrt { } 2 - 4 \sqrt { } 2 i ,$$ giving your answers in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $- \pi < \theta \leq \pi$.
3. Find the general solution of the differential equation $$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$ giving your answer in the form $y = \mathrm { f } ( x )$.
4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-01_279_524_1078_1873} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leq \theta < 2 \pi .$$ The area enclosed by the curve is $\frac { 107 } { 2 } \pi$.
Find the value of $a$.
5. $$y = \sec ^ { 2 } x$$ (a) Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x$.
(b) Find a Taylor series expansion of $\sec ^ { 2 } x$ in ascending powers of $\left( x - \frac { \pi } { 4 } \right)$, up to and including the term in $\left( x - \frac { \pi } { 4 } \right) ^ { 3 }$.
6. A transformation $T$ from the $z$-plane to the $w$-plane is given by $$w = \frac { Z } { Z + \mathrm { i } } , \quad Z \neq - \mathrm { i }$$ The circle with equation $| z | = 3$ is mapped by $T$ onto the curve $C$.
(a) Show that $C$ is a circle and find its centre and radius. The region $| z | < 3$ in the $z$-plane is mapped by $T$ onto the region $R$ in the $w$-plane.
(b) Shade the region $R$ on an Argand diagram.
7. (a) Sketch the graph of $y = \left| x ^ { 2 } - a ^ { 2 } \right|$, where $a > 1$, showing the coordinates of the points where the graph meets the axes.
(b) Solve $\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1$.
(c) Find the set of values of $x$ for which $\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1$.
8. $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \mathrm { e } ^ { - t }$$ Given that $x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 2$ at $t = 0$,
(a) find $x$ in terms of $t$. The solution to part (a) is used to represent the motion of a particle $P$ on the $x$-axis. At time $t$ seconds, where $t > 0 , P$ is $x$ metres from the origin $O$.
(b) Show that the maximum distance between $O$ and $P$ is $\frac { 2 \sqrt { 3 } } { 9 } \mathrm {~m}$ and justify that this distance is a maximum.
(7) \section*{TOTAL FOR PAPER: 75 MARKS} \section*{6668/01} \section*{Further Pure Mathematics FP2 Advanced Subsidiary} \section*{Thursday 24 June 2010 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.

(a) Express $\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }$ in partial fractions.
(b) Using your answer to part (a) and the method of differences, show that

$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$ (c) Evaluate $\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }$, giving your answer to 3 significant figures.

Edexcel FP2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-01_279_524_1078_1873} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leq \theta < 2 \pi .$$ The area enclosed by the curve is $\frac { 107 } { 2 } \pi$.
Find the value of $a$.

Edexcel FP2 Q5

5. $$y = \sec ^ { 2 } x$$

Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x$.
Find a Taylor series expansion of $\sec ^ { 2 } x$ in ascending powers of $\left( x - \frac { \pi } { 4 } \right)$, up to and including the term in $\left( x - \frac { \pi } { 4 } \right) ^ { 3 }$.

Edexcel FP2 Q6

6. A transformation $T$ from the $z$-plane to the $w$-plane is given by $$w = \frac { Z } { Z + \mathrm { i } } , \quad Z \neq - \mathrm { i }$$ The circle with equation $| z | = 3$ is mapped by $T$ onto the curve $C$.

Show that $C$ is a circle and find its centre and radius. The region $| z | < 3$ in the $z$-plane is mapped by $T$ onto the region $R$ in the $w$-plane.
Shade the region $R$ on an Argand diagram.

Edexcel FP2 Q7

7. (a) Sketch the graph of $y = \left| x ^ { 2 } - a ^ { 2 } \right|$, where $a > 1$, showing the coordinates of the points where the graph meets the axes.
(b) Solve $\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1$.
(c) Find the set of values of $x$ for which $\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1$.

Edexcel FP2 Q8

8. $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \mathrm { e } ^ { - t }$$ Given that $x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 2$ at $t = 0$,

find $x$ in terms of $t$. The solution to part (a) is used to represent the motion of a particle $P$ on the $x$-axis. At time $t$ seconds, where $t > 0 , P$ is $x$ metres from the origin $O$.
Show that the maximum distance between $O$ and $P$ is $\frac { 2 \sqrt { 3 } } { 9 } \mathrm {~m}$ and justify that this distance is a maximum.
(7) \section*{TOTAL FOR PAPER: 75 MARKS} \section*{6668/01} \section*{Further Pure Mathematics FP2 Advanced Subsidiary} \section*{Thursday 24 June 2010 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
1. (a) Express $\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }$ in partial fractions.
2. Using your answer to part (a) and the method of differences, show that
$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$
Evaluate $\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }$, giving your answer to 3 significant figures.
2. The displacement $x$ metres of a particle at time $t$ seconds is given by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$ When $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }$.
Find a Taylor series solution for $x$ in ascending powers of $t$, up to and including the term in $t ^ { 3 }$.
3. (a) Find the set of values of $x$ for which $$x + 4 > \frac { 2 } { x + 3 } .$$
Deduce, or otherwise find, the values of $x$ for which $$x + 4 > \frac { 2 } { | x + 3 | }$$ 4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
Find the modulus of $z$ and the argument of $z$. Using de Moivre's theorem,
find $z ^ { 3 }$,
find the values of $w$ such that $w ^ { 4 } = z$, giving your answers in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$.
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-04_339_488_687_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curves given by the polar equations $$\begin{array} { c c } r = 2 , & 0 \leq \theta \leq \frac { \pi } { 2 }
\text { and } r = 1.5 + \sin 3 \theta , & 0 \leq \theta \leq \frac { \pi } { 2 } \end{array}$$
Find the coordinates of the points where the curves intersect. The region $S$, between the curves, for which $r > 2$ and for which $r < ( 1.5 + \sin 3 \theta )$, is shown shaded in Figure 1.
Find, by integration, the area of the shaded region $S$, giving your answer in the form $a \pi + b \sqrt { } 3$, where $a$ and $b$ are simplified fractions.
6. A complex number $z$ is represented by the point $P$ in the Argand diagram.
Given that $| z - 6 | = | z |$, sketch the locus of $P$.
Find the complex numbers $z$ which satisfy both $| z - 6 | = | z |$ and $| z - 3 - 4 \mathrm { i } | = 5$. The transformation $T$ from the $z$-plane to the $w$-plane is given by $w = \frac { 30 } { z }$.
Show that $T$ maps $| z - 6 | = | z |$ onto a circle in the $w$-plane and give the cartesian equation of this circle.
7. (a) Show that the transformation $z = y ^ { \frac { 1 } { 2 } }$ transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$
Solve the differential equation (II) to find $z$ as a function of $x$.
Hence obtain the general solution of the differential equation (I).
8. (a) Find the value of $\lambda$ for which $y = \lambda x \sin 5 x$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at $x = 0 , y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 5$,
find the particular solution of this differential equation, giving your solution in the form $y = \mathrm { f } ( x )$.
Sketch the curve with equation $y = \mathrm { f } ( x )$ for $0 \leq x \leq \pi$.

Edexcel FP2 Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-16_435_837_721_1731} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve $C _ { 1 }$ with polar equation $r = 2 a \sin 2 \theta , 0 \leq \theta \leq \frac { \pi } { 2 }$, and the circle $C _ { 2 }$ with polar equation $r = a , 0 \leq \theta \leq 2 \pi$, where $a$ is a positive constant.

Find, in terms of $a$, the polar coordinates of the points where the curve $C _ { 1 }$ meets the circle $C _ { 2 }$. The regions enclosed by the curve $C _ { 1 }$ and the circle $C _ { 2 }$ overlap and the common region $R$ is shaded in Figure 1.
Find the area of the shaded region $R$, giving your answer in the form $\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { } 3 )$, where $p$ and $q$ are integers to be found. \section*{END} \section*{6668/01R Edexcel GCE} \section*{Further Pure Mathematics FP2 (R)} \section*{Advanced/Advanced Subsidiary} \section*{Friday 6 June 2014 - Afternoon} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question.
When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for the parts of questions are shown in round brackets, e.g. (2).
There are 8 questions in this question paper. The total mark for this paper is 75 .
There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
1. (a) Express $\frac { 2 } { 4 r ^ { 2 } - 1 }$ in partial fractions.
2. Hence use the method of differences to show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
1. Using algebra, find the set of values of $x$ for which
$$3 x - 5 < \frac { 2 } { x }$$
1. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \mathrm { e } ^ { 4 x } \cos ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ giving your answer in the form $y = \mathrm { f } ( x )$.
Find the particular solution for which $y = 1$ at $x = 0$.
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-18_311_841_251_331} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve $C$ with polar equation $$r = 2 \cos 2 \theta , \quad 0 \leq \theta \leq \frac { \pi } { 4 }$$ The line $l$ is parallel to the initial line and is a tangent to $C$.
Find an equation of $l$, giving your answer in the form $r = \mathrm { f } ( \theta )$.
5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0$$
Find an expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ 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 $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0.5$ at $x = 0$,
find a series solution for $y$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
6. The transformation $T$ maps points from the $z$-plane, where $z = x + \mathrm { i } y$, to the $w$-plane, where $w = u + \mathrm { i } v$. The transformation $T$ is given by $$w = \frac { z } { \mathrm { i } z + 1 } , \quad z \neq \mathrm { i }$$ The transformation $T$ maps the line $l$ in the $z$-plane onto the line with equation $v = - 1$ in the $w$-plane.
Find a cartesian equation of $l$ in terms of $x$ and $y$. The transformation $T$ maps the line with equation $y = \frac { 1 } { 2 }$ in the $z$-plane onto the curve $C$ in the $w$-plane.
1. Show that $C$ is a circle with centre the origin.
2. Write down a cartesian equation of $C$ in terms of $u$ and $v$.
  7. (a) Use de Moivre's theorem to show that $$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$
Hence find the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 2 } = 0$$ giving your answers to 3 decimal places where necessary.
Use the identity given in (a) to find $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta \right) d \theta$$ expressing your answer in the form $a \sqrt { } 2 + b$, where $a$ and $b$ are rational numbers.
8. (a) Show that the substitution $x = \mathrm { e } ^ { z }$ transforms the differential equation
into the equation
Find the general solution of the differential equation (II). the form $y = \mathrm { f } ( x )$. $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 3 \ln x , \quad x > 0$$ $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} z ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} z } - 2 y = 3 z$$
Hence obtain the general solution of the differential equation (I) giving your answer in Mathematical Formulae (Pink) \section*{Paper Reference(s)} 6668/01 \section*{Advanced/Advanced Subsidiary} \section*{Friday 6 June 2014 - Afternoon} Time: 1 hour 30 minutes Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have \section*{TOTAL FOR PAPER: 75 MARKS} \section*{\textbackslash section*\{END\}} retrievable mathematical formulae stored in them. Nil In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question.
When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for the parts of questions are shown in round brackets, e.g. (2).
There are 8 questions in this question paper. The total mark for this paper is 75 .
There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. \section*{P44512A} This publication may only be reproduced in accordance with Pearson Education Limited copyright policy. ©2014 Pearson Education Limited.
1. (a) Express $\frac { 2 } { ( r + 2 ) ( r + 4 ) }$ in partial fractions.
2. Hence show that
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 2 ) ( r + 4 ) } = \frac { n ( 7 n + 25 ) } { 12 ( n + 3 ) ( n + 4 ) }$$
1. Use algebra to find the set of values of $x$ for which
$$\left| 3 x ^ { 2 } - 19 x + 20 \right| < 2 x + 2$$ 3. $$y = \sqrt { 8 + \mathrm { e } ^ { x } } , \quad x \in$$ Find the series expansion for $y$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, giving each coefficient in its simplest form.
4. (a) Use de Moivre's theorem to show that
Hence show that
$\_\_\_\_$ v
s $x$ for which
3.
$$\begin{aligned} & \qquad \cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1
& \text { (b) Hence solve for } 0 \leq \theta \leq \frac { \pi } { 2 }
& \qquad 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0
& \text { giving your answers as exact multiples of } \pi \end{aligned}$$
1. (a) Find the general solution of the differential equation
$$\begin{aligned} & \qquad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 27 \mathrm { e } ^ { - x }
& \text { (b) Find the particular solution that satisfies } y = 0 \text { and } \frac { \mathrm { d } y } { \mathrm {~d} x } = 0 \text { when } x = 0 \text {. }
\hline & \end{aligned}$$
1. 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
$$w = \frac { 4 ( 1 - \mathrm { i } ) z - 8 \mathrm { i } } { 2 ( - 1 + \mathrm { i } ) z - \mathrm { i } } , \quad z \neq \frac { 1 } { 4 } - \frac { 1 } { 4 } \mathrm { i }$$ The transformation $T$ maps the points on the line $l$ with equation $y = x$ in the $z$-plane to a circle $C$ in the $w$-plane.
Show that $$w = \frac { a x ^ { 2 } + b x i + c } { 16 x ^ { 2 } + 1 }$$ where $a , b$ and $c$ are real constants to be found.
Hence show that the circle $C$ has equation $$( u - 3 ) ^ { 2 } + v ^ { 2 } = k ^ { 2 }$$ where $k$ is a constant to be found.
7. (a) Show that the substitution $v = y ^ { - 3 }$ transforms the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 2 x ^ { 4 } y ^ { 4 }$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } - \frac { 3 v } { x } = - 6 x ^ { 3 }$$
By solving differential equation (II), find a general solution of differential equation (I) in the form $y ^ { 3 } = \mathrm { f } ( x )$.
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-21_511_684_255_408} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve $C$ with polar equation $$r = 1 + \tan \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to the curve $C$ at the point $P$ is perpendicular to the initial line.
Find the polar coordinates of the point $P$. The point $Q$ lies on the curve $C$, where $\theta = \frac { \pi } { 3 }$.
The shaded region $R$ is bounded by $O P , O Q$ and the curve $C$, as shown in Figure 1.
Find the exact area of $R$, giving your answer in the form $$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$ where $p , q$ and $r$ are integers to be found.

Edexcel FP2 Q1

Find the set of values for which

$$| x - 1 | > 6 x - 1$$ [P4 January 2002 Qn 2]

Edexcel FP2 Q2

2. (a) Find the general solution of the differential equation $$t \frac { \mathrm {~d} v } { \mathrm {~d} t } - v = t , \quad t > 0$$ and hence show that the solution can be written in the form $v = t ( \ln t + c )$, where $c$ is an arbitrary constant.
(b) This differential equation is used to model the motion of a particle which has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$. When $t = 2$ the speed of the particle is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find, to 3 significant figures, the speed of the particle when $t = 4$.
[0pt] [P4 January 2002 Qn 6]

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