Questions FP2 - A-Level Maths

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 $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$

OCR MEI FP2 2009 January Q2

Write down the modulus and argument of the complex number $\mathrm { e } ^ { \mathrm { j } \pi / 3 }$.
The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number $a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }$ corresponds to the complex number $b$. Show A and the two possible positions for B in a sketch. Express $a$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Find the two possibilities for $b$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$.
Given that $z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }$, show that $z _ { 1 } ^ { 6 } = 8$. Write down, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, the other five complex numbers $z$ such that $z ^ { 6 } = 8$. Sketch all six complex numbers in a new Argand diagram. Let $w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }$.
Find $w$ in the form $x + \mathrm { j } y$, and mark this complex number on your Argand diagram.
Find $w ^ { 6 }$, expressing your answer in as simple a form as possible.

OCR MEI FP2 2013 January Q2

1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
2. The series $C$ and $S$ are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta
  & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering $C + \mathrm { j } S$, show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for $S$.
1. Express $\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }$ in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing $2 + 4 \mathrm { j }$. Obtain the complex numbers representing the other two vertices, giving your answers in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
3. Show that the length of a side of the triangle is $2 \sqrt { 15 }$.

CAIE FP2 2017 June Q4

Find the moment of inertia of this object about an axis $l$, which is perpendicular to the plane of the object and through the centre of $\operatorname { disc } A$.
The object is free to rotate about the horizontal axis $l$. It is released from rest in the position shown, with the centre of disc $B$ vertically above the centre of disc $A$.
Write down the change in the vertical position of the centre of mass of the object when the centre of disc $B$ is vertically below the centre of disc $A$. Hence find the angular velocity of the object when the centre of disc $B$ is vertically below the centre of disc $A$.

CAIE FP2 2017 June Q4

Find the tension in the string in terms of $W$.
Find the modulus of elasticity of the string in terms of $W$.
Find the angle that the force acting on the rod at $A$ makes with the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-10_442_442_260_849} A particle of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is moving in complete vertical circles with the string taut. When the particle is at the point $P$, where $O P$ makes an angle $\alpha$ with the upward vertical through $O$, its speed is $u$. When the particle is at the point $Q$, where angle $Q O P = 90 ^ { \circ }$, its speed is $v$ (see diagram). It is given that $\cos \alpha = \frac { 4 } { 5 }$.
Show that $v ^ { 2 } = u ^ { 2 } + \frac { 14 } { 5 } a g$.
The tension in the string when the particle is at $Q$ is twice the tension in the string when the particle is at $P$.
Obtain another equation relating $u ^ { 2 } , v ^ { 2 } , a$ and $g$, and hence find $u$ in terms of $a$ and $g$.
Find the least tension in the string during the motion.

CAIE FP2 2018 June Q5

Show that the moment of inertia of the object about the axis $l$ is $180 M a ^ { 2 }$.
Show that small oscillations of the object about the axis $l$ are approximately simple harmonic, and state the period.

CAIE FP2 2019 June Q4

Find the moment of inertia of the object, consisting of the rod and two spheres, about $L$.
The object is pivoted at $A$ so that it can rotate freely about $L$. The object is released from rest with the rod making an angle of $60 ^ { \circ }$ to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is $\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)$.
Find the value of $k$.

CAIE FP2 2017 November Q5

Show that the moment of inertia of the system, consisting of frame and small object, about an axis through $O$ perpendicular to the plane of the frame, is $\frac { 169 } { 3 } m a ^ { 2 }$.
Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.

CAIE FP2 2018 November Q3

Show that $u ^ { 2 } = 2 a g$.
Find the maximum tension in the string as the particle moves from $A$ to $B$.
\includegraphics[max width=\textwidth, alt={}, center]{a092cd45-dc19-476d-adf7-0198fbb2116e-06_543_807_255_669} A uniform rod $A B$ of length $2 a$ and weight $W$ rests against a smooth horizontal peg at a point $C$ on the rod, where $A C = x$. The lower end $A$ of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of $45 ^ { \circ }$ to the horizontal (see diagram). The coefficient of friction between the rod and the ground is $\mu$. The rod is about to slip at $A$.
Find an expression for $x$ in terms of $a$ and $\mu$.
Hence show that $\mu \geqslant \frac { 1 } { 3 }$.
Given that $x = \frac { 3 } { 2 } a$, find the value of $\mu$ and the magnitude of the resultant force on the rod at $A$.

CAIE FP2 2019 November Q5

Show that the moment of inertia of the object about $L$ is $\left( \frac { 408 + 7 \lambda } { 12 } \right) M a ^ { 2 }$.
The period of small oscillations of the object about $L$ is $5 \pi \sqrt { } \left( \frac { 2 a } { g } \right)$.
Find the value of $\lambda$.

CAIE FP2 2017 Specimen Q3

Find the value of $k$.
The particle $P$ is released from rest at a point between $A$ and $B$ where both strings are taut. Show that $P$ performs simple harmonic motion and state the period of the motion.
In the case where $P$ is released from rest at a distance $0.2 a \mathrm {~m}$ from $M$, the speed of $P$ is $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when $P$ is $0.05 a \mathrm {~m}$ from $M$. Find the value of $a$.

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)

OCR FP2 2010 January Q5

Using the definitions of $\sinh x$ and $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$ Deduce that $1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$.
Solve the equation $2 \tanh ^ { 2 } x - \operatorname { sech } x = 1$, giving your answer(s) in logarithmic form.
Express $\frac { 4 } { ( 1 - x ) ( 1 + x ) \left( 1 + x ^ { 2 } \right) }$ in partial fractions.
Show that $\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) + \frac { 1 } { 3 } \pi$.

OCR MEI FP2 2006 June Q5

5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where $k$ is a positive constant.

For the case $k = 1$, use your graphical calculator to sketch the curve. Describe its main features.
Sketch the curve for a value of $k$ between 0 and 1 . Describe briefly how the main features differ from those for the case $k = 1$.
For the case $k = 2$ :
(A) sketch the curve;
(B) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$;
(C) show that the width of each loop, measured parallel to the $x$-axis, is $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
Use your calculator to find, correct to one decimal place, the value of $k$ for which successive loops just touch each other.

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]

Edexcel FP2 Q3

3. (a) Show that $y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }$ is a solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$ (b) Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$ given that at $x = 0 , y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$.
[0pt] [P4 January 2002 Qn 7]

Edexcel FP2 Q4

4. The curve $C$ has polar equation $r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }$. The curve $D$ has polar equation $r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi$. Given that $a$ is a positive constant,

sketch, on the same diagram, the graphs of $C$ and $D$, indicating where each curve cuts the initial line. The graphs of $C$ intersect at the pole $O$ and at the points $P$ and $Q$.
Find the polar coordinates of $P$ and $Q$.
Use integration to find the exact value of the area enclosed by the curve $D$ and the lines $\theta = 0$ and $\theta = \frac { \pi } { 3 }$. The region $R$ contains all points which lie outside $D$ and inside $C$. Given that the value of the smaller area enclosed by the curve $C$ and the line $\theta = \frac { \pi } { 3 }$ is $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$
show that the area of $R$ is $\pi a ^ { 2 }$.
(4)
[0pt] [P4 January 2002 Qn 8]

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