Questions - Edexcel - A-Level Maths

Edexcel FP1 Q16

16. (a) Show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { 1 } { 6 } n ( n + 7 ) ( 2 n + 7 ) .$$ (b) Hence calculate the value of $\quad \sum _ { r = 10 } ^ { 40 } ( r + 1 ) ( r + 5 )$.
[0pt] [P4 June 2004 Qn 1]

Edexcel FP1 Q17

17. $$f ( x ) = 2 ^ { x } + x - 4$$ The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1,2].
Use linear interpolation on the values at the end points of this interval to find an approximation to $\alpha$.
[0pt] [*P4 June 2004 Qn 2]

Edexcel FP1 Q18

18. The complex number $z = a + \mathrm { i } b$, where $a$ and $b$ are real numbers, satisfies the equation $$z ^ { 2 } + 16 - 30 i = 0$$

Show that $a b = 15$.
Write down a second equation in $a$ and $b$ and hence find the roots of $$z ^ { 2 } + 16 - 30 i = 0$$

Edexcel FP1 Q19

Given that $z = 1 + \sqrt { } 3 \mathrm { i }$ and that $\frac { w } { z } = 2 + 2 \mathrm { i }$, find
1. $w$ in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$,
2. the argument of $w$,
3. the exact value for the modulus of $w$.
On an Argand diagram, the point $A$ represents $z$ and the point $B$ represents $w$.
Draw the Argand diagram, showing the points $A$ and $B$.
Find the distance $A B$, giving your answer as a simplified surd.

Edexcel FP1 Q20

20. Show that the normal to the rectangular hyperbola $x y = c ^ { 2 }$, at the point $P \left( c t , \frac { c } { t } \right) , t \neq 0$ has equation $$y = t ^ { 2 } x + \frac { c } { t } - c t ^ { 3 }$$ [*P5 June 2004 Qn 8]
21. Given that $z = - 2 \sqrt { } 2 + 2 \sqrt { } 2 \mathrm { i }$ and $w = 1 - \mathrm { i } \sqrt { } 3$, find

$\left| \frac { z } { w } \right|$,
$\arg \left( \frac { z } { w } \right)$.
On an Argand diagram, plot points $A , B , C$ and $D$ representing the complex numbers $z$, $w , \left( \frac { z } { w } \right)$ and 4, respectively.
Show that $\angle A O C = \angle D O B$.
Find the area of triangle $A O C$.
22. Given that - 2 is a root of the equation $z ^ { 3 } + 6 z + 20 = 0$,
find the other two roots of the equation,
show, on a single Argand diagram, the three points representing the roots of the equation,
prove that these three points are the vertices of a right-angled triangle.
23. $$f ( x ) = 1 - e ^ { x } + 3 \sin 2 x$$ The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $1.0 < x < 1.4$.
Starting with the interval (1.0, 1.4), use interval bisection three times to find the value of $\alpha$ to one decimal place.
24. $$z = - 4 + 6 i$$
Calculate arg $z$, giving your answer in radians to 3 decimal places.
(2) The complex number $w$ is given by $w = \frac { A } { 2 - \mathrm { i } }$, where $A$ is a positive constant. Given that $| w | = \sqrt { } 20$,
find $w$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are constants,
(4)
calculate arg $\frac { W } { Z }$.
(3)
[0pt] [FP1/P4 June 2005 Qn 5]
25. The point $P \left( a p ^ { 2 } , 2 a p \right)$ lies on the parabola $M$ with equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant.
Show that an equation of the tangent to $M$ at $P$ is $$p y = x + a p ^ { 2 }$$ The point $Q \left( 16 a p ^ { 2 } , 8 a p \right)$ also lies on $M$.
Write down an equation of the tangent to $M$ at $Q$.
[0pt] [*FP2/P5 June 2005 Qn 5]
26. (a) Express $\frac { 6 x + 10 } { x + 3 }$ in the form $p + \frac { q } { x + 3 }$, where $p$ and $q$ are integers to be found. The sequence of real numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } = 5.2$ and $u _ { n + 1 } = \frac { 6 u _ { n } + 10 } { u _ { n } + 3 }$.
Prove by induction that $u _ { n } > 5$, for $n \in \mathbb { Z } ^ { + }$.
[0pt] [FP3/P6 June 2005 Qn 1]
27. Prove that $\sum _ { r = 1 } ^ { n } ( r - 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n - 1 ) n ( n + 4 )$.
28. Given that $\frac { z + 2 \mathrm { i } } { z - \lambda \mathrm { i } } = \mathrm { i }$, where $\lambda$ is a positive, real constant,
show that $z = \left( \frac { \lambda } { 2 } + 1 \right) + \mathrm { i } \left( \frac { \lambda } { 2 } - 1 \right)$. Given also that $\arg z = \arctan \frac { 1 } { 2 }$, calculate
the value of $\lambda$,
the value of $| z | ^ { 2 }$.
29. The temperature $\theta ^ { \circ } \mathrm { C }$ of a room $t$ hours after a heating system has been turned on is given by $$\theta = t + 26 - 20 \mathrm { e } ^ { - 0.5 t } , \quad t \geq 0 .$$ The heating system switches off when $\theta = 20$. The time $t = \alpha$, when the heating system switches off, is the solution of the equation $\theta - 20 = 0$, where $\alpha$ lies in the interval [1.8, 2].
[0pt]
Using the end points of the interval [1.8, 2], find, by linear interpolation, an approximation to $\alpha$. Give your answer to 2 decimal places.
Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on.
30. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a constant.
Show that an equation for the normal to $C$ at the point $P \left( a p ^ { 2 } , 2 a p \right)$ is $$y + p x = 2 a p + a p ^ { 3 }$$ The normals to $C$ at the points $P \left( a p ^ { 2 } , 2 a p \right)$ and $Q \left( a q ^ { 2 } , 2 a q \right) , p \neq q$, meet at the point $R$.
Find, in terms of $a , p$ and $q$, the coordinates of $R$.
31. A transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is represented by the matrix $$\mathbf { A } = \left( \begin{array} { r r } - 4 & 2
2 & - 1 \end{array} \right) , \text { where } k \text { is a constant. }$$ Find the image under $T$ of the line with equation $y = 2 x + 1$.
[0pt] [*FP3/P6 January 2006 Qn 3]
32. Prove by induction that, for $n \in \mathbb { Z } ^ { + } , \sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 \left\{ 1 + ( n - 1 ) 2 ^ { n } \right\}$.
[0pt] [*FP3/P6 January 2006 Qn 5]
33. The complex numbers $z$ and $w$ satisfy the simultaneous equations $$\begin{aligned} 2 z + \mathrm { i } w & = - 1
z - w & = 3 + 3 \mathrm { i } \end{aligned}$$
Use algebra to find $z$, giving your answers in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.
Calculate arg $z$, giving your answer in radians to 2 decimal places.
34. $$f ( x ) = 0.25 x - 2 + 4 \sin \sqrt { } x$$
Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ between $x = 0.24$ and $x = 0.28$.
[0pt]
Starting with the interval [0.24, 0.28], use interval bisection three times to find an interval of width 0.005 which contains $\alpha$.
[0pt] [*FP1 June 2006 Qn 6]
35. (a) Find the roots of the equation $$z ^ { 2 } + 2 z + 17 = 0$$ giving your answers in the form $a + \mathrm { i } b$, where $a$ and $b$ are integers.
Show these roots on an Argand diagram.
[0pt] [FP1 January 2007 Qn 1]
36. The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $$\begin{aligned} & z _ { 1 } = 5 + 3 i
& z _ { 1 } = 1 + p i \end{aligned}$$ where $p$ is an integer.
Find $\frac { z _ { 2 } } { z _ { 1 } }$, in the form $a + \mathrm { i } b$, where $a$ and $b$ are expressed in terms of $p$. Given that $\arg \left( \frac { z _ { 2 } } { z _ { 1 } } \right) = \frac { \pi } { 4 }$,
find the value of $p$.
37. $$f ( x ) = x ^ { 3 } + 8 x - 19$$
Show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.
Show that the real root of $\mathrm { f } ( x ) = 0$ lies between 1 and 2 .
Obtain an approximation to the real root of $\mathrm { f } ( x ) = 0$ by performing two applications of the Newton-Raphson procedure to $\mathrm { f } ( x )$, using $x = 2$ as the first approximation. Give your answer to 3 decimal places.
By considering the change of sign of $\mathrm { f } ( x )$ over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places.
38. $$z = \sqrt { 3 } - i$$ $z ^ { * }$ is the complex conjugate of $z$.
Show that $\frac { z } { z * } = \frac { 1 } { 2 } - \frac { \sqrt { } 3 } { 2 } \mathrm { i }$.
Find the value of $\left| \frac { z } { z * } \right|$.
Verify, for $z = \sqrt { } 3 - \mathrm { i }$, that $\arg \frac { z } { z ^ { * } } = \arg z - \arg z ^ { * }$.
Display $z , z ^ { * }$ and $\frac { Z } { Z ^ { * } }$ on a single Argand diagram.
Find a quadratic equation with roots $z$ and $z ^ { * }$ in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are real constants to be found.
39. The points $P \left( a p ^ { 2 } , 2 a p \right)$ and $Q \left( a q ^ { 2 } , 2 a q \right) , p \neq q$, lie on the parabola $C$ with equation $y ^ { 2 } = 4 a x$, where $a$ is a constant.
Show that an equation for the chord $P Q$ is $( p + q ) y = 2 ( x + a p q )$. The normals to $C$ at $P$ and $Q$ meet at the point $R$.
Show that the coordinates of $R$ are $\left( a \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - a p q ( p + q ) \right)$.
40. Prove by induction that, for $n \in \mathbb { Z } ^ { + } , \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n ( 2 n - 1 ) ( 2 n + 1 )$.
[0pt] [FP3 June 2007 Qn 5]
41. Given that $$f ( n ) = 3 ^ { 4 n } + 2 ^ { 4 n + 2 } ,$$
show that, for $k \in \mathbb { Z } ^ { + } , \mathrm { f } ( k + 1 ) - \mathrm { f } ( k )$ is divisible by 15 ,
prove that, for $n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )$ is divisible by 5 ,
[0pt] [*FP3 June 2007 Qn 6]
42. Given that $x = - \frac { 1 } { 2 }$ is the real solution of the equation $$2 x ^ { 3 } - 11 x ^ { 2 } + 14 x + 10 = 0$$ find the two complex solutions of this equation.
43. $$f ( x ) = 3 x ^ { 2 } + x - \tan \left( \frac { x } { 2 } \right) - 2 , \quad - \pi < x < \pi$$ The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $[ 0.7,0.8 ]$. Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to $\alpha$. Give your answer to 3 decimal places.
44. $$z = - 2 + \mathrm { i }$$
Express in the form $a + \mathrm { i } b$
1. $\frac { 1 } { z }$
2. $z ^ { 2 }$.
Show that $\left| z ^ { 2 } - z \right| = 5 \sqrt { } 2$.
Find arg $\left( z ^ { 2 } - z \right)$.
Display $z$ and $z ^ { 2 } - z$ on a single Argand diagram.
45. (a) Write down the value of the real root of the equation $$x ^ { 3 } - 64 = 0 .$$
Find the complex roots of $x ^ { 3 } - 64 = 0$, giving your answers in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.
Show the three roots of $x ^ { 3 } - 64 = 0$ on an Argand diagram.
46. The complex number $z$ is defined by $$z = \frac { a + 2 \mathrm { i } } { a - \mathrm { i } } , \quad a \in \mathbb { R } , \quad a > 0$$ Given that the real part of $z$ is $\frac { 1 } { 2 }$, find
the value of $a$,
the argument of $z$, giving your answer in radians to 2 decimal places.
47. $$\mathbf { A } = \left( \begin{array} { c r } k & - 2
1 - k & k \end{array} \right) , \text { where } k \text { is constant. }$$ A transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is represented by the matrix $\mathbf { A }$.
Find the value of $k$ for which the line $y = 2 x$ is mapped onto itself under $T$.
Show that $\mathbf { A }$ is non-singular for all values of $k$.
Find $\mathbf { A } ^ { - 1 }$ in terms of $k$. A point $P$ is mapped onto a point $Q$ under $T$. The point $Q$ has position vector $\binom { 4 } { - 3 }$ relative to an origin $O$. Given that $k = 3$,
find the position vector of $P$.

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]

Edexcel FP2 Q5

5. Using algebra, find the set of values of $x$ for which $$2 x - 5 > \frac { 3 } { x }$$ [P4 June 2002 Qn 4]

Edexcel FP2 Q6

6. (a) Find the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$ (b) Show that, for $0 \leq x \leq 2 \pi$, there are two points on the $x$-axis through which all the solution curves for this differential equation pass.
(c) Sketch the graph, for $0 \leq x \leq 2 \pi$, of the particular solution for which $y = 0$ at $x = 0$.
[0pt] [P4 June 2002 Qn 6]

Edexcel FP2 Q8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{858e6727-8498-462a-8064-d65254f1fd0f-04_559_954_1091_552}

\end{figure} The curve $C$ shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi$$

Find the polar coordinates of the points $P$ and $Q$ where the tangents to $C$ are parallel to the initial line.
(6) The curve $C$ represents the perimeter of the surface of a swimming pool. The direct distance from $P$ to $Q$ is 20 m .
Calculate the value of $a$.
Find the area of the surface of the pool.
(6)
[0pt] [P4 June 2002 Qn 8]

Edexcel FP2 Q9

9. (a) The point $P$ represents a complex number $z$ in an Argand diagram. Given that $$| z - 2 \mathrm { i } | = 2 | z + \mathrm { i } |$$

find a cartesian equation for the locus of $P$, simplifying your answer.
sketch the locus of $P$.
(b) A transformation $T$ from the $z$-plane to the $w$-plane is a translation $- 7 + 11 \mathrm { i }$ followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation $T$ in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$ [P6 June 2002 Qn 3]

Edexcel FP2 Q10

10. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + y = 0$$

Find an expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$. Given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ at $x = 0$,
find the series solution for $y$, in ascending powers of $x$, up to an including the term in $x ^ { 3 }$.
Comment on whether it would be sensible to use your series solution to give estimates for $y$ at $x = 0.2$ and at $x = 50$.
[0pt] [P6 June 2002 Qn 4]

Edexcel FP2 Q11

11. $$z = 4 \left( \cos \frac { \pi } { 4 } + \mathrm { i } \sin \frac { \pi } { 4 } \right) , \text { and } w = 3 \left( \cos \frac { 2 \pi } { 3 } + \mathrm { i } \sin \frac { 2 \pi } { 3 } \right)$$ Express $z w$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi$.
[0pt] [P4 January 2003 Qn 1]

Edexcel FP2 Q12

12. (a) Express $\frac { 2 } { ( r + 1 ) ( r + 3 ) }$ in partial fractions.
(b) Hence prove that $\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } \equiv \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }$.

Edexcel FP2 Q13

13. (a) Sketch, on the same axes, the graphs with equation $y = | 2 x - 3 |$, and the line with equation $y = 5 x - 1$.
(b) Solve the inequality $| 2 x - 3 | < 5 x - 1$.

Edexcel FP2 Q14

14. (a) Use the substitution $y = v x$ to transform the equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + y ) ( x + y ) } { x ^ { 2 } } , x > 0$$ into the equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( 2 + v ) ^ { 2 }$$ (b) Solve the differential equation II to find $v$ as a function of $x$.
(c) Hence show that $$y = - 2 x - \frac { x } { \ln x + c } , \text { where } c \text { is an arbitrary constant, }$$ is a general solution of the differential equation I.
[0pt] [P4 January 2003 Qn 5]

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