Questions FP2 (1157 questions)

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Edexcel FP2 2013 June Q5
5. (a) Express \(\frac { 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
(b) Using your answer to part (a) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( n + 3 ) } { 2 ( n + 1 ) ( n + 2 ) }$$
Edexcel FP2 2013 June Q6
6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).
Edexcel FP2 2013 June Q7
7. (a) Find the value of the constant \(\lambda\) for which \(y = \lambda x \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$ (b) Hence, or otherwise, find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 y = 6 \mathrm { e } ^ { 2 x }$$
Edexcel FP2 2013 June Q8
8. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.
  1. Given that \(| z | = 1\), sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 7 \mathrm { i } } { z - 2 \mathrm { i } }$$
  2. Show that \(T\) maps \(| z | = 1\) onto a circle in the \(w\)-plane.
  3. Show that this circle has its centre at \(w = - 5\) and find its radius.
Edexcel FP2 2013 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f8a7998-613b-449b-9758-9bf105c56a8f-9_370_820_316_626} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curves given by the polar equations $$r = 1 \text { and } r = 2 - 2 \sin \theta$$
  1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r < 1\) and for which \(r < 2 - 2 \sin \theta\), is shown shaded in Figure 1.
  2. 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 rational numbers.
Edexcel FP2 2013 June Q1
  1. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
    (b) Using your answer to (a), find, in terms of \(n\),
$$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Give your answer as a single fraction in its simplest form.
Edexcel FP2 2013 June Q2
2. $$z = 5 \sqrt { } 3 - 5 i$$ Find
  1. \(| z |\),
  2. \(\arg ( z )\), in terms of \(\pi\). $$w = 2 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$ Find
  3. \(\left| \frac { w } { z } \right|\),
  4. \(\quad \arg \left( \frac { w } { z } \right)\), in terms of \(\pi\).
Edexcel FP2 2013 June Q3
3. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y - \sin x = 0$$ Given that \(y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 8 }\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 2013 June Q4
4. (a) Given that $$z = r ( \cos \theta + \mathrm { i } \sin \theta ) , \quad r \in \mathbb { R }$$ prove, by induction, that \(z ^ { n } = r ^ { n } ( \cos n \theta + \mathrm { i } \sin n \theta ) , \quad n \in \mathbb { Z } ^ { + }\) $$w = 3 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$$ (b) Find the exact value of \(w ^ { 5 }\), giving your answer in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
Edexcel FP2 2013 June Q5
  1. (a) Find the general solution of the differential equation
    (b) Find the particular solution for which \(y = 5\) at \(x = 1\), giving your answer in the form \(y = \mathrm { f } ( x )\).
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x ^ { 2 }$$ (c) (i) Find the exact values of the coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\), making your method clear.
(ii) Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the turning points.
Edexcel FP2 2013 June Q6
  1. (a) Use algebra to find the exact solutions of the equation
$$\left| 2 x ^ { 2 } + 6 x - 5 \right| = 5 - 2 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + 6 x - 5 \right|\) and the line with equation \(y = 5 - 2 x\), showing the \(x\)-coordinates of the points where the line crosses the curve.
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + 6 x - 5 \right| > 5 - 2 x$$
Edexcel FP2 2013 June Q7
  1. (a) Show that the transformation \(y = x v\) transforms the equation
$$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 8 + 4 x ^ { 2 } \right) y = x ^ { 4 }$$ into the equation $$4 \frac { \mathrm {~d} ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 4 v = x$$ (b) Solve the differential equation (II) to find \(v\) as a function of \(x\).
(c) Hence state the general solution of the differential equation (I).
Edexcel FP2 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b8b399d-ba16-4fcb-be45-0ba40a7ae09d-13_542_748_205_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).
  1. Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
  2. Find the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
  3. Use calculus to show that the exact area of \(S\) is $$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$
Edexcel FP2 2014 June Q1
  1. (a) Express \(\frac { 2 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
    (b) Hence use the method of differences to show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
Edexcel FP2 2014 June Q2
2. Using algebra, find the set of values of \(x\) for which $$3 x - 5 < \frac { 2 } { x }$$
Edexcel FP2 2014 June Q3
3. (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 \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 1\) at \(x = 0\)
Edexcel FP2 2014 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9fff982-d38b-42ff-ab4e-08008439a95b-06_456_1273_262_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with polar equation $$r = 2 \cos 2 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ 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 \(r = \mathrm { f } ( \theta )\).
Edexcel FP2 2014 June Q5
5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0$$
  1. 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 \(x = 0\),
  2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
    5. \(y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0\)
  3. 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 \(y\).
Edexcel FP2 2014 June Q6
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 $$w = \frac { z } { i z + 1 } , \quad z \neq i$$ The transformation \(T\) maps the line \(l\) in the \(z\)-plane onto the line with equation \(v = - 1\) in the \(w\)-plane.
  1. 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 \(v\).
Edexcel FP2 2014 June Q7
7. (a) Use de Moivre's theorem to show that $$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ (b) 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.
(c) Use the identity given in (a) to find $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta \right) \mathrm { d } \theta$$ expressing your answer in the form \(a \sqrt { } 2 + b\), where \(a\) and \(b\) are rational numbers.
Edexcel FP2 2014 June Q8
8. (a) Show that the substitution \(x = \mathrm { e } ^ { z }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 3 \ln x , \quad x > 0$$ into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} z ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} z } - 2 y = 3 z$$ (b) Find the general solution of the differential equation (II).
(c) Hence obtain the general solution of the differential equation (I) giving your answer in the form \(y = \mathrm { f } ( x )\).
\(\square\)
Edexcel FP2 2014 June Q1
  1. (a) Express \(\frac { 2 } { ( r + 2 ) ( r + 4 ) }\) in partial fractions.
    (b) Hence show that
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 2 ) ( r + 4 ) } = \frac { n ( 7 n + 25 ) } { 12 ( n + 3 ) ( n + 4 ) }$$
Edexcel FP2 2014 June Q2
2. Use algebra to find the set of values of \(x\) for which $$\left| 3 x ^ { 2 } - 19 x + 20 \right| < 2 x + 2$$
Edexcel FP2 2014 June Q3
3. $$y = \sqrt { 8 + \mathrm { e } ^ { x } } , \quad x \in \mathbb { R }$$ 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.
Edexcel FP2 2014 June Q4
4. (a) Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ (b) Hence solve for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) $$64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0$$ giving your answers as exact multiples of \(\pi\).