Questions — Edexcel FP2 (291 questions)

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Edexcel FP2 2009 June Q6
  1. 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 \(C\).
  1. 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.
  2. Shade the region \(R\) on an Argand diagram.
Edexcel FP2 2009 June Q7
  1. (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 2009 June 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 \(t = 0\),
  1. 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 \(O\).
  2. 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.
Edexcel FP2 2010 June Q1
  1. (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 2010 June Q2
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 \(t ^ { 3 }\).
Edexcel FP2 2010 June Q3
3. (a) Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 }$$ (b) Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$
Edexcel FP2 2010 June Q4
4. $$z = - 8 + ( 8 \sqrt { } 3 ) i$$
  1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
  2. find \(z ^ { 3 }\),
  3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
Edexcel FP2 2010 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ff7c42d-40b0-4d59-8716-14de4890ac1b-06_524_750_219_610} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curves given by the polar equations $$r = 2 , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } ,$$ and $$r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
  1. 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.
  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 simplified fractions.
Edexcel FP2 2010 June Q6
6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
  1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
  2. 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 \(w = \frac { 30 } { z }\).
  3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.
Edexcel FP2 2010 June Q7
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 $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$ (b) Solve the differential equation (II) to find \(z\) as a function of \(x\).
(c) Hence obtain the general solution of the differential equation (I).
Edexcel FP2 2010 June Q8
8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ (b) 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
(d) Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).
Edexcel FP2 2011 June Q1
  1. Find the set of values of \(x\) for which
$$\frac { 3 } { x + 3 } > \frac { x - 4 } { x }$$
Edexcel FP2 2011 June Q2
2. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { x } \left( 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right)$$
  1. Show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \mathrm { e } ^ { x } \left[ 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + k y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + 1 \right]$$ where \(k\) is a constant to be found. Given that, at \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\),
  2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 2011 June Q3
3. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = \frac { \ln x } { x } , \quad x > 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel FP2 2011 June Q4
4. Given that $$( 2 r + 1 ) ^ { 3 } = A r ^ { 3 } + B r ^ { 2 } + C r + 1 ,$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Show that $$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
  3. Using the result in part (b) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$
Edexcel FP2 2011 June Q5
  1. The point \(P\) represents the complex number \(z\) on an Argand diagram, where
$$| z - \mathrm { i } | = 2$$ The locus of \(P\) as \(z\) varies is the curve \(C\).
  1. Find a cartesian equation of \(C\).
  2. Sketch the curve \(C\). A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,
  3. show that \(Q\) lies on \(C\).
Edexcel FP2 2011 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{893efbc9-8321-469f-bd5e-89f9d5827737-09_650_937_269_482} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 5 } { 2 }\).
The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\).
Edexcel FP2 2011 June Q7
  1. (a) Use de Moivre's theorem to show that
$$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence, given also that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\),
(b) find all the solutions of $$\sin 5 \theta = 5 \sin 3 \theta$$ in the interval \(0 \leqslant \theta < 2 \pi\). Give your answers to 3 decimal places.
Edexcel FP2 2011 June Q8
  1. The differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = \cos 3 t , \quad t \geqslant 0$$ describes the motion of a particle along the \(x\)-axis.
  1. Find the general solution of this differential equation.
  2. Find the particular solution of this differential equation for which, at \(t = 0\), $$x = \frac { 1 } { 2 } \text { and } \frac { \mathrm { d } x } { \mathrm {~d} t } = 0$$ On the graph of the particular solution defined in part (b), the first turning point for \(t > 30\) is the point \(A\).
  3. Find approximate values for the coordinates of \(A\).
Edexcel FP2 2012 June Q1
  1. Find the set of values of \(x\) for which
$$\left| x ^ { 2 } - 4 \right| > 3 x$$
Edexcel FP2 2012 June Q2
2. The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.
Given that \(O\) is the pole, find the exact length of the line \(O P\).
Edexcel FP2 2012 June Q3
3. (a) Express the complex number \(- 2 + ( 2 \sqrt { 3 } ) \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
(b) Solve the equation $$z ^ { 4 } = - 2 + ( 2 \sqrt { } 3 ) i$$ giving the roots in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
Edexcel FP2 2012 June Q4
4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \cos t - \sin t$$
Edexcel FP2 2012 June Q5
5. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x + y ^ { 2 }$$
  1. Show that $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 - 2 y ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 3$$ Given that \(y = 1\) at \(x = 1\),
  2. find a series solution for \(y\) in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 3 }\).
Edexcel FP2 2012 June Q6
  1. (a) Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
    (b) Hence prove, by the method of differences, that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( a n + b ) } { 4 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( 4 n + 5 ) } { 4 ( n + 1 ) ( n + 2 ) ( 2 n + 1 ) }$$