Questions — Edexcel FP2 (291 questions)

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Edexcel FP2 Q1
  1. (a) Sketch, on the same axes, the graph with equation \(y = | 3 x - 1 |\), and the line with equation \(y = 4 x + 3\).
Show the coordinates of the points at which the graphs meet the \(x\)-axis.
(b) Solve the inequality \(| 3 x - 1 | < 4 x + 3\).
Edexcel FP2 Q2
2. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 n } { 3 ( 2 n + 3 ) }\).
Edexcel FP2 Q3
3. (a) Given that \(y = \ln ( 1 + 5 x ) , | x | < 0.2\), find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(b) Hence obtain the M aclaurin series for \(\ln ( 1 + 5 x ) , | x | < 0.2\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 Q4
4. Use the Taylor Series method to find the series solution, ascending up to and including the term in \(x ^ { 3 }\), of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } = 3 x + 4$$ given that \(\frac { \mathrm { dy } } { \mathrm { dx } } = y = 1\) at \(x = 0\).
(Total 8 marks)
Edexcel FP2 Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{affb668f-4d43-4fa8-a5b7-d536a58126b9-3_529_668_223_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has polar equation, \(r = 2 + \sin 3 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)
Use integration to calculate the exact value of the area enclosed by \(C\), the line \(\theta = 0\) and the line \(\theta = \frac { \pi } { 2 }\).
Edexcel FP2 Q6
6. (a) Use de M oivre's Theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta .$$ (b) Hence or otherwise, prove that the only real solutions of the equation $$\sin 5 \theta = 5 \sin \theta ,$$ are given by \(\theta = n \tau\), where \(n\) is an integer.
Edexcel FP2 Q7
7. A population \(P\) is growing at a rate which is modelled by the differential equation $$\frac { d P } { d t } - 0.1 P = 0.05 t$$ where \(t\) years is the time that has elapsed from the start of observations.
It is given that the population is 10000 at the start of the observations.
  1. Solve the differential equation to obtain an expression for \(P\) in terms of \(t\).
  2. Show that the population doubles between the sixth and seventh year after the observations began.
    (2)
Edexcel FP2 Q8
8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$
  1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
  2. the maximum and minimum values for \(| z |\),
  3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.
Edexcel FP2 Q9
9. Resonance in an electrical circuit is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } } + 64 V = \cos 8 t$$ where \(V\) represents the voltage in the circuit and \(t\) represents time.
  1. Find the value of \(\lambda\) for which \(\lambda\) tsin8t is a particular integral of the differential equation.
  2. Find the general solution of the differential equation. Given that \(V = 0\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 0\) when \(t = 0\),
  3. find the particular solution of the equation.
  4. Describe the behaviour of \(V\) as \(t\) becomes large, according to this model.
Edexcel FP2 Specimen Q1
  1. Find the set of values of \(x\) for which
$$\frac { x } { x - 3 } > \frac { 1 } { x - 2 }$$
Edexcel FP2 Specimen Q2
  1. (a) Express as a simplified single fraction \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } }\)
    (b) Hence prove, by the method of differences, that
$$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
Edexcel FP2 Specimen Q3
  1. (a) Show that the transformation \(T\)
$$w = \frac { z - 1 } { z + 1 }$$ maps the circle \(| z | = 1\) in the \(z\)-plane to the line \(| w - 1 | = | w + \mathrm { i } |\) in the \(w\)-plane. The transformation \(T\) maps the region \(| z | \leq 1\) in the \(z\)-plane to the region \(R\) in the \(w\)-plane.
(b) Shade the region \(R\) on an Argand diagram.
Edexcel FP2 Specimen Q4
4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } = x , \quad y = 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \text { at } x = 1$$ Find a series solution of the differential equation in ascending powers of ( \(x - 1\) ) up to and including the term in \(( x - 1 ) ^ { 3 }\).
Edexcel FP2 Specimen Q5
5. (a) Obtain the general solution of the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } - 0.1 S = t$$ (b) The differential equation in part (a) is used to model the assets, \(\pounds S\) million, of a bank \(t\) years after it was set up. Given that the initial assets of the bank were \(\pounds 200\) million, use your answer to part (a) to estimate, to the nearest \(\pounds\) million, the assets of the bank 10 years after it was set up.
Edexcel FP2 Specimen Q6
6. The curve \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta , \quad \frac { - \pi } { 4 } \leq \theta \leq \frac { \pi } { 4 }$$
  1. Sketch the curve \(C\).
  2. Find the polar coordinates of the points where tangents to \(C\) are parallel to the initial line.
  3. Find the area of the region bounded by \(C\).
Edexcel FP2 Specimen Q7
7. (a) Given that \(x = e ^ { t }\), show that
  1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \frac { \mathrm {~d} y } { \mathrm {~d} t }$$
  2. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$ (b) Use you answers to part (a) to show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = \mathrm { e } ^ { 3 t }$$ (c) Hence find the general solution of $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$
Edexcel FP2 Specimen Q8
  1. (a) Given that \(z = e ^ { i \theta }\), show that
$$z ^ { p } + \frac { 1 } { z ^ { p } } = 2 \cos p \theta$$ where \(p\) is a positive integer.
(b) Given that $$\cos ^ { 4 } \theta = A \cos 4 \theta + B \cos 2 \theta + C$$ find the values of the constants \(A , B\) and \(C\). The region \(R\) bounded by the curve with equation \(y = \cos ^ { 2 } x , - \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }\), and the \(x\)-axis is rotated through \(2 \pi\) about the \(x\)-axis.
(c) Find the volume of the solid generated.
Edexcel FP2 2006 January Q2
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Given that \(x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\) at \(t = 0\), find the particular solution of the differential equation, giving your answer in the form \(x = \mathrm { f } ( t )\).
  3. Sketch the curve with equation \(x = \mathrm { f } ( t ) , 0 \leq t \leq \pi\), showing the coordinates, as multiples of \(\pi\), of the points where the curve cuts the \(x\)-axis.
    (4)(Total 13 marks)
Edexcel FP2 2002 June Q3
  1. Show that \(y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }\) is a solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$
  2. Solve the differential equation \(\quad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }\).
    given that at \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\).
Edexcel FP2 2004 June Q2
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$
  1. Verify that \(x ^ { 3 } \mathrm { e } ^ { x }\) is an integrating factor for the differential equation.
  2. Find the general solution of the differential equation.
  3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\).
    (3)(Total 10 marks)
Edexcel FP2 2005 June Q2
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \cot 2 x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(Total 7 marks)
Edexcel FP2 2009 June Q2
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 \(- \pi < \theta \leqslant \pi\).
Edexcel FP2 2008 June Q1
\begin{enumerate} \item Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = x\) to obtain \(y\) as a function of \(x\). \item (a) Simplify the expression \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } - ( 3 x - 5 )\), giving your answer in the form \(\frac { a ( x + b ) ( x + c ) } { x - 1 }\), where \(a , b\) and \(c\) are integers.
(b) Hence, or otherwise, solve the inequality \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } > 3 x - 5 \quad\) (4)(Total \(\mathbf { 8 }\) marks) \item (a) Find the general solution of the differential equation \(3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = x ^ { 2 }\)
(b) Find the particular solution for which, at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\).(6)(Total 14 marks) \item The diagram above shows the curve \(C _ { 1 }\) which has polar equation \(\boldsymbol { r } = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \boldsymbol { \operatorname { c o s } } \boldsymbol { \theta } ) , 0 \leq \theta < 2 \pi\) and the circle \(C _ { 2 }\) with equation \(\boldsymbol { r } = \mathbf { 4 } \boldsymbol { a } , 0 \leq \theta < 2 \pi\), where \(a\) is a positive constant.
Edexcel FP2 2008 June Q5
5. (a) Find, in terms of \(k\), the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 3 x = k t + 5 , \text { where } k \text { is a constant and } t > 0 .$$ For large values of \(t\), this general solution may be approximated by a linear function.
(b) Given that \(k = 6\), find the equation of this linear function.(2)(Total 9 marks)
Edexcel FP2 2008 June Q6
6. (a) Find, in the simplest surd form where appropriate, the exact values of \(x\) for which $$\frac { x } { 2 } + 3 = \left| \frac { 4 } { x } \right|$$ (b) Sketch, on the same axes, the line with equation \(y = \frac { x } { 2 } + 3\) and the graph of \(y = \left| \frac { 4 } { x } \right| , \quad x \neq 0\).
(c) Find the set of values of \(x\) for which \(\frac { x } { 2 } + 3 > \left| \frac { 4 } { x } \right|\).
(2)(Total 10 marks)