Questions FP2 (1157 questions)

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Edexcel FP2 2017 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9019397a-a9c2-4b69-97fd-ea9eb9132244-18_364_695_260_756} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve with polar equation $$r = 6 + a \sin \theta$$ where \(0 < a < 6\) and \(0 \leqslant \theta < 2 \pi\)
The area enclosed by the curve is \(\frac { 97 \pi } { 2 }\)
Find the value of the constant \(a\).
Edexcel FP2 2017 June Q7
7. (a) Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = 2 \cos ^ { 3 } x \sin x + 1 , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 5 \sqrt { 2 }\) when \(x = \frac { \pi } { 4 }\)
(b) find the value of \(y\) when \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational numbers to be found.
Edexcel FP2 2017 June Q8
8. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 3 \mathrm { i } } { 1 + \mathrm { i } z } , \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 1\) in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
  1. Find a cartesian equation of the line \(l\). The circle \(| z - a - b \mathrm { i } | = c\) in the \(z\)-plane is mapped by \(T\) onto the circle \(| w | = 5\) in the \(w\)-plane.
  2. Find the exact values of the real constants \(a\), \(b\) and \(c\).
    END
Edexcel FP2 2018 June Q1
  1. (a) Express \(\frac { 1 } { ( r + 3 ) ( r + 4 ) }\) in partial fractions.
    (b) Hence, using the method of differences, show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 4 ) } = \frac { n } { a ( n + a ) }$$ where \(a\) is a constant to be found.
(c) Find the exact value of \(\sum _ { r = 15 } ^ { 30 } \frac { 1 } { ( r + 3 ) ( r + 4 ) }\) uestion 1 continued
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□ D D D " "
Edexcel FP2 2018 June Q2
2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).
Edexcel FP2 2018 June Q3
3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that
  1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
  2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\)
    (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)
Edexcel FP2 2018 June Q4
4. Use algebra to find the set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 4 x$$
Edexcel FP2 2018 June Q5
5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 0$$ Given that at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
  1. show that, at \(x = 0 , \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 3 } { 2 }\)
  2. Find a series solution for \(y\) up to and including the term in \(x ^ { 3 }\)
Edexcel FP2 2018 June Q6
6. (a) Find the general solution of the differential equation $$6 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 6 y = x - 6 x ^ { 2 }$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 }\) when \(x = 0\)
Edexcel FP2 2018 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-24_508_896_212_525} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \sqrt { 3 } \cos \theta , \quad 0 \leqslant \theta < 2 \pi$$ The tangent to \(C\) at the point \(P\) is parallel to the initial line.
  1. Show that \(O P = \frac { 1 } { 2 } ( 3 + \sqrt { 7 } )\)
  2. Find the exact area enclosed by the curve \(C\).
Edexcel FP2 2018 June Q8
8. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int 2 x ^ { 5 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). Given that \(y = 0\) when \(x = 1\)
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-32_2632_1826_121_121}
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.