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Edexcel F2 2022 January Q1
7 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Express the complex number
    $$- 4 - 4 \sqrt { 3 } i$$ in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
  2. Solve the equation $$z ^ { 3 } + 4 + 4 \sqrt { 3 } i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
Edexcel F2 2022 January Q2
6 marks Standard +0.3
2. Determine the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \mathrm { e } ^ { 3 x }$$
Edexcel F2 2022 January Q3
11 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d458344-42cb-48d1-90b3-e071df8ea7bb-08_693_987_116_482} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \frac { 4 x } { 4 - | x | }$$ and the curve \(C _ { 2 }\) with equation $$y = x ^ { 2 } - 8 x$$ For \(x > 0 , C _ { 1 }\) has equation \(y = \frac { 4 x } { 4 - x }\)
  1. Use algebra to show that \(C _ { 1 }\) touches \(C _ { 2 }\) at a point \(P\), stating the coordinates of \(P\)
  2. Hence or otherwise, using algebra, solve the inequality $$x ^ { 2 } - 8 x > \frac { 4 x } { 4 - | x | }$$
Edexcel F2 2022 January Q4
10 marks Challenging +1.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d458344-42cb-48d1-90b3-e071df8ea7bb-12_897_1040_205_534} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the curve with polar equation $$r = 4 - \frac { 3 } { 2 } \cos 6 \theta \quad 0 \leqslant \theta < 2 \pi$$
  1. Sketch, on the polar grid in Figure 2,
    1. the rest of the curve with equation $$r = 4 - \frac { 3 } { 2 } \cos 6 \theta \quad 0 \leqslant \theta < 2 \pi$$
    2. the polar curve with equation $$r = 1$$ $$0 \leqslant \theta < 2 \pi$$ A spare copy of the grid is given on page 15. In part (b) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  2. Determine the exact area enclosed between the two curves defined in part (a). Only use this grid if you need to redraw your answer to part (a) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0d458344-42cb-48d1-90b3-e071df8ea7bb-15_901_1042_1651_532} \captionsetup{labelformat=empty} \caption{Copy of Figure 2}
    \end{figure}
Edexcel F2 2022 January Q5
8 marks Standard +0.8
5. $$y = \sqrt { 4 + \ln x } \quad x > \frac { 1 } { 2 }$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 9 + 2 \ln x } { 4 x ^ { 2 } ( 4 + \ln x ) ^ { \frac { 3 } { 2 } } }$$
  2. Hence, or otherwise, determine the Taylor series expansion about \(x = 1\) for \(y\), in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 2 }\), giving each coefficient in simplest form.
Edexcel F2 2022 January Q6
11 marks Challenging +1.2
6. Given that \(A > B > 0\), by letting \(x = \arctan A\) and \(y = \arctan B\)
  1. prove that $$\arctan A - \arctan B = \arctan \left( \frac { A - B } { 1 + A B } \right)$$
  2. Show that when \(A = r + 2\) and \(B = r\) $$\frac { A - B } { 1 + A B } = \frac { 2 } { ( 1 + r ) ^ { 2 } }$$
  3. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \arctan \frac { 2 } { ( 1 + r ) ^ { 2 } } = \arctan ( n + p ) + \arctan ( n + q ) - \arctan 2 - \frac { \pi } { 4 }$$ where \(p\) and \(q\) are integers to be determined.
  4. Hence, making your reasoning clear, determine $$\sum _ { r = 1 } ^ { \infty } \arctan \left( \frac { 2 } { ( 1 + r ) ^ { 2 } } \right)$$ giving the answer in the form \(k \pi - \arctan 2\), where \(k\) is a constant.
Edexcel F2 2022 January Q7
8 marks Challenging +1.2
7. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { ( 1 + \mathrm { i } ) z + 2 ( 1 - \mathrm { i } ) } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps points on the imaginary axis in the \(z\)-plane onto a line in the \(w\)-plane.
  1. Find an equation for this line. The transformation maps points on the real axis in the \(z\)-plane onto a circle in the \(w\)-plane.
  2. Find the centre and radius of this circle.
Edexcel F2 2022 January Q8
14 marks Challenging +1.3
  1. (a) Show that the transformation \(v = y - 2 x\) transforms the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y x ( y - 4 x ) = 2 - 8 x ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } = - 2 x v ^ { 2 }$$ (b) Solve the differential equation (II) to determine \(v\) as a function of \(x\) (c) Hence obtain the general solution of the differential equation (I).
(d) Sketch the solution curve that passes through the point \(( - 1 , - 1 )\). On your sketch show clearly the equation of any horizontal or vertical asymptotes.
You do not need to find the coordinates of any intercepts with the coordinate axes or the coordinates of any stationary points.
\includegraphics[max width=\textwidth, alt={}]{0d458344-42cb-48d1-90b3-e071df8ea7bb-32_2817_1962_105_105}
Edexcel F2 2023 January Q1
8 marks Standard +0.3
  1. Given that \(y = \ln ( 5 + 3 x )\)
    1. determine, in simplest form, \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
    2. Hence determine the Maclaurin series expansion of \(\ln ( 5 + 3 x )\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
    3. Hence write down the Maclaurin series expansion of \(\ln ( 5 - 3 x )\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
    4. Use the answers to parts (b) and (c) to determine the first 2 non-zero terms, in ascending powers of \(x\), of the Maclaurin series expansion of
    $$\ln \left( \frac { 5 + 3 x } { 5 - 3 x } \right)$$
Edexcel F2 2023 January Q2
6 marks Standard +0.3
  1. (a) Express
$$\frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { n ( n + 2 ) } { a ( 2 n + b ) ( 2 n + c ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2023 January Q3
9 marks Standard +0.8
  1. (a) Show that the transformation \(y = \frac { 1 } { z }\) transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 2 y ^ { 2 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { z } { x } = - \frac { 2 } { x ^ { 2 } }$$ (b) Solve differential equation (II) to determine \(z\) in terms of \(x\).
(c) Hence determine the particular solution of differential equation (I) for which \(y = - \frac { 3 } { 8 }\) at \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel F2 2023 January Q4
7 marks Challenging +1.8
4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } - x$$
  1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = A y \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + B \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(A\) and \(B\) are integers to be determined. Given that \(y = 1\) at \(x = - 1\)
  2. determine the Taylor series solution for \(y\), in ascending powers of \(( x + 1 )\) up to and including the term in \(( x + 1 ) ^ { 4 }\), giving each coefficient in simplest form.
Edexcel F2 2023 January Q5
6 marks Challenging +1.2
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the set of values of \(x\) for which $$\frac { x ^ { 2 } - 9 } { | x + 8 | } > 6 - 2 x$$
Edexcel F2 2023 January Q6
8 marks Challenging +1.2
  1. A complex number \(z\) is represented by the point \(P\) in an Argand diagram.
Given that $$| z - 2 i | = | z - 3 |$$
  1. sketch the locus of \(P\). You do not need to find the coordinates of any intercepts. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { \mathrm { i } z } { z - 2 \mathrm { i } } \quad z \neq 2 \mathrm { i }$$ Given that \(T\) maps \(| z - 2 i | = | z - 3 |\) to a circle \(C\) in the \(w\)-plane,
  2. find the equation of \(C\), giving your answer in the form $$| w - ( p + q \mathrm { i } ) | = r$$ where \(p , q\) and \(r\) are real numbers to be determined.
Edexcel F2 2023 January Q7
8 marks Challenging +1.8
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
  2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.
Edexcel F2 2023 January Q8
10 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed3689f7-b3f0-447b-baa5-e44b8d8342d0-28_522_1084_260_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 - \sin \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\), such that the tangent to \(C\) at \(P\) is parallel to the initial line.
  1. Use calculus to determine the polar coordinates of \(P\) The finite region \(R\), shown shaded in Figure 1, is bounded by
    • the line with equation \(\theta = \frac { \pi } { 2 }\)
    • the tangent to \(C\) at \(P\)
    • part of the curve \(C\)
    • the initial line
    • Use algebraic integration to show that the area of \(R\) is
    $$\frac { 1 } { 32 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2023 January Q9
13 marks Challenging +1.2
  1. (a) Given that \(x = t ^ { \frac { 1 } { 2 } }\), determine, in terms of \(y\) and \(t\),
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) (b) Hence show that the transformation \(x = t ^ { \frac { 1 } { 2 } }\), where \(t > 0\), transforms the differential equation
    $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 6 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 9 x ^ { 3 } y = x ^ { 5 }$$ into the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 12 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = t$$ (c) Solve differential equation (II) to determine a general solution for \(y\) in terms of \(t\).
    (d) Hence determine the general solution of differential equation (I).
Edexcel F2 2024 January Q1
5 marks Moderate -0.3
  1. Using algebra, solve the inequality
$$\frac { 1 } { x + 2 } > 2 x + 3$$
Edexcel F2 2024 January Q2
8 marks Standard +0.3
2. $$z = 6 - 6 \sqrt { 3 } i$$
    1. Determine the modulus of \(z\)
    2. Show that the argument of \(z\) is \(- \frac { \pi } { 3 }\) Using de Moivre's theorem, and making your method clear,
  2. determine, in simplest form, \(z ^ { 4 }\)
  3. Determine the values of \(w\) such that \(w ^ { 2 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
Edexcel F2 2024 January Q3
7 marks Challenging +1.2
  1. (a) Show that for \(r \geqslant 1\)
$$\frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } \equiv A ( \sqrt { r ( r + 1 ) } - \sqrt { r ( r - 1 ) } )$$ where \(A\) is a constant to be determined.
(b) Hence use the method of differences to determine a simplified expression for $$\sum _ { r = 1 } ^ { n } \frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } }$$ (c) Determine, as a surd in simplest form, the constant \(k\) such that $$\sum _ { r = 1 } ^ { n } \frac { k r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } = \sqrt { \sum _ { r = 1 } ^ { n } r }$$
Edexcel F2 2024 January Q4
9 marks Challenging +1.2
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Determine, in ascending powers of \(\left( x - \frac { \pi } { 6 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 6 } \right) ^ { 3 }\), the Taylor series expansion about \(\frac { \pi } { 6 }\) of $$y = \tan \left( \frac { 3 x } { 2 } \right)$$ giving each coefficient in simplest form.
  2. Hence show that $$\tan \frac { 3 \pi } { 8 } \approx 1 + \frac { \pi } { 4 } + \frac { \pi ^ { 2 } } { A } + \frac { \pi ^ { 3 } } { B }$$ where \(A\) and \(B\) are integers to be determined.
Edexcel F2 2024 January Q5
9 marks
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5eb0ca8-92ba-466f-84f5-8fc36c168695-16_669_817_296_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = 10 \cos \theta + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on the curve where \(\theta = \frac { \pi } { 3 }\) The region \(R\), shown shaded in Figure 1, is bounded by the initial line, the curve and the line \(O P\), where \(O\) is the pole. Use algebraic integration to show that the exact area of \(R\) is $$\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2024 January Q6
14 marks Standard +0.8
  1. The differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 8 \mathrm { e } ^ { - 3 t } \quad t \geqslant 0$$ describes the motion of a particle along the \(x\)-axis.
  1. Determine the general solution of this differential equation. Given that the motion of the particle satisfies \(x = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\) when \(t = 0\)
  2. determine the particular solution for the motion of the particle. On the graph of the particular solution found in part (b), the first turning point for \(t > 0\) occurs at \(x = a\).
  3. Determine, to 3 significant figures, the value of \(a\).
    [0pt] [Solutions relying entirely on calculator technology are not acceptable.]
Edexcel F2 2024 January Q7
10 marks Challenging +1.2
  1. A 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
$$w = \frac { z - 3 } { 2 \mathrm { i } - z } \quad z \neq 2 \mathrm { i }$$ The line in the \(z\)-plane with equation \(y = x + 3\) is mapped by \(T\) onto a circle \(C\) in the \(w\)-plane.
  1. Determine
    1. the coordinates of the centre of \(C\)
    2. the exact radius of \(C\) The region \(y > x + 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  2. On a single Argand diagram
    1. sketch the circle \(C\)
    2. shade and label the region \(R\)
Edexcel F2 2024 January Q8
13 marks Challenging +1.8
  1. (a) For all the values of \(x\) where the identity is defined, prove that
$$\cot 2 x + \tan x \equiv \operatorname { cosec } 2 x$$ (b) Show that the substitution \(y ^ { 2 } = w \sin 2 x\), where \(w\) is a function of \(x\), transforms the differential equation $$y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } \tan x = \sin x \quad 0 < x < \frac { \pi } { 2 }$$ into the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} x } + 2 w \operatorname { cosec } 2 x = \sec x \quad 0 < x < \frac { \pi } { 2 }$$ (c) By solving differential equation (II), determine a general solution of differential equation (I) in the form \(y ^ { 2 } = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a function in terms of \(\cos x\) $$\text { [You may use without proof } \left. \int \operatorname { cosec } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \ln | \tan x | \text { (+ constant) } \right]$$