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Edexcel F2 2021 June Q2
2. The 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 $$w = \frac { z + 2 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 2\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane.
Find (i) the centre of \(C\),
(ii) the radius of \(C\).
Edexcel F2 2021 June Q3
  1. The curve \(C\), with pole \(O\), has polar equation
$$r = 1 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the tangent to \(C\) is parallel to the initial line.
  1. Find the polar coordinates of \(A\).
  2. Find the finite area enclosed by the initial line, the line \(O A\) and the curve \(C\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational constants to be found.
Edexcel F2 2021 June Q4
4. Given that $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y = 0$$
  1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 28 } { y ^ { 2 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 3 } - \frac { 24 } { y } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 8\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
  2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients where possible.
Edexcel F2 2021 June Q5
  1. Use algebra to find the set of values of \(x\) for which
$$\left| 2 x ^ { 2 } + x - 3 \right| > 3 ( 1 - x )$$ [Solutions based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-21_2647_1840_118_111}
Edexcel F2 2021 June Q6
6. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = 2 x ^ { 2 } + x$$ (b) Find the particular solution of this differential equation for which \(y = 1\) and $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 0 \text { when } x = 0$$
Edexcel F2 2021 June Q7
  1. (a) Use de Moivre's theorem to show that
$$\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$ (b) Use the identity given in part (a) to find the 2 positive roots of $$x ^ { 4 } + 2 x ^ { 3 } - 6 x ^ { 2 } - 2 x + 1 = 0$$ giving your answers to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-29_2255_50_314_35}
Edexcel F2 2021 June Q8
8. (a) Show that the substitution \(v = y ^ { - 2 }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 6 x y = 3 x \mathrm { e } ^ { x ^ { 2 } } y ^ { 3 } \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} x } - 12 v x = - 6 x \mathrm { e } ^ { x ^ { 2 } } \quad x > 0$$ (b) Hence find the general solution of the differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
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Q8

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Edexcel F2 2022 June Q1
  1. Given that
$$\frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } \equiv \frac { A } { n ^ { 2 } } + \frac { B } { ( n + 1 ) ^ { 2 } }$$
  1. determine the value of \(A\) and the value of \(B\)
  2. Hence show that, for \(n \geqslant 5\) $$\sum _ { r = 5 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } + a n + b } { c ( n + 1 ) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2022 June Q2
  1. (a) Use algebra to determine the set of values of \(x\) for which
$$x - 5 < \frac { 9 } { x + 3 }$$ (b) Hence, or otherwise, determine the set of values of \(x\) for which $$x - 5 < \frac { 9 } { | x + 3 | }$$
Edexcel F2 2022 June Q3
  1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the circle \(C\) Determine
  1. a Cartesian equation of \(C\)
  2. the centre and radius of \(C\)
Edexcel F2 2022 June Q4
  1. (a) Determine the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y \tan x = \mathrm { e } ^ { 4 x } \sec ^ { 3 } x$$ giving your answer in the form \(y = \mathrm { f } ( x )\)
(b) Determine the particular solution for which \(y = 4\) at \(x = 0\)
Edexcel F2 2022 June Q5
  1. Given that
$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 y = 0 \quad y > 0$$
  1. determine \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
  2. determine a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in its simplest form.
Edexcel F2 2022 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff9ff379-78d8-41c0-a177-ec346e359249-20_497_1196_260_520} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve shown in Figure 1 has polar equation $$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$ where \(a\) is a positive constant.
The tangent to the curve at the point \(A\) is parallel to the initial line.
  1. Show that the polar coordinates of \(A\) are \(\left( 6 a , \frac { \pi } { 3 } \right)\) The point \(B\) lies on the curve such that angle \(A O B = \frac { \pi } { 6 }\)
    The finite region \(R\), shown shaded in Figure 1, is bounded by the line \(A B\) and the curve.
  2. Use calculus to determine the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )\), where \(n , p\) and \(q\) are integers.
Edexcel F2 2022 June Q7
  1. (a) Show that the transformation \(y = x v\) transforms the equation
$$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 6 } { x } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 6 y } { x ^ { 2 } } + 3 y = x ^ { 2 } \quad x \neq 0$$ into the equation $$3 \frac { \mathrm {~d} ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 3 v = x$$ (b) Hence obtain the general solution of the differential equation (I), giving your answer in the form \(y = \mathrm { f } ( x )\)
Edexcel F2 2022 June Q8
  1. (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 determine the five distinct solutions of the equation $$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 5 } = 0$$ giving your answers to 3 decimal places.
(c) Use the identity given in part (a) to show that $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta - 6 \sin \theta \right) \mathrm { d } \theta = a \sqrt { 2 } + b$$ where \(a\) and \(b\) are rational numbers to be determined.
Edexcel F2 2023 June Q1
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Show that, for \(r \geqslant 2\) $$\frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = \sqrt { r } - \sqrt { r - 2 }$$
  2. Hence use the method of differences to determine $$\sum _ { r = 2 } ^ { n } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } }$$ giving your answer in simplest form.
  3. Hence show that $$\sum _ { r = 4 } ^ { 50 } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = A + B \sqrt { 2 } + C \sqrt { 3 }$$ where \(A\), \(B\) and \(C\) are integers to be determined.
Edexcel F2 2023 June Q2
  1. The complex number \(z _ { 1 }\) is defined as
$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$
  1. Without using your calculator show that $$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
  2. Shade, on a single Argand diagram, the region \(R\) defined by $$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$ Given that the complex number \(z\) lies in \(R\)
  3. determine the smallest possible positive value of \(\arg z\)
Edexcel F2 2023 June Q3
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where \(k\) is a positive constant,
  1. show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where \(p , q\) and \(r\) are expressions in terms of \(k\) to be determined.
  2. Hence, or otherwise, determine the values for \(x\) for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$
Edexcel F2 2023 June Q4
  1. (a) Determine the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 48 x ^ { 2 } - 34$$ Given that \(y = 4\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 21\) at \(x = 0\)
(b) determine the particular solution of the differential equation.
(c) Hence find the value of \(y\) at \(x = - 2\), giving your answer in the form \(p \mathrm { e } ^ { q } + r\) where \(p , q\) and \(r\) are integers to be determined.
Edexcel F2 2023 June Q5
  1. The 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
$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.
  1. Show that \(C\) has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where \(k\) is a constant to be determined.
  2. Hence determine
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\)
Edexcel F2 2023 June Q6
  1. Given that \(y = \sec x\)
    1. show that
    $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sec x \tan x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
  2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of sec \(x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
  3. Use the answer to part (b) to determine, to four significant figures, an approximate value of \(\sec \left( \frac { 7 \pi } { 24 } \right)\)
Edexcel F2 2023 June Q7
  1. (a) Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y + 4 x ^ { 2 } y ^ { 3 } \ln x = 0 \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { 2 z } { x } = 8 x \ln x \quad x > 0$$ (b) By solving differential equation (II), determine the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)
Edexcel F2 2023 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{709ed2f1-f81c-4820-ac31-e1f86baae9d7-28_552_759_246_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,
  1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
  2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
  3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.
Edexcel F2 2024 June Q1
  1. The complex number \(z = x + i y\) satisfies the equation
$$| z - 3 - 4 i | = | z + 1 + i |$$
  1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.
Edexcel F2 2024 June Q2
2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$
  1. Show that $$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined. Given that \(y = 1\) at \(x = 2\)
  2. determine the Taylor series expansion for \(y\) in ascending powers of \(( x - 2 )\), up to and including the term in \(( x - 2 ) ^ { 3 }\), giving each coefficient in simplest form.