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Edexcel F2 2024 June Q3
  1. (a) Express
$$\frac { 1 } { ( n + 3 ) ( n + 5 ) }$$ in partial fractions.
(b) Hence, using the method of differences, show that for all positive integer values of \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 5 ) } = \frac { n ( p n + q ) } { 40 ( n + 4 ) ( n + 5 ) }$$ where \(p\) and \(q\) are integers to be determined.
(c) Use the answer to part (b) to determine, as a simplified fraction, the value of $$\frac { 1 } { 9 \times 11 } + \frac { 1 } { 10 \times 12 } + \ldots + \frac { 1 } { 24 \times 26 }$$
Edexcel F2 2024 June Q4
  1. (a) Show that the substitution \(y ^ { 2 } = \frac { 1 } { t }\) transforms the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y = x y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } - 2 t = - 2 x$$ (b) Solve differential equation (II) and determine \(y ^ { 2 }\) in terms of \(x\).
Edexcel F2 2024 June Q5
  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 values of \(x\) for which $$\frac { x + 1 } { ( x - 3 ) ( x + 2 ) } \leqslant 1 - \frac { 2 } { x - 3 }$$
Edexcel F2 2024 June Q6
  1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\)
(ii) the radius of \(C\)
Edexcel F2 2024 June Q7
  1. Given that \(y = \mathrm { e } ^ { x } \sin x\)
    1. show that
    $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = k \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(k\) is a constant to be determined.
  2. Hence determine the first 5 non-zero terms in the Maclaurin series expansion for \(y\), giving each coefficient in simplest form.
Edexcel F2 2024 June Q8
  1. (a) Given that \(t = \ln x\), where \(x > 0\), show that
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$ (b) Hence show that the transformation \(t = \ln x\), where \(x > 0\), transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 y = 1 + 4 \ln x - 2 ( \ln x ) ^ { 2 }$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } - 2 y = 1 + 4 t - 2 t ^ { 2 }$$ (c) Solve differential equation (II) to determine \(y\) in terms of \(t\).
(d) Hence determine the general solution of differential equation (I).
Edexcel F2 2024 June Q9
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
  2. Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.
Edexcel F2 2024 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09582a82-cd57-4c2f-aefa-8412d4f4cb64-32_497_919_292_573} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 1 + \cos \theta \quad 0 \leqslant \theta \leqslant \pi$$ and the line \(l\) with polar equation $$r = k \sec \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ where \(k\) is a positive constant.
Given that
  • \(\quad C\) and \(l\) intersect at the point \(P\)
  • \(O P = 1 + \frac { \sqrt { 3 } } { 2 }\)
    1. determine the exact value of \(k\).
The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\), the initial line and \(l\).
  • Use algebraic integration to show that the area of \(R\) is $$p \pi + q \sqrt { 3 } + r$$ where \(p , q\) and \(r\) are simplified rational numbers to be determined.
    •
    Edexcel F2 2021 October Q1
    1. Solve the equation
    $$z ^ { 5 } - 32 i = 0$$ giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < 2 \pi\)
    Edexcel F2 2021 October Q2
    2. Use algebra to determine the set of values of \(x\) for which $$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$ (Solutions relying entirely on graphical methods are not acceptable.)
    (8)
    Edexcel F2 2021 October Q3
    3. A transformation maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation is given by $$w = \frac { ( 2 + \mathrm { i } ) z + 4 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps the imaginary axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
    Determine a Cartesian equation of \(l\), giving your answer in the form \(a u + b v + c = 0\) where \(a , b\) and \(c\) are integers to be found.
    (6)
    Edexcel F2 2021 October Q4
    4. (a) Determine the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = \mathrm { e } ^ { 3 x } \quad x > - 1$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
    (b) Determine the particular solution of the differential equation for which \(y = 5\) when \(x = 0\)
    Edexcel F2 2021 October Q5
    5. Given that \(y = \tan ^ { 2 } x\)
    1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \tan x \sec ^ { 2 } x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
    2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of \(\tan ^ { 2 } x\) 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.
      \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-19_33_407_306_258}
      \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-19_58_458_2752_150}
    Edexcel F2 2021 October Q6
    6. The complex number \(z\) on an Argand diagram is represented by the point \(P\) where $$| z + 1 - 13 i | = 3 | z - 7 - 5 i |$$ Given that the locus of \(P\) is a circle,
    1. determine the centre and radius of this circle. The complex number \(w\), on the same Argand diagram, is represented by the point \(Q\), where $$\arg ( w - 8 - 6 \mathrm { i } ) = - \frac { 3 \pi } { 4 }$$ Given that the locus of \(P\) intersects the locus of \(Q\) at the point \(R\),
    2. determine the complex number representing \(R\).
    Edexcel F2 2021 October Q7
    7. (a) Show that the transformation \(x = t ^ { 2 }\) transforms the differential equation $$4 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 1 + 2 \sqrt { x } ) \frac { \mathrm { d } y } { \mathrm {~d} x } - 15 y = 15 x$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } - 15 y = 15 t ^ { 2 }$$ (b) Solve differential equation (II) to determine \(y\) in terms of \(t\).
    (c) Hence determine the general solution of differential equation (I).
    \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-24_2258_53_308_1980}
    Edexcel F2 2021 October Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-28_735_892_264_529} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 + \sin \theta \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) such that the tangent to \(C\) at \(P\) is perpendicular to the initial line.
    1. Use calculus to determine the polar coordinates of \(P\). The tangent to \(C\) at the point \(Q\) where \(\theta = \frac { \pi } { 2 }\) is parallel to the initial line.
      The tangent to \(C\) at \(Q\) meets the tangent to \(C\) at \(P\) at the point \(S\), as shown in Figure 1.
      The finite region \(R\), shown shaded in Figure 1, is bounded by the line segments \(Q S , S P\) and the curve \(C\).
    2. Use algebraic integration to show that the area of \(R\) is $$\frac { 1 } { 32 } ( a \sqrt { 3 } + b \pi )$$ where \(a\) and \(b\) are integers to be determined.
      (6)
    Edexcel F2 2021 October Q9
    1. (a) Show that
    $$n ^ { 5 } - ( n - 1 ) ^ { 5 } \equiv 5 n ^ { 4 } - 10 n ^ { 3 } + 10 n ^ { 2 } - 5 n + 1$$ (b) Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 4 } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
    Edexcel F2 2018 Specimen Q2
    1. (a) Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
      (b) Hence show that
    $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.
    VIIN SIHI NI JIIIM IONOOVIIV SIHI NI III HM ION OOVI4V SIHI NI JIIIM IONOO
    Edexcel F2 2018 Specimen Q3
    1. (a) Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation
    $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 x y = x \mathrm { e } ^ { - x ^ { 2 } } y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 x z = - 2 x \mathrm { e } ^ { - x ^ { 2 } }$$ (b) Solve differential equation (II) to find \(z\) as a function of \(x\).
    (c) Hence find the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
    Edexcel F2 2018 Specimen Q4
    1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
    $$w = \frac { z - 1 } { z + 1 } , \quad z \neq - 1$$ The line in the \(z\)-plane with equation \(y = 2 x\) is mapped by \(T\) onto the curve \(C\) in the \(w\)-plane.
    1. Show that \(C\) is a circle and find its centre and radius. The region \(y < 2 x\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
    2. Sketch circle \(C\) on an Argand diagram and shade and label region \(R\).
      VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVEXV SIHIL NI JIIIM ION OO
    Edexcel F2 2018 Specimen Q5
    1. Given that \(y = \cot x\),
      1. show that
      $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \cot x + 2 \cot ^ { 3 } x$$
    2. Hence show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = p \cot ^ { 4 } x + q \cot ^ { 2 } x + r$$ where \(p , q\) and \(r\) are integers to be found.
    3. Find the Taylor series expansion of \(\cot x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\).
      VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
    Edexcel F2 2018 Specimen Q6
    1. (a) Find the general solution of the differential equation
    $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \sin x$$ Given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\)
    (b) find the particular solution of differential equation (I).
    Edexcel F2 2018 Specimen Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b197811e-1df5-4937-b0d8-f98f82412c76-24_480_926_217_511} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the two curves given by the polar equations $$\begin{array} { l l } r = \sqrt { 3 } \sin \theta , & 0 \leqslant \theta \leqslant \pi
    r = 1 + \cos \theta , & 0 \leqslant \theta \leqslant \pi \end{array}$$
    1. Verify that the curves intersect at the point \(P\) with polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right)\). The region \(R\), bounded by the two curves, is shown shaded in Figure 1.
    2. Use calculus to find the exact area of \(R\), giving your answer in the form \(a ( \pi - \sqrt { 3 } )\), where \(a\) is a constant to be found.
      VIIIV SIHI NI JIIIM ION OCVIIIV SIHI NI JIHM I I ON OCVI4V SIHI NI JIIYM IONOO
    Edexcel F2 2018 Specimen Q8
    1. (a) Show that
    $$\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 } = z ^ { 6 } - \frac { 1 } { z ^ { 6 } } - k \left( z ^ { 2 } - \frac { 1 } { z ^ { 2 } } \right)$$ where \(k\) is a constant to be found. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(\theta\) is real,
    (b) show that
    1. \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
    2. \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 i \sin n \theta\)
      (c) Hence show that $$\cos ^ { 3 } \theta \sin ^ { 3 } \theta = \frac { 1 } { 32 } \quad ( 3 \sin 2 \theta - \sin 6 \theta )$$ (d) Find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 8 } } \cos ^ { 3 } \theta \sin ^ { 3 } \theta d \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{b197811e-1df5-4937-b0d8-f98f82412c76-32_227_148_2524_1797}
    Edexcel F2 Specimen Q4
    4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
    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 \(a , b \in \mathbb { R }\).