Questions F2 (137 questions)

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Edexcel F2 2021 January Q1
  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
$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$ The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).
Edexcel F2 2021 January Q2
2. (a) Show that, for \(r > 0\) $$\frac { r + 2 } { r ( r + 1 ) } - \frac { r + 3 } { ( r + 1 ) ( r + 2 ) } = \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) }$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { c ( n + 1 ) ( n + 2 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2021 January Q3
3. Use algebra to obtain the set of values of \(x\) for which $$\left| x ^ { 2 } + x - 2 \right| < \frac { 1 } { 2 } ( x + 5 )$$
Edexcel F2 2021 January Q4
4. (a) Show that the substitution \(y ^ { 2 } = \frac { 1 } { z }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = 3 x y ^ { 3 } \quad y \neq 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 z = - 6 x$$ (b) Obtain the general solution of differential equation (II).
(c) Hence obtain the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)
Edexcel F2 2021 January Q5
5. Given that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 3 y$$
  1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 1 } { \left( 2 - x ^ { 2 } \right) } \left( 2 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \left( 1 - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) - 5 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }\) at \(x = 0\)
  2. obtain a series solution for \(y\) in ascending powers of \(x\) with simplified coefficients, up to and including the term in \(x ^ { 3 }\)
Edexcel F2 2021 January Q6
6. (a) Determine the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 6 \cos x$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at \(x = 0\)
Edexcel F2 2021 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d3e1c8e-c659-4cfe-82ac-5bfce0f58ba3-24_445_597_248_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of curve \(C\) with polar equation $$r = 3 \sin 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) on \(C\) has polar coordinates \(( R , \phi )\). The tangent to \(C\) at \(P\) is perpendicular to the initial line.
  1. Show that \(\tan \phi = \frac { 1 } { \sqrt { 2 } }\)
  2. Determine the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and the line \(O P\), where \(O\) is the pole.
  3. Use calculus to show that the exact area of \(S\) is $$p \arctan \frac { 1 } { \sqrt { 2 } } + q \sqrt { 2 }$$ where \(p\) and \(q\) are constants to be determined. Solutions relying entirely on calculator technology are not acceptable.
Edexcel F2 2021 January Q8
8. Given that \(z = e ^ { \mathrm { i } \theta }\)
  1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
    where \(n\) is a positive integer.
  2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
  3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
  4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.
Edexcel F2 2022 January Q1
  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
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
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
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
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
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
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
  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
  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
  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
  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
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
  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
  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
  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
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
  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).