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Edexcel F2 2016 June Q9
7 marks Challenging +1.2
9. The complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that \(\arg \left( \frac { z - 5 } { z - 2 } \right) = \frac { \pi } { 4 }\)
  1. sketch the locus of \(P\) as \(z\) varies,
  2. find the exact maximum value of \(| z |\).
    VILM SIHI NITIIIUMI ON OC
    VILV SIHI NI III HM ION OC
    VALV SIHI NI JIIIM ION OO
Edexcel F2 2017 June Q1
5 marks Moderate -0.8
  1. Solve the equation
$$z ^ { 5 } = 32$$ Give your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\)
Edexcel F2 2017 June Q2
9 marks Moderate -0.3
  1. Use algebra to find the set of values of \(x\) for which
$$\frac { x - 4 } { ( x + 3 ) } \leqslant \frac { 5 } { x ( x + 3 ) }$$
Edexcel F2 2017 June Q3
6 marks Standard +0.3
3. (a) Show that \(r ^ { 3 } - ( r - 1 ) ^ { 3 } \equiv 3 r ^ { 2 } - 3 r + 1\) (b) Hence prove by the method of differences that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$$ [You may use \(\sum _ { r = 1 } ^ { n } r = \frac { n ( n + 1 ) } { 2 }\) without proof.]
Edexcel F2 2017 June Q4
13 marks Standard +0.8
4. $$y = 3 \mathrm { e } ^ { - x } \cos 3 x + A \mathrm { e } ^ { - x } \sin 3 x$$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 40 \mathrm { e } ^ { - x } \sin 3 x$$ where \(A\) is a constant.
  1. Find the value of \(A\).
  2. Hence find the general solution of this differential equation.
  3. Find the particular solution of this differential equation for which both \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)
Edexcel F2 2017 June Q5
7 marks Standard +0.3
5. $$y = \mathrm { e } ^ { \cos ^ { 2 } x }$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { \cos ^ { 2 } x } \left( \sin ^ { 2 } 2 x - 2 \cos 2 x \right)$$
  2. Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { \cos ^ { 2 } x }\) up to and including the term in \(x ^ { 2 }\)
Edexcel F2 2017 June Q6
8 marks Standard +0.8
  1. Find the general solution of the differential equation
$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = \left( \cos ^ { 2 } x \right) \ln x , \quad 0 < x < \frac { \pi } { 2 }$$ Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel F2 2017 June Q7
15 marks Hard +2.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2026c49f-243b-497a-b702-e40d012ad308-20_465_1070_255_507} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with polar equation $$r = 4 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 } \text { and } \frac { 3 \pi } { 4 } \leqslant \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , Q R , R S\) and \(S P\) are tangents to \(C\), where \(Q R\) and \(S P\) are parallel to the initial line and \(P Q\) and \(R S\) are perpendicular to the initial line.
  1. Find the polar coordinates of the points where the tangent SP touches the curve. Give the values of \(\theta\) to 3 significant figures.
  2. Find the exact area of the finite region bounded by the curve \(C\), shown unshaded in Figure 1.
  3. Find the area enclosed by the rectangle \(P Q R S\) but outside the curve \(C\), shown shaded in Figure 1.
Edexcel F2 2017 June Q8
12 marks Challenging +1.2
  1. (a) Use de Moivre's theorem to
    1. show that
    $$\cos 5 \theta \equiv \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$
  2. find an expression for \(\sin 5 \theta\) in terms of \(\cos \theta\) and \(\sin \theta\) (b) Hence show that $$\tan 5 \theta = \frac { t ^ { 5 } - 10 t ^ { 3 } + 5 t } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$ where \(t = \tan \theta\) and \(\cos 5 \theta \neq 0\) (c) Hence find a quadratic equation whose roots \(\operatorname { are } ^ { 2 } \tan ^ { 2 } \frac { \pi } { 5 }\) and \(\tan ^ { 2 } \frac { 2 \pi } { 5 }\) Give your answer in the form \(a x ^ { 2 } + b x + c = 0\) where \(a , b\) and \(c\) are integers to be found.
    (d) Deduce that \(\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }\)
    END
Edexcel F2 2020 June Q2
9 marks Standard +0.3
2. (a) Write \(\frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }\) in partial fractions.
(b) Hence find $$\sum _ { r = 2 } ^ { n } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) } \quad n \geqslant 2$$ giving your answer in the form $$\frac { a n ^ { 2 } + b n + c } { 2 n ( n + 1 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(c) Hence determine the exact value of $$\sum _ { r = 15 } ^ { 20 } \frac { 3 r + 1 } { r ( r - 1 ) ( r + 1 ) }$$
VIXV SIHII NI JIIIM ION OCVIAN SIHI NI JYHM ION OOVAYV SIHI NI JIIIM ION OO
Edexcel F2 2020 June Q3
9 marks Challenging +1.2
3. Use algebra to obtain the set of values of \(x\) for which $$\left| \frac { x ^ { 2 } + 3 x + 10 } { x + 2 } \right| < 7 - x$$
Edexcel F2 2020 June Q4
8 marks Standard +0.3
4. (a) Express the complex number \(18 \sqrt { 3 } - 18 \mathrm { i }\) in the form $$r ( \cos \theta + \mathrm { i } \sin \theta ) \quad - \pi < \theta \leqslant \pi$$ (b) Solve the equation $$z ^ { 4 } = 18 \sqrt { 3 } - 18 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(- \pi < \theta \leqslant \pi\)
VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F2 2020 June Q5
7 marks Challenging +1.2
5. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z - 3 \mathrm { i } } { z + 2 \mathrm { i } } \quad z \neq - 2 \mathrm { i }$$ The circle with equation \(| z | = 1\) in the \(z\)-plane is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane. Determine
  1. the centre of \(C\),
  2. the radius of \(C\).
Edexcel F2 2020 June Q6
8 marks Challenging +1.2
6. Obtain the general solution of the equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( x \cot x + 2 ) x y = 4 \sin x \quad 0 < x < \pi$$ Give your answer in the form \(y = \mathrm { f } ( x )\) (8)
Edexcel F2 2020 June Q7
13 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17b48fd7-5e88-4a62-beb9-8590a363e70f-20_476_972_251_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has polar equation $$r = 2 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ where \(a\) is a positive constant. The tangent to \(C\) at the point \(A\) is parallel to the initial line.
  1. Determine the polar coordinates of \(A\). The point \(B\) on the curve has polar coordinates \(\quad a ( 2 + \sqrt { 3 } ) , \frac { \pi } { 6 }\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the line \(A B\).
  2. Use calculus to determine the exact area of the shaded region \(R\). Give your answer in the form $$\frac { a ^ { 2 } } { 4 } ( d \pi - e + f \sqrt { 3 } )$$ where \(d , e\) and \(f\) are integers.
Edexcel F2 2020 June Q8
14 marks Challenging +1.2
8. (a) Show that the transformation \(x = \mathrm { e } ^ { u }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 8 y = 4 \ln x \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} u } - 8 y = 4 u$$ (b) Determine the general solution of differential equation (II), expressing \(y\) as a function of \(u\).
(c) Hence obtain the general solution of differential equation (I).
VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F2 2021 June Q1
8 marks Standard +0.8
  1. (a) Express \(\frac { 2 } { r \left( r ^ { 2 } - 1 \right) }\) in partial fractions.
    (b) Hence find, in terms of \(n\),
$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r \left( r ^ { 2 } - 1 \right) }$$ Give your answer in the form $$\frac { n ^ { 2 } + A n + B } { C n ( n + 1 ) }$$ where \(A\), \(B\) and \(C\) are constants to be found.
Edexcel F2 2021 June Q2
8 marks Challenging +1.2
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
10 marks Standard +0.8
  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
9 marks
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
7 marks Standard +0.8
  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
13 marks Standard +0.8
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
9 marks Challenging +1.2
  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
11 marks Challenging +1.2
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
5 marks Standard +0.8
  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.