Questions — Edexcel F2 (137 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel F2 2024 January Q1
  1. Using algebra, solve the inequality
$$\frac { 1 } { x + 2 } > 2 x + 3$$
Edexcel F2 2024 January Q2
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
  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
  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
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
  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
  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
  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 \(\cos x\) $$\text { [You may use without proof } \left. \int \operatorname { cosec } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \ln | \tan x | \text { (+ constant) } \right]$$
Edexcel F2 2014 June Q1
  1. (a) Show that
$$\frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { 1 } { 2 ( r + 1 ) ( r + 2 ) } - \frac { 1 } { 2 ( r + 2 ) ( r + 3 ) }$$ (b) Hence, or otherwise, find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) }$$ giving your answer as a single fraction in its simplest form.
Edexcel F2 2014 June Q2
2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 } { x - 3 } \leqslant x + 2$$
Edexcel F2 2014 June Q3
3. Solve the equation $$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(\theta\) is in terms of \(\pi\) and \(0 \leqslant \theta < 2 \pi\).
Edexcel F2 2014 June Q4
4. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z } { z + 3 } , \quad z \neq - 3$$ Under this transformation, the circle \(| z | = 2\) in the \(z\)-plane is mapped onto a circle \(C\) in the \(w\)-plane. Determine the centre and the radius of the circle \(C\).
Edexcel F2 2014 June Q5
5. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 0$$
  1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \left( a x ^ { 2 } + b \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\) and \(b\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) at \(x = 0\)
  2. find a series solution for \(y\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\)
  3. use your series to estimate the value of \(y\) at \(x = - 0.2\), giving your answer to four decimal places.
Edexcel F2 2014 June Q6
6. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 - 2 x ) y = x , \quad x > 0$$ Find the general solution of the differential equation, giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel F2 2014 June Q7
7. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z + 1 | = | 2 z - 1 |$$ and the point \(Q\) represents a complex number \(w\) on the Argand diagram, where $$| w | = | w - 1 + \mathrm { i } |$$ Find the exact coordinates of the points where the locus of \(P\) intersects the locus of \(Q\).
Edexcel F2 2014 June Q8
8. (a) Show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = 0 , \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 0$$ (b) Hence find the general solution of the differential equation (I).
Edexcel F2 2014 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{77d00a35-e947-41ef-8d80-5a573702ed39-14_643_1274_251_342} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant.
  1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\) The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
  2. Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are integers to be found.
Edexcel F2 2015 June Q1
  1. Using algebra, find the set of values of \(x\) for which
$$\frac { x } { x + 2 } < \frac { 2 } { x + 5 }$$
Edexcel F2 2015 June 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.
Edexcel F2 2015 June 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 \(y ^ { 2 } = \mathrm { f } ( x )\).
Edexcel F2 2015 June 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 \(R\).
Edexcel F2 2015 June 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 \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\).
Edexcel F2 2015 June 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 2015 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee9a9df3-f7a4-41d0-bf8b-e44340c401d6-13_458_933_251_504} \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.
Edexcel F2 2015 June 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. \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\)
    (c) Hence show that $$\cos ^ { 3 } \theta \sin ^ { 3 } \theta = \frac { 1 } { 32 } ( 3 \sin 2 \theta - \sin 6 \theta )$$ (d) Find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 8 } } \cos ^ { 3 } \theta \sin ^ { 3 } \theta d \theta$$