Questions FP2 (1157 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 FP2 2011 June Q4
4. Given that $$( 2 r + 1 ) ^ { 3 } = A r ^ { 3 } + B r ^ { 2 } + C r + 1 ,$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Show that $$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
  3. Using the result in part (b) and the method of differences, show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$
Edexcel FP2 2011 June Q5
  1. The point \(P\) represents the complex number \(z\) on an Argand diagram, where
$$| z - \mathrm { i } | = 2$$ The locus of \(P\) as \(z\) varies is the curve \(C\).
  1. Find a cartesian equation of \(C\).
  2. Sketch the curve \(C\). A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,
  3. show that \(Q\) lies on \(C\).
Edexcel FP2 2011 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{893efbc9-8321-469f-bd5e-89f9d5827737-09_650_937_269_482} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(A\) on \(C\), the value of \(r\) is \(\frac { 5 } { 2 }\).
The point \(N\) lies on the initial line and \(A N\) is perpendicular to the initial line.
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(A N\). Find the exact area of the shaded region \(R\).
Edexcel FP2 2011 June Q7
  1. (a) Use de Moivre's theorem to show that
$$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence, given also that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\),
(b) find all the solutions of $$\sin 5 \theta = 5 \sin 3 \theta$$ in the interval \(0 \leqslant \theta < 2 \pi\). Give your answers to 3 decimal places.
Edexcel FP2 2011 June Q8
  1. The differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = \cos 3 t , \quad t \geqslant 0$$ describes the motion of a particle along the \(x\)-axis.
  1. Find the general solution of this differential equation.
  2. Find the particular solution of this differential equation for which, at \(t = 0\), $$x = \frac { 1 } { 2 } \text { and } \frac { \mathrm { d } x } { \mathrm {~d} t } = 0$$ On the graph of the particular solution defined in part (b), the first turning point for \(t > 30\) is the point \(A\).
  3. Find approximate values for the coordinates of \(A\).
Edexcel FP2 2012 June Q1
  1. Find the set of values of \(x\) for which
$$\left| x ^ { 2 } - 4 \right| > 3 x$$
Edexcel FP2 2012 June Q2
2. The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.
Given that \(O\) is the pole, find the exact length of the line \(O P\).
Edexcel FP2 2012 June Q3
3. (a) Express the complex number \(- 2 + ( 2 \sqrt { 3 } ) \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
(b) Solve the equation $$z ^ { 4 } = - 2 + ( 2 \sqrt { } 3 ) i$$ giving the roots in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
Edexcel FP2 2012 June Q4
4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \cos t - \sin t$$
Edexcel FP2 2012 June Q5
5. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x + y ^ { 2 }$$
  1. Show that $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 - 2 y ) \frac { \mathrm { d } y } { \mathrm {~d} x } = 3$$ Given that \(y = 1\) at \(x = 1\),
  2. find a series solution for \(y\) in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 3 }\).
Edexcel FP2 2012 June Q6
  1. (a) Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
    (b) Hence prove, by the method of differences, that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( a n + b ) } { 4 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( 4 n + 5 ) } { 4 ( n + 1 ) ( n + 2 ) ( 2 n + 1 ) }$$
Edexcel FP2 2012 June Q7
  1. (a) Show that the substitution \(y = v x\) transforms the differential equation
$$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 3 } + y ^ { 3 }$$ into the differential equation $$3 v ^ { 2 } x \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1 - 2 v ^ { 3 }$$ (b) By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = \mathrm { f } ( x )\). Given that \(y = 2\) at \(x = 1\),
(c) find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 1\)
Edexcel FP2 2012 June Q8
  1. The point \(P\) represents a complex number \(z\) on an Argand diagram such that
$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$$
  2. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.
  3. Find the complex number for which both \(| z - 6 \mathrm { i } | = 2 | z - 3 |\) and \(\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }\)
Edexcel FP2 2013 June Q1
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z + 2 \mathrm { i } } { \mathrm { i } z } \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line.
Edexcel FP2 2013 June Q2
2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 x } { 3 - x } > \frac { 1 } { x + 1 }$$
Edexcel FP2 2013 June Q3
3. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }$$ (c) Evaluate \(\sum _ { r = 10 } ^ { 100 } \frac { 2 } { ( r + 1 ) ( r + 3 ) }\), giving your answer to 3 significant figures.
Edexcel FP2 2013 June Q4
  1. Given that
$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + 5 y = 0$$
  1. find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(x = 0\)
  2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 2013 June Q5
  1. (a) Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 2\) at \(x = \frac { \pi } { 3 }\)
(b) find the value of \(y\) at \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + k \ln b\), where \(a\) and \(b\) are integers and \(k\) is rational.
Edexcel FP2 2013 June Q6
6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.
  1. Use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ where \(n\) is a positive integer.
  2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
  3. Hence find all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)
Edexcel FP2 2013 June Q7
  1. (a) Find the value of \(\lambda\) for which \(\lambda t ^ { 2 } \mathrm { e } ^ { 3 t }\) is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 6 \mathrm { e } ^ { 3 t } , \quad t \geqslant 0$$ (b) Hence find the general solution of this differential equation. Given that when \(t = 0 , y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 4\)
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( t )\).
Edexcel FP2 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6d1485a6-e52b-4492-8d3b-eadca26962db-14_360_1109_237_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 ( \cos 2 \theta ) ^ { \frac { 1 } { 2 } } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leqslant \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } < \theta \leqslant \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1.
  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.
Edexcel FP2 2013 June Q1
1. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \cos x$$
  1. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(x , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\)
  2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\)
  3. Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 2013 June Q2
2. (a) Sketch, on the same axes,
  1. \(y = | 2 x - 3 |\)
  2. \(y = 4 - x ^ { 2 }\)
    (b) Find the set of values of \(x\) for which $$4 - x ^ { 2 } > | 2 x - 3 |$$
Edexcel FP2 2013 June Q3
3. $$f ( x ) = \ln ( 1 + \sin k x )$$ where \(k\) is a constant, \(x \in \mathbb { R }\) and \(- \frac { \pi } { 2 } < k x < \frac { 3 \pi } { 2 }\)
  1. Find f \({ } ^ { \prime } ( x )\)
  2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { - k ^ { 2 } } { 1 + \sin k x }\)
  3. Find the Maclaurin series of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 2013 June Q4
4. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( 1 + x \cot x ) y = \sin x , \quad 0 < x < \pi$$ giving your answer in the form \(y = \mathrm { f } ( x )\).