Questions — Edexcel (9685 questions)

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Edexcel FP2 2015 June Q4
7 marks Standard +0.8
4. (a) Show that $$r ^ { 2 } ( r + 1 ) ^ { 2 } - ( r - 1 ) ^ { 2 } r ^ { 2 } \equiv 4 r ^ { 3 }$$ Given that \(\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )\) (b) use the identity in (a) and the method of differences to show that $$\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + \ldots + n ^ { 3 } \right) = ( 1 + 2 + 3 + \ldots + n ) ^ { 2 }$$
Edexcel FP2 2015 June Q5
10 marks Challenging +1.2
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z } { z + 3 \mathrm { i } } , \quad z \neq - 3 \mathrm { i }$$ The circle with equation \(| z | = 2\) is mapped by \(T\) onto the curve \(C\).
    1. Show that \(C\) is a circle.
    2. Find the centre and radius of \(C\). The region \(| z | \leqslant 2\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  1. Shade the region \(R\) on an Argand diagram.
Edexcel FP2 2015 June Q6
11 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{49da3c56-ccd1-4599-95d8-d1395461bcca-11_451_1063_237_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has polar equation $$r = 3 a ( 1 + \cos \theta ) , \quad 0 \leqslant \theta < \pi$$ The tangent to \(C\) at the point \(A\) is parallel to the initial line.
  1. Find the polar coordinates of \(A\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(O A\).
  2. Use calculus to find the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are rational numbers.
Edexcel FP2 2015 June Q7
11 marks Challenging +1.2
7. $$y = \tan ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\)
  2. Hence show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \sec ^ { 2 } x \tan x \left( A \sec ^ { 2 } x + B \right)\), where \(A\) and \(B\) are constants to be found.
  3. Find the Taylor series expansion of \(\tan ^ { 2 } x\), in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\)
Edexcel FP2 2015 June Q8
14 marks Challenging +1.2
  1. (a) Show that the transformation \(x = \mathrm { e } ^ { u }\) transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 7 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 2 \ln x , \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 16 y = 2 u$$ (b) Find the general solution of the differential equation (II), expressing \(y\) as a function of \(u\).
(c) Hence obtain the general solution of the differential equation (I).
Edexcel FP2 2016 June Q1
6 marks Standard +0.3
  1. Use algebra to find the set of values of \(x\) for which
$$\frac { x } { x + 1 } < \frac { 2 } { x + 2 }$$
Edexcel FP2 2016 June Q2
7 marks Standard +0.8
2. (a) Show that, for \(r > 0\) $$r - 3 + \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) }$$ (b) Hence prove, using the method of differences, that $$\sum _ { r = 1 } ^ { n } \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) } = \frac { n \left( n ^ { 2 } + a n + b \right) } { 2 ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
Edexcel FP2 2016 June Q3
7 marks Standard +0.3
3. (a) Find the four roots of the equation \(z ^ { 4 } = 8 ( \sqrt { 3 } + \mathrm { i } )\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\) (b) Show these roots on an Argand diagram.
Edexcel FP2 2016 June Q4
12 marks Standard +0.3
4. (i) $$p \frac { \mathrm {~d} x } { \mathrm {~d} t } + q x = r \quad \text { where } p , q \text { and } r \text { are constants }$$ Given that \(x = 0\) when \(t = 0\)
  1. find \(x\) in terms of \(t\)
  2. find the limiting value of \(x\) as \(t \rightarrow \infty\) (ii) $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + 2 y = \sin \theta$$ Given that \(y = 0\) when \(\theta = 0\), find \(y\) in terms of \(\theta\)
Edexcel FP2 2016 June Q5
10 marks Challenging +1.2
5. (a) Use de Moivre's theorem to show that $$\sin ^ { 5 } \theta \equiv a \sin 5 \theta + b \sin 3 \theta + c \sin \theta$$ where \(a\), \(b\) and \(c\) are constants to be found.
(b) Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }\) VILM SIHI NITIIIUMI ON OC
VILV SIHI NI III HM ION OC
VALV SIHI NI JIIIM ION OO
Edexcel FP2 2016 June Q6
9 marks Standard +0.8
6. (a) Find the Taylor series expansion about \(\frac { \pi } { 4 }\) of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
(b) Deduce that an approximation for \(\tan \frac { 5 \pi } { 12 }\) is \(1 + \frac { \pi } { 3 } + \frac { \pi ^ { 2 } } { 18 } + \frac { \pi ^ { 3 } } { 81 }\)
Edexcel FP2 2016 June Q7
14 marks Challenging +1.2
7. (a) Show that the substitution \(x = e ^ { u }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = - x ^ { - 2 } , \quad x > 0$$ into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 2 y = - \mathrm { e } ^ { - 2 u }$$ (b) Find the general solution of the differential equation (II).
(c) Hence obtain the general solution of the differential equation (I) giving your answer in the form \(y = \mathrm { f } ( x )\)
Edexcel FP2 2016 June Q8
10 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{18620cc5-2377-480b-b815-63bfc6a9760a-15_618_942_255_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C _ { 1 }\) with equation $$r = 7 \cos \theta , \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ and the curve \(C _ { 2 }\) with equation $$r = 3 ( 1 + \cos \theta ) , \quad - \pi < \theta \leqslant \pi$$ are shown on Figure 1.
The curves \(C _ { 1 }\) and \(C _ { 2 }\) both pass through the pole and intersect at the point \(P\) and the point \(Q\).
  1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\). The regions enclosed by the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) overlap, and the common region \(R\) is shaded in Figure 1.
  2. Find the area of \(R\).
Edexcel FP2 2017 June Q1
7 marks Challenging +1.2
  1. (a) Show that, for \(r > 0\)
$$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$ (b) Hence prove that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$ (c) Show that, for \(n \in \mathbb { N } , n > 1\) $$\sum _ { r = n } ^ { 3 n } \frac { 6 r + 3 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { a n ^ { 2 } + b n + c } { n ^ { 2 } ( 3 n + 1 ) ^ { 2 } }$$ where \(a , b\) and \(c\) are constants to be found.
Edexcel FP2 2017 June Q2
9 marks Standard +0.3
2. Use algebra to find the set of values of \(x\) for which $$\frac { x - 2 } { 2 ( x + 2 ) } \leqslant \frac { 12 } { x ( x + 2 ) }$$ "
Edexcel FP2 2017 June Q3
6 marks Standard +0.3
3. Solve the equation $$z ^ { 3 } + 32 + 32 i \sqrt { 3 } = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
Edexcel FP2 2017 June Q4
10 marks Standard +0.3
4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
  2. Hence, or otherwise, find the series expansion of \(\ln \left( \frac { 1 } { 1 - 2 x } \right)\) about \(x = 0\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
  3. Use your expansion to find an approximate value for \(\ln \left( \frac { 3 } { 2 } \right)\), giving your answer
    to 3 decimal places.
Edexcel FP2 2017 June Q5
13 marks Standard +0.8
5. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 26 \sin 3 x$$ (b) Find the particular solution of this differential equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\) when \(x = 0\)
Edexcel FP2 2017 June Q6
8 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9019397a-a9c2-4b69-97fd-ea9eb9132244-18_364_695_260_756} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve with polar equation $$r = 6 + a \sin \theta$$ where \(0 < a < 6\) and \(0 \leqslant \theta < 2 \pi\) The area enclosed by the curve is \(\frac { 97 \pi } { 2 }\) Find the value of the constant \(a\).
Edexcel FP2 2017 June Q7
11 marks Standard +0.8
7. (a) Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y \sin x = 2 \cos ^ { 3 } x \sin x + 1 , \quad 0 < x < \frac { \pi } { 2 }$$ Given that \(y = 5 \sqrt { 2 }\) when \(x = \frac { \pi } { 4 }\) (b) find the value of \(y\) when \(x = \frac { \pi } { 6 }\), giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational numbers to be found.
Edexcel FP2 2017 June Q8
11 marks Challenging +1.8
8. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 3 \mathrm { i } } { 1 + \mathrm { i } z } , \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 1\) in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
  1. Find a cartesian equation of the line \(l\). The circle \(| z - a - b \mathrm { i } | = c\) in the \(z\)-plane is mapped by \(T\) onto the circle \(| w | = 5\) in the \(w\)-plane.
  2. Find the exact values of the real constants \(a\), \(b\) and \(c\).
    END
Edexcel FP2 2018 June Q1
8 marks Standard +0.3
  1. (a) Express \(\frac { 1 } { ( r + 3 ) ( r + 4 ) }\) in partial fractions.
    (b) Hence, using the method of differences, show that
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 3 ) ( r + 4 ) } = \frac { n } { a ( n + a ) }$$ where \(a\) is a constant to be found.
(c) Find the exact value of \(\sum _ { r = 15 } ^ { 30 } \frac { 1 } { ( r + 3 ) ( r + 4 ) }\) uestion 1 continued \includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_29_40_182_1914} \includegraphics[max width=\textwidth, alt={}, center]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-05_33_37_201_1914}
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Edexcel FP2 2018 June Q2
4 marks Standard +0.8
2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).
Edexcel FP2 2018 June Q3
9 marks Standard +0.3
3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that
  1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
  2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\) (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)
Edexcel FP2 2018 June Q4
7 marks Challenging +1.2
4. Use algebra to find the set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 4 x$$