Questions — Edexcel (10514 questions)

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Edexcel FP1 2024 June Q7
7 marks Challenging +1.2
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
  1. Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that $$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$ where \(a\), \(b\) and \(c\) are constants to be determined.
  2. Hence show that $$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$
Edexcel FP1 2024 June Q8
8 marks Challenging +1.2
  1. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).
  1. Use calculus to show that an equation of the tangent to \(P\) at \(A\) is $$y t = x + a t ^ { 2 }$$ The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
    The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
    Given that the area of the triangle \(B C D\) is 432
  2. determine the coordinates of \(B\) and the coordinates of \(C\).
Edexcel FP1 2024 June Q9
10 marks Standard +0.8
  1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 4 \\ - 1 \end{array} \right)\) The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13 \\ 5 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right)\) where \(\lambda\) and \(\mu\) are scalar parameters.
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
    1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
    2. determine a Cartesian equation for \(\Pi\).
    3. Determine a Cartesian equation for each of the two lines that
Edexcel FP1 2024 June Q10
12 marks Challenging +1.3
  1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t ( t + 1 ) \frac { \mathrm { d } x } { \mathrm {~d} t } + 2 ( t + 1 ) x = 8 t ^ { 3 } \mathrm { e } ^ { t }$$ where \(P\) has displacement \(x\) metres from the origin \(O\) at time \(t\) minutes, \(t > 0\)
  1. Show that the transformation \(x = t u\) transforms the differential equation (I) into the differential equation $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} u } { \mathrm {~d} t } = 8 \mathrm { e } ^ { t }$$ Given that \(P\) is at \(O\) when \(t = \ln 3\) and when \(t = \ln 5\)
  2. determine the particular solution of the differential equation (I)
Edexcel FP1 Specimen Q1
5 marks Moderate -0.8
  1. Use Simpson's Rule with 6 intervals to estimate
$$\int _ { 1 } ^ { 4 } \sqrt { 1 + x ^ { 3 } } d x$$
Edexcel FP1 Specimen Q2
4 marks Challenging +1.8
  1. Given \(k\) is a constant and that
$$y = x ^ { 3 } \mathrm { e } ^ { k x }$$ use Leibnitz theorem to show that $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = k ^ { n - 3 } \mathrm { e } ^ { k x } \left( k ^ { 3 } x ^ { 3 } + 3 n k ^ { 2 } x ^ { 2 } + 3 n ( n - 1 ) k x + n ( n - 1 ) ( n - 2 ) \right)$$
Edexcel FP1 Specimen Q3
14 marks Challenging +1.2
  1. A vibrating spring, fixed at one end, has an external force acting on it such that the centre of the spring moves in a straight line. At time \(t\) seconds, \(t \geqslant 0\), the displacement of the centre \(C\) of the spring from a fixed point \(O\) is \(x\) micrometres.
The displacement of \(C\) from \(O\) is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 2 + t ^ { 2 } \right) x = t ^ { 4 }$$
  1. Show that the transformation \(x = t v\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} t ^ { 2 } } + v = t$$
  2. Hence find the general equation for the displacement of \(C\) from \(O\) at time \(t\) seconds.
    1. State what happens to the displacement of \(C\) from \(O\) as \(t\) becomes large.
    2. Comment on the model with reference to this long term behaviour.
Edexcel FP1 Specimen Q4
9 marks Challenging +1.2
4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$
  1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a x \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$$ where \(a\) and \(b\) are integers to be found.
  2. Hence find a series solution, in ascending powers of \(x\), as far as the term in \(x ^ { 5 }\), of the differential equation (I) where \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
Edexcel FP1 Specimen Q5
9 marks Challenging +1.2
  1. The normal to the parabola \(y ^ { 2 } = 4 a x\) at the point \(P \left( a p ^ { 2 } , 2 a p \right)\) passes through the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\).
The line \(O P\) is perpendicular to the line \(O Q\), where \(O\) is the origin.
Prove that \(p ^ { 2 } = 2\)
Edexcel FP1 Specimen Q6
11 marks Challenging +1.2
  1. A tetrahedron has vertices \(A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )\) and \(D ( 10,5,5 )\).
Find
  1. a Cartesian equation of the plane \(A B C\).
  2. the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) has equation \(2 x - 3 y + 3 = 0\) The point \(E\) lies on the line \(A C\) and the point \(F\) lies on the line \(A D\).
    Given that \(\Pi\) contains the point \(B\), the point \(E\) and the point \(F\),
  3. find the value of \(k\) such that \(\overrightarrow { A E } = k \overrightarrow { A C }\). Given that \(\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }\)
  4. show that the volume of the tetrahedron \(A B C D\) is 45 times the volume of the tetrahedron \(A B E F\).
Edexcel FP1 Specimen Q7
8 marks Challenging +1.8
  1. \(P\) and \(Q\) are two distinct points on the ellipse described by the equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)
The line \(l\) passes through the point \(P\) and the point \(Q\).
The tangent to the ellipse at \(P\) and the tangent to the ellipse at \(Q\) intersect at the point \(( r , s )\).
Show that an equation of the line \(l\) is $$4 s y + r x = 4$$
Edexcel FP1 Specimen Q8
15 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_407_1100_201_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of the function \(\mathrm { h } ( x )\) with equation $$h ( x ) = 45 + 15 \sin x + 21 \sin \left( \frac { x } { 2 } \right) + 25 \cos \left( \frac { x } { 2 } \right) \quad x \in [ 0,40 ]$$
  1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 6 t - 17 \right) \left( 9 t ^ { 2 } + 4 t - 3 \right) } { 2 \left( 1 + t ^ { 2 } \right) ^ { 2 } }$$ where \(t = \tan \left( \frac { x } { 4 } \right)\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_581_1403_1263_331} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Source: \({ } ^ { 1 }\) Data taken on 29th December 2016 from \href{http://www.ukho.gov.uk/easytide/EasyTide}{http://www.ukho.gov.uk/easytide/EasyTide} Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 08:00 on the 3rd January 2017 to the end of the 4th January \(2017 { } ^ { 1 }\). The graph of \(k \mathrm {~h} ( x )\), where \(k\) is a constant and \(x\) is the number of hours after 08:00 on 3rd of January, can be used to model the predicted tide heights, in metres, for this period of time.
    1. Suggest a value of \(k\) that could be used for the graph of \(k \mathrm {~h} ( x )\) to form a suitable model.
    2. Why may such a model be suitable to predict the times when the tide heights are at their peaks, but not to predict the heights of these peaks?
  3. Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of the highest tide height on the 4th January 2017.
Edexcel FP2 2019 June Q1
5 marks Standard +0.3
  1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.
Given that $$| z - 3 | = 4 | z + 1 |$$
  1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
  2. Hence find the maximum value of \(| z |\)
Edexcel FP2 2019 June Q2
11 marks Challenging +1.2
  1. The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2 \\ - 2 & 3 & - 1 \\ 2 & - 1 & 3 \end{array} \right)$$
  1. Show that 2 is a repeated eigenvalue of \(\mathbf { A }\) and find the other eigenvalue.
  2. Hence find three non-parallel eigenvectors of \(\mathbf { A }\).
  3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix.
Edexcel FP2 2019 June Q3
8 marks Standard +0.3
  1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.
Given that \(V _ { n }\) is the number of visits to the website in month \(n\),
  1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
    • \(k = 0.24\)
    • In month 1 , there were 65 visits to the website.
    • In month 2 , there were 71 visits to the website.
    • Show that
    $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
  2. Find the value of \(N\).
Edexcel FP2 2019 June Q4
12 marks Standard +0.3
  1. Use Fermat's Little Theorem to find the least positive residue of \(6 ^ { 542 }\) modulo 13
  2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if
  1. there are no restrictions where they sit in the row,
  2. Alan, Enid, Felix and Graham sit together,
  3. Brenda sits at one end of the row and Graham sits at the other end of the row,
  4. Charles and Devindra do not sit together.
Edexcel FP2 2019 June Q5
8 marks Challenging +1.8
5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$
  1. Prove that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
  2. Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$
Edexcel FP2 2019 June Q6
12 marks Challenging +1.2
  1. A binary operation * is defined on positive real numbers by $$a * b = a + b + a b$$ Prove that the operation * is associative.
  2. The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7
  1. Show that \(G\) is cyclic. The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
  2. List all the subgroups of \(H\).
  3. Describe an isomorphism between \(G\) and \(H\).
Edexcel FP2 2019 June Q7
6 marks Challenging +1.2
  1. A transformation from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$
  1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
  2. Sketch \(C\) and \(D\) on Argand diagrams.
Edexcel FP2 2019 June Q8
13 marks Challenging +1.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ba4a815-f53d-4de2-810b-b06e145f457b-24_547_629_242_717} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point \(A\) is vertically above the point \(B\) and the height of the spinning top is 5 cm . The line \(C D\) is perpendicular to \(A B\) such that \(C D\) is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through \(2 \pi\) radians about the initial line.
  1. Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where \(k\) is a constant to be determined.
  2. Show that, according to the model, the length \(C D\) is \(\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}\).
Edexcel FP2 2020 June Q1
6 marks Moderate -0.3
  1. A small sports club has 12 adult members and 14 junior members.
The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if
  1. there are no restrictions on the team,
  2. the team must contain 4 adults and 4 juniors,
  3. more than half the team must be adults.
Edexcel FP2 2020 June Q2
9 marks Challenging +1.2
  1. Solve the recurrence system
$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$
Edexcel FP2 2020 June Q3
10 marks Standard +0.3
3. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & - 2 \\ 2 & - 4 & 1 \\ 1 & 2 & 3 \end{array} \right)$$ where \(k\) is a constant.
  1. Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that \(\operatorname { det } \mathbf { M } = 5\)
    1. find the value of \(k\)
    2. use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).
Edexcel FP2 2020 June Q4
10 marks Challenging +1.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_213_684_257_221} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_193_736_258_1053} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross-section with vertical ends and its length is 30 cm . A side profile of the speed bump is shown in Figure 2. The curve \(C\) shown in Figure 2 is modelled by the polar equation $$r = 30 \left( 1 - \theta ^ { 2 } \right) \quad 0 \leqslant \theta \leqslant 1$$ The units for \(r\) are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in \(\mathrm { cm } ^ { 2 }\), the area that is to be painted, according to the model.
Edexcel FP2 2020 June Q5
10 marks Challenging +1.2
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).
  1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
  2. Hence find
    1. a Cartesian equation for \(C\),
    2. the centre and radius of \(C\).