Questions — Edexcel FP2 (291 questions)

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Edexcel FP2 2008 June Q7
7. (a) Show that the substitution \(y = v x\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { y } + \frac { 3 y } { x } , \quad x > 0 , \quad y > 0$$ into the differential equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 v + \frac { 1 } { v } .$$ (b) By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = \mathrm { f } ( x )\). Given that \(y = 3\) at \(x = 1\),
(c)find the particular solution of differential equation (I).(2)
Edexcel FP2 2008 June Q8
8. The curve \(C\) shown in the diagram above has polar equation $$r = 4 ( 1 - \cos \theta ) , 0 \leq \theta \leq \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
  1. Show that \(P\) has polar coordinates \(\left( 2 , \frac { \pi } { 3 } \right)\). The curve \(C\) meets the line \(\theta = \frac { \pi } { 2 }\) at the point \(A\). The tangent to \(C\) at the initial line at the point \(N\). The finite region \(R\), shown shaded in
    \includegraphics[max width=\textwidth, alt={}]{863ef52d-ae75-450c-9eab-8102804868f5-2_737_561_1395_1329} the diagram above, is bounded by the initial line, the line \(\theta = \frac { \pi } { 2 }\), the arc \(A P\) of \(C\) and the line \(P N\).
  2. Calculate the exact area of \(R\).
Edexcel FP2 2008 June Q9
9. $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } + ( 1 - 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
  1. By differentiating equation (I) with respect to \(x\), show that $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = ( 1 - 4 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 4 y - 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\),
  2. find the series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x _ { 3 }\).(4)
  3. Use your series to estimate the value of \(y\) at \(x = - 0.5\), giving your answer to two decimal places.(1)
Edexcel FP2 2008 June Q10
10. The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$| z - 3 | = 2 | z |$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle.(5) The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$| z + 3 | = | z - i \sqrt { } 3 |$$
  2. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
  3. On your diagram shade the region which satisfies $$| z - 3 | \geq 2 | z | \text { and } | z + 3 | \geq | z - i \sqrt { } 3 |$$
Edexcel FP2 2008 June Q11
  1. De Moivre's theorem states that \(\quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta\) for \(n \in \Re\)
    1. Use induction to prove de Moivre's theorem for \(n \in \mathbb { Z } ^ { + }\).
    2. Show that \(\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta\)
    3. Hence show that \(2 \cos \frac { \pi } { 10 }\) is a root of the equation
    $$x ^ { 4 } - 5 x ^ { 2 } + 5 = 0$$
Edexcel FP2 2019 June Q1
  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
  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
  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
    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,
    3. Alan, Enid, Felix and Graham sit together,
    4. Brenda sits at one end of the row and Graham sits at the other end of the row,
    5. Charles and Devindra do not sit together.
Edexcel FP2 2019 June Q5
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
    1. A binary operation * is defined on positive real numbers by
$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) 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
  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
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
  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
  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
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
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
  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 \(C\).
Edexcel FP2 2020 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-20_371_328_255_870} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).
  1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6
    3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
  2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
  3. Explain why each of the following statements is false, making your reasoning clear.
    1. \(G\) has a subgroup of order 4
    2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
  4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
    \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
    \section*{Diagram 1} \section*{Spare copy of Diagram 1}
    \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
    Only use this diagram if you need to redraw your answer to part (d).
Edexcel FP2 2020 June Q7
7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$
  1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
  2. Find \(I _ { 2 }\)
Edexcel FP2 2020 June Q8
  1. The four digit number \(n = a b c d\) satisfies the following properties:
    (1) \(n \equiv 3 ( \bmod 7 )\)
    (2) \(n\) is divisible by 9
    (3) the first two digits have the same sum as the last two digits
    (4) the digit \(b\) is smaller than any other digit
    (5) the digit \(c\) is even
    1. Use property (1) to explain why \(6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )\)
    2. Use properties (2), (3) and (4) to show that \(a + b = 9\)
    3. Deduce that \(c \equiv 5 ( a - 1 ) ( \bmod 7 )\)
    4. Hence determine the number \(n\), verifying that it is unique. You must make your reasoning clear.
Edexcel FP2 2021 June Q1
  1. In this question you must show detailed reasoning.
Without performing any division, explain why \(n = 20210520\) is divisible by 66
Edexcel FP2 2021 June Q2
  1. A binary operation ★ on the set of non-negative integers, \(\mathbb { Z } _ { 0 } ^ { + }\), is defined by
$$m \star n = | m - n | \quad m , n \in \mathbb { Z } _ { 0 } ^ { + }$$
  1. Explain why \(\mathbb { Z } _ { 0 } ^ { + }\)is closed under the operation
  2. Show that 0 is an identity for \(\left( \mathbb { Z } _ { 0 } ^ { + } , \star \right)\)
  3. Show that all elements of \(\mathbb { Z } _ { 0 } ^ { + }\)have an inverse under ★
  4. Determine if \(\mathbb { Z } _ { 0 } ^ { + }\)forms a group under ★, giving clear justification for your answer.
Edexcel FP2 2021 June Q3
  1. (a) Use the Euclidean Algorithm to find integers \(a\) and \(b\) such that
$$125 a + 87 b = 1$$ (b) Hence write down a multiplicative inverse of 87 modulo 125
(c) Solve the linear congruence $$87 x \equiv 16 ( \bmod 125 )$$
Edexcel FP2 2021 June Q4
  1. Let \(G\) be a group of order \(46 ^ { 46 } + 47 ^ { 47 }\)
Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of \(G\)
  1. 11
  2. 21