Questions — Edexcel (9671 questions)

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Edexcel FP2 AS 2019 June Q3
10 marks Standard +0.8
  1. A curve \(C\) in the complex plane is described by the equation
$$| z - 1 - 8 i | = 3 | z - 1 |$$
  1. Show that \(C\) is a circle, and find its centre and radius.
  2. Using the answer to part (a), determine whether \(z = 3 - 3 \mathrm { i }\) satisfies the inequality $$| z - 1 - 8 i | \geqslant 3 | z - 1 |$$
  3. Shade, on an Argand diagram, the set of points that satisfies both $$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$
Edexcel FP2 AS 2019 June Q4
7 marks Standard +0.3
  1. The set \(\{ e , p , q , r , s \}\) forms a group, \(A\), under the operation *
Given that \(e\) is the identity element and that $$p ^ { * } p = s \quad s ^ { * } s = r \quad p ^ { * } p ^ { * } p = q$$
  1. show that
    1. \(p ^ { * } q = r\)
    2. \(s ^ { * } p = q\)
  2. Hence complete the Cayley table below.
    *\(e\)\(\boldsymbol { p }\)\(\boldsymbol { q }\)\(r\)\(s\)
    \(e\)
    \(\boldsymbol { p }\)
    \(\boldsymbol { q }\)
    \(\boldsymbol { r }\)
    \(S\)
    A spare table can be found on page 11 if you need to rewrite your Cayley table.
  3. Use your table to find \(p ^ { * } q ^ { * } r ^ { * } s\) A student states that there is a subgroup of \(A\) of order 3
  4. Comment on the validity of this statement, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{989d779e-c40a-4658-ad98-17a37ab1d9e1-11_2464_74_304_36}
    Only use this grid if you need to rewrite the Cayley table.
    *\(e\)\(\boldsymbol { p }\)\(\boldsymbol { q }\)\(r\)\(s\)
    \(e\)
    \(\boldsymbol { p }\)
    \(\boldsymbol { q }\)
    \(\boldsymbol { r }\)
    \(S\)
Edexcel FP2 AS 2019 June Q5
11 marks Standard +0.3
  1. On Jim's 11 th birthday his parents invest \(\pounds 1000\) for him in a savings account.
The account earns 2\% interest each year.
On each subsequent birthday, Jim's parents add another \(\pounds 500\) to this savings account.
Let \(U _ { n }\) be the amount of money that Jim has in his savings account \(n\) years after his 11th birthday, once the interest for the previous year has been paid and the \(\pounds 500\) has been added.
  1. Explain, in the context of the problem, why the amount of money that Jim has in his savings account can be modelled by the recurrence relation of the form $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
  2. State an assumption that must be made for this model to be valid.
  3. Solve the recurrence relation $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ Jim hopes to be able to buy a car on his 18th birthday.
  4. Use the answer to part (c) to find out whether Jim will have enough money in his savings account to buy a car that costs \(\pounds 4500\)
Edexcel FP2 AS 2020 June Q1
8 marks Standard +0.8
  1. The set \(G = \{ 1,3,7,9,11,13,17,19 \}\) under the binary operation of multiplication modulo 20 forms a group.
    1. Find the inverse of each element of \(G\).
    2. Find the order of each element of \(G\).
    3. Find a subgroup of \(G\) of order 4
    4. Explain how the subgroup you found in part (c) satisfies Lagrange's theorem.
Edexcel FP2 AS 2020 June Q2
6 marks Standard +0.3
  1. The highest common factor of 963 and 657 is \(c\).
    1. Use the Euclidean algorithm to find the value of \(c\).
    2. Hence find integers \(a\) and \(b\) such that
    $$963 a + 657 b = c$$
Edexcel FP2 AS 2020 June Q3
10 marks Standard +0.3
$$A = \left( \begin{array} { r r } 1 & - 2 \\ 1 & 4 \end{array} \right)$$
  1. Show that the characteristic equation for \(\mathbf { A }\) is \(\lambda ^ { 2 } - 5 \lambda + 6 = 0\)
  2. Use the Cayley-Hamilton theorem to find integers \(p\) and \(q\) such that $$\mathbf { A } ^ { 3 } = p \mathbf { A } + q \mathbf { I }$$ (ii) Given that the \(2 \times 2\) matrix \(\mathbf { M }\) has eigenvalues \(- 1 + \mathrm { i }\) and \(- 1 - \mathrm { i }\), with eigenvectors \(\binom { 1 } { 2 - \mathrm { i } }\) and \(\binom { 1 } { 2 + \mathrm { i } }\) respectively, find the matrix \(\mathbf { M }\).
Edexcel FP2 AS 2020 June Q4
10 marks Standard +0.3
  1. Sam borrows \(\pounds 10000\) from a bank to pay for an extension to his house.
The bank charges \(5 \%\) annual interest on the portion of the loan yet to be repaid. Immediately after the interest has been added at the end of each year and before the start of the next year, Sam pays the bank a fixed amount, \(\pounds F\). Given that \(\pounds A _ { n }\) (where \(A _ { n } \geqslant 0\) ) is the amount owed at the start of year \(n\),
  1. write down an expression for \(A _ { n + 1 }\) in terms of \(A _ { n }\) and \(F\),
  2. prove, by induction that, for \(n \geqslant 1\) $$A _ { n } = ( 10000 - 20 F ) 1.05 ^ { n - 1 } + 20 F$$
  3. Find the smallest value of \(F\) for which Sam can repay all of the loan by the start of year 16 .
Edexcel FP2 AS 2020 June Q5
6 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d0194d2-7958-4699-9c5c-02e815ac433c-18_510_714_251_689} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.
  1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
  2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.
Edexcel FP2 AS 2022 June Q1
4 marks Standard +0.8
  1. Sketch on an Argand diagram the region defined by
$$z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg ( z + 2 ) < \frac { \pi } { 4 } \cap \{ z \in \mathbb { C } : - 1 < \operatorname { Re } ( z ) \leqslant 1 \}$$ On your sketch
  • shade the part of the diagram that is included in the region
  • use solid lines to show the parts of the boundary that are included in the region
  • use dashed lines to show the parts of the boundary that are not included in the region
Edexcel FP2 AS 2022 June Q2
7 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
$$\mathbf { M } = \left( \begin{array} { r r } 4 & 2 \\ 3 & - 1 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$$
Edexcel FP2 AS 2022 June Q3
9 marks Challenging +1.2
  1. (i) Let \(G\) be a group of order 5291848
Without performing any division, use proof by contradiction to show that \(G\) cannot have a subgroup of order 11
(ii) (a) Complete the following Cayley table for the set \(X = \{ 2,4,8,14,16,22,26,28 \}\) with the operation of multiplication modulo 30
\(\times _ { 30 }\)2481416222628
24816282142226
4822814
8162814
1428221684
16241416
2214264216
26221448
282614288
A copy of this table is given on page 11 if you need to rewrite your Cayley table.
(b) Hence determine whether the set \(X\) with the operation of multiplication modulo 30 forms a group.
[0pt] [You may assume multiplication modulo \(n\) is an associative operation.] Only use this grid if you need to rewrite your Cayley table.
\(\times _ { 30 }\)2481416222628
24816282142226
4822814
8162814
1428221684
16241416
2214264216
26221448
282614288
(Total for Question 3 is 9 marks)
Edexcel FP2 AS 2022 June Q4
11 marks Standard +0.8
4. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. (a) Use the Euclidean algorithm to find the highest common factor \(h\) of 416 and 72
    (b) Hence determine integers \(a\) and \(b\) such that $$416 a + 72 b = h$$ (c) Determine the value \(c\) in the set \(\{ 0,1,2 \ldots , 415 \}\) such that $$23 \times 72 \equiv c ( \bmod 416 )$$
  2. Evaluate \(5 ^ { 10 } ( \bmod 13 )\) giving your answer as the smallest positive integer solution.
Edexcel FP2 AS 2022 June Q5
9 marks Standard +0.3
  1. A person takes a course of a particular vitamin.
Before the course there was none of the vitamin in the person's body.
During the course, vitamin tablets are taken at the same time each day.
Initially two tablets are taken and on each following day only one tablet is taken.
Each tablet contains 10 mg of the vitamin.
Between doses the amount of the vitamin in the person's body decreases naturally by 60\% Let \(u _ { n } \mathrm { mg }\) be the amount of the vitamin in the person's body immediately after a tablet is taken, \(n\) days after the initial two tablets were taken.
  1. Explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { 0 } = 20 \quad u _ { n + 1 } = 0.4 u _ { n } + 10$$ The general solution to this recurrence relation has the form \(u _ { n } = a ( 0.4 ) ^ { n } + b\)
  2. Determine the value of \(a\) and the value of \(b\). The course is only effective if the amount of the vitamin in the person's body remains above 6 mg at all times throughout the course.
  3. Determine whether this course of the vitamin will be effective for this person, giving a reason for your answer.
Edexcel FP2 AS 2023 June Q1
8 marks Standard +0.3
  1. The operation * is defined on the set \(G = \{ 0,1,2,3 \}\) by
$$x ^ { * } y \equiv x + y - 2 x y ( \bmod 4 )$$
  1. Complete the Cayley table below.
    *0123
    0
    1
    2
    3
  2. Show that \(G\) is a group under the operation *
    (You may assume the associative law is satisfied.)
  3. State the order of each element of \(G\).
  4. State whether \(G\) is a cyclic group, giving a reason for your answer.
Edexcel FP2 AS 2023 June Q2
8 marks Standard +0.3
  1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix
$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,
  1. determine
    1. the value of \(k\)
    2. the eigenvalue.
  2. Hence determine a Cartesian equation of the invariant line under \(T\).
Edexcel FP2 AS 2023 June Q3
7 marks Challenging +1.2
  1. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.
Given that $$\arg \left( \frac { z - 4 - i } { z - 2 - 7 i } \right) = \frac { \pi } { 2 }$$
  1. sketch the locus of \(P\) as \(z\) varies,
  2. determine the exact maximum possible value of \(| z |\)
Edexcel FP2 AS 2023 June Q4
9 marks Standard +0.3
  1. A student takes out a loan for \(\pounds 1000\) from a bank.
The bank charges \(0.5 \%\) monthly interest on the amount of the loan yet to be repaid.
At the end of each month
  • the interest is added to the loan
  • the student then repays \(\pounds 50\)
Let \(U _ { n }\) be the amount of money owed \(n\) months after the loan was taken out.
The amount of money owed by the student is modelled by the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ where \(A\) is a constant.
    1. State the value of the constant \(A\).
    2. Explain, in the context of the problem, the value 1.005 Using the value of \(A\) found in part (a)(i),
  2. solve the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
  3. Hence determine, according to the model, the number of months it will take to completely repay the loan.
Edexcel FP2 AS 2023 June Q5
8 marks Standard +0.3
    1. Making your reasoning clear and using modulo arithmetic, show that
$$214 ^ { 6 } \text { is divisible by } 8$$ (ii) The following 7-digit number has four unknown digits $$a 5 \square b \square a b 0$$ Given that the number is divisible by 11
  1. determine the value of the digit \(a\). Given that the number is also divisible by 3
  2. determine the possible values of the digit \(b\).
Edexcel FP2 AS 2024 June Q1
9 marks Standard +0.3
    1. The table below is a Cayley table for the group \(G\) with operation ∘
。\(а\)\(b\)\(c\)\(d\)\(e\)\(f\)
\(a\)\(d\)c\(b\)\(a\)\(f\)\(e\)
\(b\)\(e\)\(f\)\(a\)\(b\)\(c\)\(d\)
\(c\)\(f\)\(e\)\(d\)\(c\)\(b\)\(a\)
\(d\)\(а\)\(b\)\(c\)\(d\)\(e\)\(f\)
\(e\)\(b\)\(а\)\(f\)\(e\)\(d\)\(c\)
\(f\)c\(d\)\(e\)\(f\)\(а\)\(b\)
  1. State which element is the identity of the group.
  2. Determine the inverse of the element ( \(b \circ c\) )
  3. Give a reason why the set \(\{ a , b , e , f \}\) cannot be a subgroup of \(G\). You must justify your answer.
  4. Show that the set \(\{ b , d , f \}\) is a subgroup of \(G\).
    (ii) Given that \(H\) is a group with an element \(x\) of order 3 and an element \(y\) of order 6 satisfying $$y x = x y ^ { 5 }$$ show that \(y ^ { 3 } x y ^ { 3 } x ^ { 2 }\) is the identity element.
    \includegraphics[max width=\textwidth, alt={}, center]{7d269bf1-f481-46bd-b9d3-fea211b186cf-02_2270_54_309_1980}
Edexcel FP2 AS 2024 June Q2
6 marks Standard +0.3
  1. Tiles are sold in boxes with 21 tiles in each box.
The tiles are laid out in \(x\) rows of 5 tiles and \(y\) rows of 6 tiles.
All the tiles from a box are used before the next box is opened.
When all the rows of tiles have been laid, there are \(n\) tiles left in the last opened box.
  1. Write down a congruence expression for \(n\) in the form $$a x + b y ( \bmod c )$$ where \(a\), \(b\) and \(c\) are integers. Given that
    • exactly 43 rows of tiles are laid
    • there are no tiles left in the last opened box
    • use your congruence expression to determine the minimum number of rows of 6 tiles laid.
Edexcel FP2 AS 2024 June Q3
7 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
$$\mathbf { A } = \left( \begin{array} { r r } 3 & k \\ - 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0 \\ 0 & - 3 \end{array} \right)$$
  1. show that \(k = - 6\)
  2. determine a suitable matrix \(\mathbf { P }\)
Edexcel FP2 AS 2024 June Q4
9 marks Challenging +1.8
  1. A circle \(C\) in the complex plane has equation
$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$ where \(\alpha\) is a real constant with \(\alpha > 1\)
Given that the imaginary axis is a tangent to \(C\)
  1. sketch, on an Argand diagram, the circle \(C\)
  2. explain why the value of \(\alpha\) is 3 The circle \(C\) is contained in the region $$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
  3. Determine the maximum value of \(\beta\) Give your answer in radians to 3 significant figures.
Edexcel FP2 AS 2024 June Q5
9 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d269bf1-f481-46bd-b9d3-fea211b186cf-14_317_1557_255_255} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the first three stages of a pattern that is created by a recursive process.
The process starts with a square and proceeds as follows
  • each square is replaced by 5 smaller squares each \(\frac { 1 } { 9 }\) th the size of the square being replaced
  • the 5 smaller squares are the ones in each corner and the one in the centre
  • once each of the squares has been replaced, the square immediately to the right and above the centre square of the pattern is then removed
Let \(u _ { n }\) be the number of squares in the pattern in stage \(n\), where stage 1 is the original square.
  1. Explain why \(u _ { n }\) satisfies the recurrence system $$u _ { 1 } = 1 \quad u _ { n + 1 } = 5 u _ { n } - 1 \quad ( n = 1,2,3 , \ldots )$$
  2. Solve this recurrence system. Given that the initial square has area 25
  3. determine the total area of all the squares in stage 8 of the pattern, giving your answer to 2 significant figures.
Edexcel FP2 AS Specimen Q1
5 marks Standard +0.3
  1. Given that
$$A = \left( \begin{array} { l l } 3 & 1 \\ 6 & 4 \end{array} \right)$$
  1. find the characteristic equation of the matrix \(\mathbf { A }\).
  2. Hence show that \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).
Edexcel FP2 AS Specimen Q2
6 marks Moderate -0.3
  1. (i) Without performing any division, explain why 8184 is divisible by 6
    (ii) Use the Euclidean algorithm to find integers \(a\) and \(b\) such that
$$27 a + 31 b = 1$$