Questions — Edexcel (10514 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
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. 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
  2. (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 $$214 ^ { 6 } \text { is divisible by } 8$$
  2. 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 ∘
    。\(a\)\(b\)\(c\)\(d\)\(e\)\(f\)
    \(a\)\(d\)c\(b\)\(a\)\(f\)\(e\)
    \(b\)\(e\)\(f\)\(a\)\(b\)\(c\)\(d\)
    \(c\)\(f\)\(e\)\(d\)\(c\)\(b\)\(a\)
    \(d\)\(a\)\(b\)\(c\)\(d\)\(e\)\(f\)
    \(e\)\(b\)\(a\)\(f\)\(e\)\(d\)\(c\)
    \(f\)c\(d\)\(e\)\(f\)\(a\)\(b\)
    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 \(G\).
    5. 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. Without performing any division, explain why 8184 is divisible by 6
  2. Use the Euclidean algorithm to find integers \(a\) and \(b\) such that $$27 a + 31 b = 1$$
Edexcel FP2 AS Specimen Q3
8 marks Standard +0.3
  1. A curve \(C\) is described by the equation
$$| z - 9 + 12 i | = 2 | z |$$
  1. Show that \(C\) is a circle, and find its centre and radius.
  2. Sketch \(C\) on an Argand diagram. Given that \(w\) lies on \(C\),
  3. find the largest value of \(a\) and the smallest value of \(b\) that must satisfy $$a \leqslant \operatorname { Re } ( w ) \leqslant b$$
Edexcel FP2 AS Specimen Q4
11 marks Standard +0.3
  1. The operation * is defined on the set \(S = \{ 0,2,3,4,5,6 \}\) by \(x ^ { * } y = x + y = x y ( \bmod 7 )\)
*023456
0
20
35
4
54
6
    1. Complete the Cayley table shown above
    2. Show that \(S\) is a group under the operation *
      (You may assume the associative law is satisfied.)
  2. Show that the element 4 has order 3
  3. Find an element which generates the group and express each of the elements in terms of this generator.
Edexcel FP2 AS Specimen Q5
10 marks Standard +0.3
  1. A population of deer on a large estate is assumed to increase by \(10 \%\) during each year due to natural causes.
The population is controlled by removing a constant number, \(Q\), of the deer from the estate at the end of each year. At the start of the first year there are 5000 deer on the estate.
Let \(P _ { n }\) be the population of deer at the end of year \(n\).
  1. Explain, in the context of the problem, the reason that the deer population is modelled by the recurrence relation $$P _ { n } = 1.1 P _ { n - 1 } - Q , \quad P _ { 0 } = 5000 , \quad n \in \mathbb { Z } ^ { + }$$
  2. Prove by induction that \(P _ { n } = ( 1.1 ) ^ { n } ( 5000 - 10 Q ) + 10 Q , \quad n \geqslant 0\)
  3. Explain how the long term behaviour of this population varies for different values of \(Q\).
Edexcel FS1 AS 2018 June Q1
10 marks Standard +0.3
  1. A researcher is investigating the distribution of orchids in a field. He believes that the Poisson distribution with a mean of 1.75 may be a good model for the number of orchids in each square metre. He randomly selects 150 non-overlapping areas, each of one square metre, and counts the number of orchids present in each square.
The results are recorded in the table below.
Number of orchids in
each square metre
0123456
Number of squares304235261160
He calculates the expected frequencies as follows
Number of orchids in
each square metre
012345More than 5
Number of squares26.0745.6239.9123.2810.193.57\(r\)
  1. Find the value of \(r\) giving your answer to 2 decimal places. The researcher will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution with mean 1.75
  2. State clearly the hypotheses required to test whether or not this Poisson distribution is a suitable model for these data. The test statistic for this test is 2.0 and the number of degrees of freedom to be used is 4
  3. Explain fully why there are 4 degrees of freedom.
  4. Stating your critical value clearly, determine whether or not these data support the researcher's belief. The researcher works in another field where the number of orchids in each square metre is known to have a Poisson distribution with mean 1.5 He randomly selects 200 non-overlapping areas, each of one square metre, in this second field, and counts the number of orchids present in each square.
  5. Using a Poisson approximation, show that the probability that he finds at least one square with exactly 6 orchids in it is 0.506 to 3 decimal places.
Edexcel FS1 AS 2018 June Q2
11 marks Standard +0.3
The number of heaters, \(H\), bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7
  1. Calculate \(\mathrm { P } ( H \geqslant 2 )\) The number of heaters, \(G\), bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where \(G\) and \(H\) are independent.
  2. Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places.
  3. Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days. December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14
  4. Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FS1 AS 2018 June Q3
12 marks Challenging +1.2
  1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4
The random variable \(B\) represents the score when the black die is rolled.
  1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\) The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by
    \(w\)1245\(c\)
    \(\mathrm { P } ( W = w )\)\(a + b\)\(a\)0.3\(a\)\(b\)
    Greg and Nilaya play a game with these dice.
    Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
    The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
  2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\) Given that \(\mathrm { E } ( X ) = 2.6\)
  3. find the exact value of \(\operatorname { Var } ( X )\)
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC