Questions — Edexcel P2 (158 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 P2 2019 June Q1
4 marks Easy -1.3
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { n + 1 } & = 4 - a _ { n } \\ a _ { 1 } & = 3 \end{aligned}$$ Find the value of
    1. \(a _ { 2 }\)
    2. \(a _ { 107 }\)
  2. \(\sum _ { n = 1 } ^ { 200 } \left( 2 a _ { n } - 1 \right)\)
Edexcel P2 2019 June Q2
7 marks Moderate -0.3
2. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 10 y - 21 = 0$$ Find
    1. the coordinates of the centre of \(C\),
    2. the exact value of the radius of \(C\). The point \(P ( 5,4 )\) lies on \(C\).
  2. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
Edexcel P2 2019 June Q3
4 marks Moderate -0.8
3. (i) Use algebra to prove that for all real values of \(x\) $$( x - 4 ) ^ { 2 } \geqslant 2 x - 9$$ (ii) Show that the following statement is untrue. $$2 ^ { n } + 1 \text { is a prime number for all values of } n , n \in \mathbb { N }$$
Edexcel P2 2019 June Q4
7 marks Moderate -0.8
4. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 }$$ (b) Given that \(x\) is small, so terms in \(x ^ { 4 }\) and higher powers of \(x\) may be ignored, show $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 } + \left( 2 + \frac { 1 } { 4 } x \right) ^ { 6 } = a + b x ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.
Edexcel P2 2019 June Q5
8 marks Moderate -0.3
5. A company makes a particular type of watch. The annual profit made by the company from sales of these watches is modelled by the equation $$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$ where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch. According to this model,
  1. find, using calculus, the maximum possible annual profit.
  2. Justify, also using calculus, that the profit you have found is a maximum.
Edexcel P2 2019 June Q6
8 marks Standard +0.3
6. \(\mathrm { f } ( x ) = k x ^ { 3 } - 15 x ^ { 2 } - 32 x - 12\) where \(k\) is a constant Given ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(k = 9\)
  2. Using algebra and showing each step of your working, fully factorise \(\mathrm { f } ( x )\).
  3. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \cos ^ { 3 } \theta - 15 \cos ^ { 2 } \theta - 32 \cos \theta - 12 = 0$$ giving your answers to one decimal place.
Edexcel P2 2019 June Q7
9 marks Moderate -0.3
7. Kim starts working for a company.
  • In year 1 her annual salary will be \(\pounds 16200\)
  • In year 10 her annual salary is predicted to be \(\pounds 31500\)
Model \(A\) assumes that her annual salary will increase by the same amount each year.
  1. According to model \(A\), determine Kim's annual salary in year 2 . Model \(B\) assumes that her annual salary will increase by the same percentage each year.
  2. According to model \(B\), determine Kim's annual salary in year 2 . Give your answer to the nearest \(\pounds 10\)
  3. Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10
Edexcel P2 2019 June Q8
9 marks Moderate -0.3
8. (i) Find the exact solution of the equation $$8 ^ { 2 x + 1 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.
(ii) Using the laws of logarithms, solve $$\log _ { 5 } ( 7 - 2 y ) = 2 \log _ { 5 } ( y + 1 ) - 1$$
Edexcel P2 2019 June Q9
8 marks Standard +0.3
9. (a) Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written in the form $$5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0$$ (b) Hence solve, for \(0 \leqslant x < \frac { \pi } { 2 }\) $$\cos 2 x - 1 = 4 \sin 2 x \tan 2 x$$ giving your answers, where appropriate, to 2 decimal places.
Edexcel P2 2019 June Q10
11 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc9cd828-f9bc-4cad-8a70-4214697b1c6a-11_707_855_255_539} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 36 } { x ^ { 2 } } + 2 x - 13 \quad x > 0$$ Using calculus,
  1. find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing,
  2. show that \(\int _ { 2 } ^ { 9 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x = 0\) The point \(P ( 2,0 )\) and the point \(Q ( 6,0 )\) lie on \(C\).
    Given \(\int _ { 2 } ^ { 6 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x = - 8\)
    1. state the value of \(\int _ { 6 } ^ { 9 } \left( \frac { 36 } { x ^ { 2 } } + 2 x - 13 \right) \mathrm { d } x\)
    2. find the value of the constant \(k\) such that \(\int _ { 2 } ^ { 6 } \left( \frac { 36 } { x ^ { 2 } } + 2 x + k \right) \mathrm { d } x = 0\)
Edexcel P2 2021 June Q1
7 marks Easy -1.2
  1. Adina is saving money to buy a new computer. She saves \(\pounds 5\) in week \(1 , \pounds 5.25\) in week 2 , \(\pounds 5.50\) in week 3 and so on until she has enough money, in total, to buy the computer.
She decides to model her savings using either an arithmetic series or a geometric series.
Using the information given,
    1. state with a reason whether an arithmetic series or a geometric series should be used,
    2. write down an expression, in terms of \(n\), for the amount, in pounds ( \(\pounds\) ), saved in week \(n\). Given that the computer Adina wants to buy costs \(\pounds 350\)
  2. find the number of weeks it will take for Adina to save enough money to buy the computer.
    VIAV SIHI NI III IM ION OCVIIN SIHI NI III M M O N OOVIAV SIHI NI IIIIM I ION OC
Edexcel P2 2021 June Q2
8 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-04_1001_1481_267_221} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
  1. On Diagram 1, sketch the curve with equation
    1. \(y = 2 ^ { x }\)
    2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
  2. Using algebra, find the exact coordinates of \(P\).
    \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
    \section*{Diagram 1}
Edexcel P2 2021 June Q3
5 marks Moderate -0.3
3. (i) Prove that for all single digit prime numbers, \(p\), $$p ^ { 3 } + p \text { is a multiple of } 10$$ (ii) Show, using algebra, that for \(n \in \mathbb { N }\) $$( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is not a multiple of } 3$$
Edexcel P2 2021 June Q4
8 marks Moderate -0.3
  1. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of
$$\left( 2 + \frac { x } { 8 } \right) ^ { 13 }$$ fully simplifying each coefficient.
(b) Use the answer to part (a) to find an approximation for \(2.0125 ^ { 13 }\) Give your answer to 3 decimal places. Without calculating \(2.0125 { } ^ { 13 }\) (c) state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.
Edexcel P2 2021 June Q5
10 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-14_547_1084_269_420} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\) The curves intersect when \(x = 2.5\) and when \(x = 4\) A table of values for some points on the curve \(C _ { 1 }\) is shown below, with \(y\) values given to 3 decimal places as appropriate.
\(x\)2.52.7533.253.53.754
\(y\)5.4537.7649.3759.9649.3677.6265
Using the trapezium rule with all the values of \(y\) in the table,
  1. find, to 2 decimal places, an estimate for the area bounded by the curve \(C _ { 1 }\), the line with equation \(x = 2.5\), the \(x\)-axis and the line with equation \(x = 4\) The curve \(C _ { 2 }\) has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x + 9 \quad x > 0$$
  2. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x + 9 \right) \mathrm { d } x\) The region \(R\), shown shaded in Figure 2, is bounded by the curves \(C _ { 1 }\) and \(C _ { 2 }\)
  3. Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region \(R\).
    (3)
Edexcel P2 2021 June Q6
7 marks Standard +0.3
  1. A circle has equation
$$x ^ { 2 } - 6 x + y ^ { 2 } + 8 y + k = 0$$ where \(k\) is a positive constant. Given that the \(x\)-axis is a tangent to this circle,
  1. find the value of \(k\). The circle meets the coordinate axes at the points \(R , S\) and \(T\).
  2. Find the exact area of the triangle \(R S T\). \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-21_2647_1840_118_111}
Edexcel P2 2021 June Q7
10 marks Standard +0.3
7. (a) Given that $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$ show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 30 x + 56 = 0$$ (b) Show that - 4 is a root of this cubic equation.
(c) Hence, using algebra and showing each step of your working, solve $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$
Edexcel P2 2021 June Q8
10 marks Standard +0.3
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
  2. (a) Show that the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ can be written in the form $$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$ where \(a , b\) and \(c\) are constants to be found.
    (b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-27_2644_1840_118_111}
Edexcel P2 2021 June Q9
10 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-30_469_863_251_593} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of a square based, open top box.
The height of the box is \(h \mathrm {~cm}\), and the base edges each have length \(l \mathrm {~cm}\).
Given that the volume of the box is \(250000 \mathrm {~cm} ^ { 3 }\)
  1. show that the external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the box is given by $$S = \frac { 250000 } { h } + 2000 \sqrt { h }$$
  2. Use algebraic differentiation to show that \(S\) has a stationary point when \(h = 250 ^ { k }\) where \(k\) is a rational constant to be found.
  3. Justify by further differentiation that this value of \(h\) gives the minimum external surface area of the box.
    \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-32_2647_1838_118_116}
Edexcel P2 2023 June Q1
3 marks Easy -1.3
  1. The continuous curve \(C\) has equation \(y = \mathrm { f } ( x )\).
A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below.
\(x\)4.04.24.44.64.85.0
\(y\)9.28.45563.85125.03427.82978.6
Use the trapezium rule with all the values of \(y\) in the table to find an approximation for $$\int _ { 4 } ^ { 5 } f ( x ) d x$$ giving your answer to 3 decimal places.
Edexcel P2 2023 June Q2
6 marks Moderate -0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } + 5 x + a$$ where \(a\) is a constant.
Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { f } ( x )\),
  1. use the factor theorem to show that \(a = - 3\)
  2. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
Edexcel P2 2023 June Q3
7 marks Moderate -0.8
  1. A circle \(C\) has centre \(( 2,5 )\)
Given that the point \(P ( 8 , - 3 )\) lies on \(C\)
    1. find the radius of \(C\)
    2. find an equation for \(C\)
  2. Find the equation of the tangent to \(C\) at \(P\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
Edexcel P2 2023 June Q4
7 marks Moderate -0.3
  1. The binomial expansion, in ascending powers of \(x\), of
$$( 3 + p x ) ^ { 5 }$$ where \(p\) is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.
  1. Find the value of \(A\) Given that
    • \(B = 18 D\)
    • \(p < 0\)
    • find
      1. the value of \(p\)
      2. the value of \(C\)
Edexcel P2 2023 June Q5
5 marks Moderate -0.3
  1. Use the laws of logarithms to solve
$$\log _ { 2 } ( 16 x ) + \log _ { 2 } ( x + 1 ) = 3 + \log _ { 2 } ( x + 6 )$$
Edexcel P2 2023 June Q6
9 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where \(k\) is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,
  1. show that \(k ^ { 2 } - 7 k - 60 = 0\)
  2. predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the \(N\) th month.
  3. Calculate the value of \(N\) according to the model.