Questions P2 (867 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 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.
Edexcel P2 2023 June Q7
8 marks Standard +0.3
  1. The height of a river above a fixed point on the riverbed was monitored over a 7-day period.
The height of the river, \(H\) metres, \(t\) days after monitoring began, was given by $$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$ Given that \(H\) has a stationary value at \(t = \alpha\)
  1. use calculus to show that \(\alpha\) satisfies the equation $$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
  2. Hence find the value of \(\alpha\), giving your answer to 3 decimal places.
  3. Use further calculus to prove that \(H\) is a maximum at this value of \(\alpha\).
Edexcel P2 2023 June Q8
5 marks Moderate -0.8
  1. (i) A student writes the following statement:
    "When \(a\) and \(b\) are consecutive prime numbers, \(a ^ { 2 } + b ^ { 2 }\) is never a multiple of 10 "
    Prove by counter example that this statement is not true.
    (ii) Given that \(x\) and \(y\) are even integers greater than 0 and less than 6 , prove by exhaustion, that
$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$
Edexcel P2 2023 June Q9
7 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.
  1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
  2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.
Edexcel P2 2023 June Q10
10 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$ where \(k\) is a positive constant.
  1. Show that $$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$ where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0e3b364c-151b-471d-acb6-01afb018fb75-26_645_670_904_699} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) and the line \(l\).
    Given that \(l\) intersects \(C\) at the point \(A ( 1,9 )\) and at the point \(B ( 16 , q )\) where \(q\) is a constant,
  2. show that \(k = 4\) The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and \(l\) Using the answers to parts (a) and (b),
  3. find the area of region \(R\)
Edexcel P2 2023 June Q11
8 marks Standard +0.3
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n } \\ u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.
  1. Find, in terms of \(a\) and \(b\),
    1. \(u _ { 2 }\)
    2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
  2. \(b = a + 9\)
  3. show that
    4. $$a ^ { 2 } - 5 a - 66 = 0$$
  4. Hence find the larger possible value of \(u _ { 2 }\)
Edexcel P2 2024 June Q1
5 marks Moderate -0.3
  1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.
Edexcel P2 2024 June Q2
7 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.
In an arithmetic series,
  • the sixth term is 2
  • the sum of the first ten terms is - 80
For this series,
  1. find the value of the first term and the value of the common difference.
  2. Hence find the smallest value of \(n\) for which $$S _ { n } > 8000$$
Edexcel P2 2024 June Q3
6 marks Moderate -0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Using the laws of logarithms, solve
    $$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
  2. Find the value of $$\log _ { \sqrt { a } } a ^ { 6 }$$ where \(a\) is a positive constant greater than 1
Edexcel P2 2024 June Q4
8 marks Moderate -0.8
4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where \(k\) is a constant.
  1. State the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) Given that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\)
  2. show that \(k = 11\)
  3. Hence
    1. fully factorise \(\mathrm { f } ( x )\),
    2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.
Edexcel P2 2024 June Q5
6 marks Standard +0.3
  1. In this question you must show detailed reasoning.
    1. Given that \(x\) and \(y\) are positive numbers such that
    $$( x - y ) ^ { 3 } > x ^ { 3 } - y ^ { 3 }$$ prove that $$y > x$$
  2. Using a counter example, show that the result in part (a) is not true for all real numbers.
Edexcel P2 2024 June Q6
9 marks Standard +0.3
  1. (a) Sketch the curve with equation
$$y = a ^ { x } + 4$$ where \(a\) is a positive constant greater than 1
On your sketch, show
  • the coordinates of the point of intersection of the curve with the \(y\)-axis
  • the equation of the asymptote of the curve
\(x\)22.32.62.93.23.5
\(y\)00.32460.86291.66432.78964.3137
The table shows corresponding values of \(x\) and \(y\) for $$y = 2 ^ { x } - 2 x$$ with the values of \(y\) given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x\), giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate
  1. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x\)
  2. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x\)
Edexcel P2 2024 June Q7
6 marks Moderate -0.3
  1. The circle \(C _ { 1 }\) has equation
$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$
    1. Find the coordinates of the centre of \(C _ { 1 }\)
    2. Find the exact value of the radius of \(C _ { 1 }\) In part (b) you must show detailed reasoning.
      The circle \(C _ { 2 }\) has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
  2. Prove that the circles \(C _ { 1 }\) and \(C _ { 2 }\) neither touch nor intersect.
Edexcel P2 2024 June Q8
12 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.
    1. Solve, for \(0 < x \leqslant \pi\), the equation
    $$5 \sin x \tan x + 13 = \cos x$$ giving your answer in radians to 3 significant figures.
  2. The temperature inside a greenhouse is monitored on one particular day. The temperature, \(H ^ { \circ } \mathrm { C }\), inside the greenhouse, \(t\) hours after midnight, is modelled by the equation $$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(k\) is a constant.
    Use the equation of the model to answer parts (a) to (c).
    Given that
    • the temperature inside the greenhouse was \(20 ^ { \circ } \mathrm { C }\) at 6 am
    • \(0 < k < 20\) (a) find all possible values for \(k\), giving each answer to 2 decimal places.
    Given further that \(0 < k < 10\) (b) find the maximum temperature inside the greenhouse,
    (c) find the time of day at which this maximum temperature occurs. Give your answer to the nearest minute.
Edexcel P2 2024 June Q9
8 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b822842d-ee62-40ce-a8de-967e556a80a8-26_915_912_255_580} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).
  1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
    The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
  2. find, using calculus, the exact value of \(k\).