Questions Paper 2 (402 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
Edexcel Paper 2 2020 October Q2
  1. Relative to a fixed origin, points \(P , Q\) and \(R\) have position vectors \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) respectively.
Given that
  • \(\quad P , Q\) and \(R\) lie on a straight line
  • \(Q\) lies one third of the way from \(P\) to \(R\)
    show that
$$\mathbf { q } = \frac { 1 } { 3 } ( \mathbf { r } + 2 \mathbf { p } )$$
Edexcel Paper 2 2020 October Q3
  1. (a) Given that
$$2 \log ( 4 - x ) = \log ( x + 8 )$$ show that $$x ^ { 2 } - 9 x + 8 = 0$$ (b) (i) Write down the roots of the equation $$x ^ { 2 } - 9 x + 8 = 0$$ (ii) State which of the roots in (b)(i) is not a solution of $$2 \log ( 4 - x ) = \log ( x + 8 )$$ giving a reason for your answer.
Edexcel Paper 2 2020 October Q4
  1. In the binomial expansion of
    \(( a + 2 x ) ^ { 7 } \quad\) where \(a\) is a constant
    the coefficient of \(x ^ { 4 }\) is 15120
    Find the value of \(a\).
Edexcel Paper 2 2020 October Q5
  1. The curve with equation \(y = 3 \times 2 ^ { x }\) meets the curve with equation \(y = 15 - 2 ^ { x + 1 }\) at the point \(P\). Find, using algebra, the exact \(x\) coordinate of \(P\).
Edexcel Paper 2 2020 October Q6
  1. (a) Given that
$$\frac { x ^ { 2 } + 8 x - 3 } { x + 2 } \equiv A x + B + \frac { C } { x + 2 } \quad x \in \mathbb { R } \quad x \neq - 2$$ find the values of the constants \(A , B\) and \(C\)
(b) Hence, using algebraic integration, find the exact value of $$\int _ { 0 } ^ { 6 } \frac { x ^ { 2 } + 8 x - 3 } { x + 2 } d x$$ giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers to be found.
Edexcel Paper 2 2020 October Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-16_621_799_246_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 4 x ^ { 2 } + x } { 2 \sqrt { x } } - 4 \ln x \quad x > 0$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 x ^ { 2 } + x - 16 \sqrt { x } } { 4 x \sqrt { x } }$$ The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).
  2. Show that the \(x\) coordinate of \(P\) is a solution of $$x = \left( \frac { 4 } { 3 } - \frac { \sqrt { x } } { 12 } \right) ^ { \frac { 2 } { 3 } }$$
  3. Use the iteration formula $$x _ { n + 1 } = \left( \frac { 4 } { 3 } - \frac { \sqrt { x _ { n } } } { 12 } \right) ^ { \frac { 2 } { 3 } } \quad \text { with } x _ { 1 } = 2$$ to find (i) the value of \(x _ { 2 }\) to 5 decimal places,
    (ii) the \(x\) coordinate of \(P\) to 5 decimal places.
Edexcel Paper 2 2020 October Q8
  1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\)
Given that
  • \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + a x - 23\) where \(a\) is a constant
  • the \(y\) intercept of \(C\) is - 12
  • ( \(x + 4\) ) is a factor of \(\mathrm { f } ( x )\)
    find, in simplest form, \(\mathrm { f } ( x )\)
Edexcel Paper 2 2020 October Q9
  1. A quantity of ethanol was heated until it reached boiling point.
The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that
  • the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
  • after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
    1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.
Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)
  • Use this information to evaluate the model.
    •
    Edexcel Paper 2 2020 October Q10
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Show that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$
    2. Hence solve, for \(- 90 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), the equation $$1 - \cos 3 x = \sin ^ { 2 } x$$
    Edexcel Paper 2 2020 October Q11
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-30_677_817_251_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.
    1. Find the coordinates of \(P\).
    2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
      Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
    3. find the range of possible values of \(a\), writing your answer in set notation.
    Edexcel Paper 2 2020 October Q12
    12. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_396_515_251_772} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The curve shown in Figure 3 has parametric equations $$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
      1. Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
      2. Hence show, by algebraic integration, that the area of \(R\) is exactly 20 \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742} \captionsetup{labelformat=empty} \caption{Figure 4}
        \end{figure} Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that
        • \(x\) and \(y\) are in metres
    2. the vertical wall of the dam is 4.2 metres high
    3. there is a horizontal walkway of width \(M N\) along the top of the dam
    4. calculate the width of the walkway.
    Edexcel Paper 2 2020 October Q13
    1. The function \(g\) is defined by
    $$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$ where \(k\) is a constant.
    1. Deduce the value of \(k\).
    2. Prove that $$\mathrm { g } ^ { \prime } ( x ) > 0$$ for all values of \(x\) in the domain of g .
    3. Find the range of values of \(a\) for which $$g ( a ) > 0$$
    Edexcel Paper 2 2020 October Q14
    1. A circle \(C\) with radius \(r\)
    • lies only in the 1st quadrant
    • touches the \(x\)-axis and touches the \(y\)-axis
    The line \(l\) has equation \(2 x + y = 12\)
    1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
    2. find the two possible values of \(r\), giving your answers as fully simplified surds.
    Edexcel Paper 2 2020 October Q15
    1. In this question you must show all stages of your working.
    \section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio \(r\) and first term \(a\).
    Given \(r \neq 1\) and \(a \neq 0\)
    1. prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that \(S _ { 10 }\) is four times \(S _ { 5 }\)
    2. find the exact value of \(r\).
    Edexcel Paper 2 2020 October Q16
    1. Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3
    Edexcel Paper 2 2021 October Q1
    1. In an arithmetic series
    • the first term is 16
    • the 21 st term is 24
      1. Find the common difference of the series.
      2. Hence find the sum of the first 500 terms of the series.
    Edexcel Paper 2 2021 October Q2
    1. The functions f and g are defined by
    $$\begin{aligned} & f ( x ) = 7 - 2 x ^ { 2 } \quad x \in \mathbb { R }
    & \operatorname { g } ( x ) = \frac { 3 x } { 5 x - 1 } \quad x \in \mathbb { R } \quad x \neq \frac { 1 } { 5 } \end{aligned}$$
    1. State the range of f
    2. Find gf (1.8)
    3. Find \(\mathrm { g } ^ { - 1 } ( x )\)
    Edexcel Paper 2 2021 October Q3
    1. Using the laws of logarithms, solve the equation
    $$\log _ { 3 } ( 12 y + 5 ) - \log _ { 3 } ( 1 - 3 y ) = 2$$
    Edexcel Paper 2 2021 October Q4
    1. Given that \(\theta\) is small and measured in radians, use the small angle approximations to show that
    $$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
    Edexcel Paper 2 2021 October Q5
    1. The curve \(C\) has equation
    $$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$
    1. Find
      1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
      2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
      1. Verify that \(C\) has a stationary point at \(x = 1\)
      2. Show that this stationary point is a point of inflection, giving reasons for your answer.
    Edexcel Paper 2 2021 October Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-12_487_784_292_644} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
    In the design
    • \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
    • sector \(O F E\) is congruent to sector \(O A B\)
    • \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
    • \(A O F\) is a straight line
    Given that the size of angle \(C O D\) is \(\theta\) radians,
    1. write down, in terms of \(\theta\), the size of angle \(A O B\)
    2. Show that the area of the logo is $$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
    3. Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).
    Edexcel Paper 2 2021 October Q7
    1. In this question you should show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-16_805_1041_388_511} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\)
    The line \(l\) is the tangent to \(C\) at \(P\)
    1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
    2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
    3. Use algebraic integration to find the exact area of \(R\).
    Edexcel Paper 2 2021 October Q8
    1. The curve \(C\) has equation
    $$p x ^ { 3 } + q x y + 3 y ^ { 2 } = 26$$ where \(p\) and \(q\) are constants.
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a p x ^ { 2 } + b q y } { q x + c y }$$ where \(a\), \(b\) and \(c\) are integers to be found. Given that
      • the point \(P ( - 1 , - 4 )\) lies on \(C\)
      • the normal to \(C\) at \(P\) has equation \(19 x + 26 y + 123 = 0\)
      • find the value of \(p\) and the value of \(q\).
    Edexcel Paper 2 2021 October Q9
    1. Show that
    $$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$
    VI4V SIHI NI SIIIM ION OCVIAV SIHI NI III IM I ON OCVJ4V SIHI NI IMIMM ION OC
    Edexcel Paper 2 2021 October Q10
    1. The time, \(T\) seconds, that a pendulum takes to complete one swing is modelled by the formula
    $$T = a l ^ { b }$$ where \(l\) metres is the length of the pendulum and \(a\) and \(b\) are constants.
    1. Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-26_581_888_749_625} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} A student carried out an experiment to find the values of the constants \(a\) and \(b\).
      The student recorded the value of \(T\) for different values of \(l\).
      Figure 3 shows the linear relationship between \(\log _ { 10 } l\) and \(\log _ { 10 } T\) for the student's data.
      The straight line passes through the points \(( - 0.7,0 )\) and \(( 0.21,0.45 )\)
      Using this information,
    2. find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of \(a\) and the value of \(b\), each to 3 significant figures.
    3. With reference to the model, interpret the value of the constant \(a\).