Questions — Edexcel (10514 questions)

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Edexcel Paper 2 2024 June Q5
3 marks Standard +0.3
  1. Given that \(\theta\) is small and in radians, use the small angle approximations to find an approximate numerical value of
$$\frac { \theta \tan 2 \theta } { 1 - \cos 3 \theta }$$
Edexcel Paper 2 2024 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-12_518_670_248_740} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curves with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) where $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 4 x ^ { 2 } - 1 } & x > 0 \\ \mathrm {~g} ( x ) = 8 \ln x & x > 0 \end{array}$$
  1. Find
    1. \(\mathrm { f } ^ { \prime } ( x )\)
    2. \(\mathrm { g } ^ { \prime } ( x )\) Given that \(\mathrm { f } ^ { \prime } ( x ) = \mathrm { g } ^ { \prime } ( x )\) at \(x = \alpha\)
  2. show that \(\alpha\) satisfies the equation $$4 x ^ { 2 } + 2 \ln x - 1 = 0$$ The iterative formula $$x _ { n + 1 } = \sqrt { \frac { 1 - 2 \ln x _ { n } } { 4 } }$$ is used with \(x _ { 1 } = 0.6\) to find an approximate value for \(\alpha\)
  3. Calculate, giving each answer to 4 decimal places,
    1. the value of \(x _ { 2 }\)
    2. the value of \(\alpha\)
Edexcel Paper 2 2024 June Q7
5 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-16_330_654_246_751} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the straight line \(l\).
Line \(l\) passes through the points \(A\) and \(B\).
Relative to a fixed origin \(O\)
  • the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
  • the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }\)
    1. Find \(\overrightarrow { A B }\)
Given that a point \(P\) lies on \(l\) such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$
  • find the possible position vectors of \(P\).
    •
    Edexcel Paper 2 2024 June Q8
    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. Prove that $$\frac { 1 } { \operatorname { cosec } \theta - 1 } + \frac { 1 } { \operatorname { cosec } \theta + 1 } \equiv 2 \tan \theta \sec \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$
    2. Hence solve, for \(0 < x < 90 ^ { \circ }\), the equation $$\frac { 1 } { \operatorname { cosec } 2 x - 1 } + \frac { 1 } { \operatorname { cosec } 2 x + 1 } = \cot 2 x \sec 2 x$$ Give each answer, in degrees, to one decimal place.
    Edexcel Paper 2 2024 June Q9
    7 marks Moderate -0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-22_595_1058_248_466} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The graph in Figure 3 shows the path of a small ball.
    The ball travels in a vertical plane above horizontal ground.
    The ball is thrown from the point represented by \(A\) and caught at the point represented by \(B\). The height, \(H\) metres, of the ball above the ground has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was thrown. With respect to a fixed origin \(O\), the point \(A\) has coordinates \(( 0,2 )\) and the point \(B\) has coordinates (20, 0.8), as shown in Figure 3. The ball reaches its maximum height when \(x = 9\) A quadratic function, linking \(H\) with \(x\), is used to model the path of the ball.
    1. Find \(H\) in terms of \(x\).
    2. Give one limitation of the model. Chandra is standing directly under the path of the ball at a point 16 m horizontally from \(O\). Chandra can catch the ball if the ball is less than 2.5 m above the ground.
    3. Use the model to determine if Chandra can catch the ball.
    Edexcel Paper 2 2024 June Q10
    6 marks Moderate -0.3
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-26_707_992_246_539} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = ( t + 3 ) ^ { 2 } \quad y = 1 - t ^ { 3 } \quad - 2 \leqslant t \leqslant 1$$ The point \(P\) with coordinates \(( 4,2 )\) lies on \(C\).
    1. Using parametric differentiation, show that the tangent to \(C\) at \(P\) has equation $$3 x + 4 y = 20$$ The curve \(C\) is used to model the profile of a slide at a water park.
      Units are in metres, with \(y\) being the height of the slide above water level.
    2. Find, according to the model, the greatest height of the slide above water level.
    Edexcel Paper 2 2024 June Q11
    5 marks Standard +0.8
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-28_668_743_251_662} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = 8 x ^ { 2 } \mathrm { e } ^ { - 3 x } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 5, is bounded by
    • the curve \(C\)
    • the line with equation \(x = 1\)
    • the \(x\)-axis
    Find the exact area of \(R\), giving your answer in the form $$A + B \mathrm { e } ^ { - 3 }$$ where \(A\) and \(B\) are rational numbers to be found.
    Edexcel Paper 2 2024 June Q12
    12 marks Standard +0.3
    1. Express \(\frac { 1 } { V ( 25 - V ) }\) in partial fractions. The volume, \(V\) microlitres, of a plant cell \(t\) hours after the plant is watered is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1 } { 10 } V ( 25 - V )$$ The plant cell has an initial volume of 20 microlitres.
    2. Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.
    3. Show that $$V = \frac { A } { \mathrm { e } ^ { - k t } + B }$$ where \(A , B\) and \(k\) are constants to be found. The model predicts that there is an upper limit, \(L\) microlitres, on the volume of the plant cell.
    4. Find the value of \(L\), giving a reason for your answer.
    Edexcel Paper 2 2024 June Q13
    9 marks Standard +0.3
    1. The world human population, \(P\) billions, is modelled by the equation
    $$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after 2004
    Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points lie approximately on a straight line with
    • gradient 0.0054
    • intercept 0.81 on the \(\log _ { 10 } P\) axis
      1. Estimate, to 3 decimal places, the value of \(a\) and the value of \(b\).
    In the context of the model,
    1. interpret the value of the constant \(a\),
    2. interpret the value of the constant \(b\).
  • Use the model to estimate the world human population in 2030
  • Comment on the reliability of the answer to part (c).
    •
    Edexcel Paper 2 2024 June Q14
    8 marks Standard +0.3
    1. The circle \(C _ { 1 }\) has equation
    $$x ^ { 2 } + y ^ { 2 } - 6 x + 14 y + 33 = 0$$
    1. Find
      1. the coordinates of the centre of \(C _ { 1 }\)
      2. the radius of \(C _ { 1 }\) A different circle \(C _ { 2 }\)
        Given that \(C _ { 1 }\) and \(C _ { 2 }\) intersect at 2 distinct points,
    2. find the range of values of \(k\), writing your answer in set notation.
    Edexcel Paper 2 2024 June Q15
    12 marks Challenging +1.2
    1. The curve \(C\) has equation
    $$( x + y ) ^ { 3 } = 3 x ^ { 2 } - 3 y - 2$$
    1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 1,0 )\) lies on \(C\).
    2. Show that the normal to \(C\) at \(P\) has equation $$y = - 2 x + 2$$
    3. Prove that the normal to \(C\) at \(P\) does not meet \(C\) again. You should use algebra for your proof and make your reasoning clear.
    Edexcel Paper 2 2020 October Q1
    5 marks Moderate -0.3
    1 The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { \frac { x } { 1 + x } }\) The values of \(y\) are given to 4 significant figures.
    \(x\)0.511.522.5
    \(y\)0.57740.70710.77460.81650.8452
    1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { x } { 1 + x } } \mathrm {~d} x$$ giving your answer to 3 significant figures.
    2. Using your answer to part (a), deduce an estimate for \(\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x\) Given that $$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x = 4.535 \text { to } 4 \text { significant figures }$$
    3. comment on the accuracy of your answer to part (b).
    Edexcel Paper 2 2020 October Q2
    3 marks Easy -1.2
    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
    5 marks Moderate -0.3
    1. Given that $$2 \log ( 4 - x ) = \log ( x + 8 )$$ show that $$x ^ { 2 } - 9 x + 8 = 0$$
      1. Write down the roots of the equation $$x ^ { 2 } - 9 x + 8 = 0$$
      2. 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
    3 marks Moderate -0.5
    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
    4 marks Standard +0.3
    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
    7 marks Standard +0.3
    1. 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 \(C\)
    2. 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
    10 marks Standard +0.3
    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
    6 marks Standard +0.3
    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
    6 marks Moderate -0.3
    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
    8 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 $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$
    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
    7 marks Standard +0.3
    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
    11 marks Standard +0.3
    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. \begin{enumerate}[label=(\alph*)] \item
    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
    Edexcel Paper 2 2020 October Q13
    6 marks Standard +0.8
    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
    7 marks Standard +0.8
    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.