Questions — Edexcel PURE (39 questions)

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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
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Edexcel PURE 2024 October Q1
Moderate -0.8
  1. The line \(l _ { 1 }\) passes through the point \(A ( - 5,20 )\) and the point \(B ( 3 , - 4 )\).
    1. Find an equation for \(l _ { 1 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\)
  2. Find an equation for \(l _ { 2 }\) giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
Edexcel PURE 2024 October Q2
Easy -1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
  2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.
Edexcel PURE 2024 October Q3
Standard +0.3
  1. In this question you must show all stages of your working.
$$f ( x ) = \frac { ( x + 5 ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$
  1. Find \(\int f ( x ) d x\)
    1. Show that when \(\mathrm { f } ^ { \prime } ( x ) = 0\) $$3 x ^ { 2 } + 10 x - 25 = 0$$
    2. Hence state the value of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) = 0$$
Edexcel PURE 2024 October Q4
Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-10_812_853_255_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)
  • has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
  • cuts the \(x\)-axis at the origin and at \(x = 4\)
  • has a minimum turning point at ( \(2 , - 4.8\) )
    1. find \(\mathrm { f } ( x )\)
Given that \(C _ { 2 }\)
The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.
  • Use algebra to find the coordinates of \(P\).
    •
    Edexcel PURE 2024 October Q5
    Standard +0.8
    1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).
    Given
    • \(O A = O B = 5 \mathrm {~km}\)
    • angle \(A O B = 1.2\) radians
      1. find the perimeter of the plot of land.
        (2)
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-14_609_650_664_705} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$
  • show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)
  • Find the length of \(A P\), giving your answer to the nearest 100 m .
    •
    Edexcel PURE 2024 October Q6
    Standard +0.8
    1. In this question you must show all stages of your working.
    \section*{Solutions relying on calculator technology are not acceptable.}
    1. Sketch the curve \(C\) with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
      • the equation of the vertical asymptote
      • the coordinates of the intersection of \(C\) with the \(y\)-axis
      The straight line \(l\) has equation \(y = k x - 4\), where \(k\) is a constant.
      Given that \(l\) cuts \(C\) at least once,
      1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
      2. find the range of possible values for \(k\).
    Edexcel PURE 2024 October Q7
    Easy -1.2
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-22_841_999_251_534} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = - 4 \cos x$$ where \(x\) is measured in radians.
    Points \(P\) and \(Q\) lie on the curve and are shown in Figure 3.
    1. State
      1. the coordinates of \(P\)
      2. the coordinates of \(Q\) The curve \(C _ { 2 }\) has equation \(y = - 4 \cos x + k\) where \(x\) is measured in radians and \(k\) is a constant. Given that \(C _ { 2 }\) has a maximum \(y\) value of 11
      1. state the value of \(k\)
      2. state the coordinates of the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
    3. Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
    Edexcel PURE 2024 October Q8
    Standard +0.3
    1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\).
    The point \(P\) with \(x\) coordinate 3 lies on \(C\) \section*{Given}
    • \(\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { 2 } + k x + 3\) where \(k\) is a constant
    • the normal to \(C\) at \(P\) has equation \(y = - \frac { 1 } { 24 } x + 5\)
      1. show that \(k = - 5\)
      2. Hence find \(\mathrm { f } ( x )\).
    Edexcel PURE 2024 October Q9
    Moderate -0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-26_732_730_251_669} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$
    1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
    2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
      Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
      (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
      1. \(y = \mathrm { f } ( x - 3 )\)
      2. \(y = 4 \mathrm { f } ( x )\)
    Edexcel PURE 2024 October Q1
    Moderate -0.8
    1. A continuous curve has equation \(y = \mathrm { f } ( x )\).
    A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below.
    \(x\)0.51.7534.255.5
    \(y\)3.4796.1017.4486.8235.182
    Using the trapezium rule with all the values of \(y\) in the given table,
    1. find an estimate for $$\int _ { 0.5 } ^ { 5.5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to one decimal place.
    2. Using your answer to part (a) and making your method clear, estimate $$\int _ { 0.5 } ^ { 5.5 } ( \mathrm { f } ( x ) + 4 x ) \mathrm { d } x$$
    Edexcel PURE 2024 October Q2
    Standard +0.8
    1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
    $$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where \(k\) is a constant.
    1. Show that \(u _ { 5 } = 7\) Given that \(\sum _ { r = 1 } ^ { 4 } u _ { r } = 30\)
    2. find the value of \(k\).
    3. Hence find the value of \(\sum _ { r = 1 } ^ { 150 } u _ { r }\)
    Edexcel PURE 2024 October Q3
    Moderate -0.8
    3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where \(A\) and \(B\) are integers.
    Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 3 )\) the remainder is 55
    1. show that $$3 A - B = - 118$$ Given also that \(( 2 x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
    2. find the value of \(A\) and the value of \(B\).
    3. Hence find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x - 7\) )
    Edexcel PURE 2024 October Q4
    Moderate -0.8
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
    2. Hence find the \(x\) coordinate of the stationary point of \(C\).
      1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) giving each term in simplest form.
      2. Hence determine the nature of the stationary point of \(C\), giving a reason for your answer.
    4. State the range of values of \(x\) for which \(y\) is decreasing.
    Edexcel PURE 2024 October Q5
    Standard +0.3
    1. (a) Find, in terms of \(a\), the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
    $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = \left( 3 + \frac { 1 } { x } \right) ^ { 2 } ( 2 + a x ) ^ { 6 }$$ Given that the constant term in the expansion of \(\mathrm { f } ( x )\) is 576
    (b) find the value of \(a\).
    Edexcel PURE 2024 October Q6
    Standard +0.3
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    Using the laws of logarithms, solve $$\log _ { 4 } ( 12 - 2 x ) = 2 + 2 \log _ { 4 } ( x + 1 )$$
    Edexcel PURE 2024 October Q7
    Moderate -0.8
    1. Jem pays money into a savings scheme, \(A\), over a period of 300 months.
    Jem pays \(\pounds 20\) into scheme \(A\) in month \(1 , \pounds 20.50\) in month \(2 , \pounds 21\) in month 3 and so on, so that the amounts Jem pays each month form an arithmetic sequence.
    1. Show that Jem pays \(\pounds 69.50\) into scheme \(A\) in month 100
    2. Find the total amount that Jem pays into scheme \(A\) over the period of 300 months. Kim pays money into a different savings scheme, \(B\), over the same period of 300 months. In a model, the amounts Kim pays into scheme \(B\) increase by the same percentage each month, so that the amounts Kim pays each month form a geometric sequence. Given that Kim pays
      • \(\pounds 20\) into scheme \(B\) in month 1
      • \(\pounds 250\) into scheme \(B\) in month 300
      • use the model to calculate, to the nearest \(\pounds 10\), the difference between the total amount paid into scheme \(A\) and the total amount paid into scheme \(B\) over the period of 300 months.
    Edexcel PURE 2024 October Q8
    Standard +0.3
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e7412e14-6a5a-4545-8d6b-4bceb141cc15-20_762_851_376_607} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = x ^ { 2 } + 3 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 13 - \frac { 9 } { x ^ { 2 } } \quad x > 0$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\) as shown in Figure 1 .
    1. Use algebra to find the \(x\) coordinate of \(P\) and the \(x\) coordinate of \(Q\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
    2. Use algebraic integration to find the exact area of \(R\).
    Edexcel PURE 2024 October Q9
    Standard +0.3
    1. In this question you must show detailed reasoning.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Show that the equation $$2 \tan \theta = 3 \cos \theta$$ can be written as $$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$
    2. Hence solve, for \(- \pi < x < \pi\), the equation $$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$ giving your answers to 3 significant figures.
    Edexcel PURE 2024 October Q10
    Standard +0.8
    1. The circle \(C\) has equation
    $$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$
    1. Find
      1. the coordinates of the centre of \(C\),
      2. the exact value of the radius of \(C\). The line \(L\) has equation \(y = m x + 1\), where \(m\) is a constant.
        Given that \(L\) is the tangent to \(C\) at the point \(P\),
    2. show that $$2 m ^ { 2 } - 7 m - 22 = 0$$
    3. Hence find the possible pairs of coordinates of \(P\).
    Edexcel PURE 2024 October Q11
    Moderate -0.5
    1. (i) Prove by counter example that the statement
      "If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number." is false.
      (ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then
    $$m ^ { 2 } - 1$$ is divisible by 3
    Edexcel PURE 2024 October Q1
    Standard +0.3
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$3 \tan ^ { 2 } \theta + 7 \sec \theta - 3 = 0$$ giving your answers to one decimal place.
    Edexcel PURE 2024 October Q2
    Moderate -0.8
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-04_761_758_251_657} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation $$x = 2 y ^ { 2 } + 5 y - 6$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). The point \(P\) lies on the curve and is shown in Figure 1.
      Given that the tangent to the curve at \(P\) is parallel to the \(y\)-axis,
    2. find the coordinates of \(P\).
    Edexcel PURE 2024 October Q3
    Standard +0.8
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-06_638_643_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$
    1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
    2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$
    Edexcel PURE 2024 October Q4
    Moderate -0.3
    1. The number of bacteria on a surface is being monitored.
    The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).
    1. Find the initial number of bacteria on the surface.
    2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
    3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.
    Edexcel PURE 2024 October Q5
    Challenging +1.2
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
      1. Show that \(\sin 3 x\) can be written in the form
      $$P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be found.
    2. Hence or otherwise, solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$2 \sin 3 \theta = 5 \sin 2 \theta$$ giving your answers, in degrees, to one decimal place as appropriate.