Questions (33218 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 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. 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
    2. 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. Prove by counter example that the statement
      "If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number." is false.
    2. 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 M1 2024 October Q1
    Moderate -0.3
    1. Particle \(A\) has mass \(4 m\) and particle \(B\) has mass \(3 m\).
    The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of \(A\) is \(2 x\) and the speed of \(B\) is \(x\).
    Immediately after the collision, the speed of \(A\) is \(y\) and the speed of \(B\) is \(5 y\).
    The direction of motion of each particle is reversed as a result of the collision.
    1. Show that \(y = \frac { 5 } { 11 } x\).
    2. Find, in terms of \(m\) and \(x\), the magnitude of the impulse received by \(A\) in the collision.
    Edexcel M1 2024 October Q2
    Standard +0.3
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-04_282_1075_296_495} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A non-uniform beam \(A B\) has length 6 m and mass 50 kg . The beam rests horizontally on two supports at \(C\) and \(D\), where \(A C = 0.9 \mathrm {~m}\) and \(D B = 1.8 \mathrm {~m}\). A child of mass 25 kg stands on the beam at \(E\), where \(A E = E B = 3 \mathrm {~m}\), as shown in Figure 1. The beam is in equilibrium.
    The magnitude of the normal reaction between the beam and the support at \(C\) is \(R _ { C }\) newtons. The magnitude of the normal reaction between the beam and the support at \(D\) is \(R _ { D }\) newtons. The beam is modelled as a rod and the child is modelled as a particle.
    The centre of mass of the beam is between \(C\) and \(D\) and is a distance \(x\) metres from \(D\).
    Given that \(2 R _ { D } = 3 R _ { C }\)
    1. show that \(x = 1.38\) The child remains at \(E\) and a block of mass \(M \mathrm {~kg}\) is placed on the beam at \(B\).
      The block is modelled as a particle.
      Given that the beam is on the point of tilting,
    2. find the value of \(M\).
    Edexcel M1 2024 October Q3
    Moderate -0.8
    1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin.]
    A ship \(A\) is moving with constant velocity.
    At 1 pm , the position vector of \(A\) is \(( 25 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\).
    At 3 pm , the position vector of \(A\) is \(( 55 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\).
    At time \(t\) hours after 1 pm , the position vector of \(A\) is \(\mathbf { r } _ { A } \mathrm {~km}\).
    1. Show that \(\mathbf { r } _ { A } = ( 25 + 15 t ) \mathbf { i } + ( 10 + 12 t ) \mathbf { j }\) The speed of \(A\) is \(V \mathrm {~ms} ^ { - 1 }\)
    2. Find the value of \(V\). A ship \(B\) is moving with constant velocity \(( 20 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At 1 pm , the position vector of \(B\) is \(( 35 \mathbf { i } + 51 \mathbf { j } ) \mathrm { km }\).
      At 2:30 pm, \(B\) passes through the point \(P\).
    3. Show that \(A\) also passes through \(P\).
    Edexcel M1 2024 October Q4
    Moderate -0.8
    1. The points \(A\) and \(B\) lie on the same straight horizontal road.
    Figure 2, on page 11, shows the speed-time graph of a cyclist \(P\), for his journey from \(A\) to \(B\).
    At time \(t = 0 , P\) starts from rest at \(A\) and accelerates uniformly for 9 seconds until his speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) He then travels at constant speed \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 42\), cyclist \(P\) passes \(B\).
    Given that the distance \(A B\) is 120 m ,
    1. show that \(V = 3.2\)
    2. Find the acceleration of cyclist \(P\) between \(t = 0\) and \(t = 9\) Cyclist \(P\) continues to cycle along the road in the same direction at the same constant speed, \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 6\), a second cyclist \(Q\) sets off from \(A\) and travels in the same direction as \(P\) along the same road. She accelerates for \(T\) seconds until her speed is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) She then travels at constant speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Cyclist \(Q\) catches up with \(P\) when \(t = 54\)
    3. On Figure 2, on page 11, sketch a speed-time graph showing the journeys of both cyclists, for the interval \(0 \leqslant t \leqslant 54\)
    4. Find the value of \(T\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-11_661_1509_292_278} \captionsetup{labelformat=empty} \caption{Figure 2}
      \end{figure} A copy of Figure 2 is on page 13 if you need to redraw your answer to part (c). Only use this copy of Figure 2 if you need to redraw your answer to part (c). \includegraphics[max width=\textwidth, alt={}, center]{2f2f89a6-cec4-444d-95d9-0112887d87eb-13_666_1509_374_278} \section*{Copy of Figure 2}
    Edexcel M1 2024 October Q5
    Standard +0.3
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-14_588_908_292_794} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.
    1. Write down an equation of motion for \(P\).
    2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
    3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
      At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
    4. Find the speed of \(Q\) at the instant it hits the ground.
    Edexcel M1 2024 October Q6
    Standard +0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-18_335_682_296_696} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A particle \(P\) of mass 5 kg lies on the surface of a rough plane.
    The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The particle is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The horizontal force acts in a vertical plane containing a line of greatest slope of the inclined plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)
    1. Find the smallest possible value of \(H\). The horizontal force is now removed, and \(P\) starts to slide down the slope.
      In the first \(T\) seconds after \(P\) is released from rest, \(P\) slides 1.5 m down the slope.
    2. Find the value of \(T\).
    Edexcel M1 2024 October Q7
    Moderate -0.3
    7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
    The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
    Ball \(A\) is modelled as a particle moving freely under gravity.
    1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
      Given that the mass of \(A\) is 0.1 kg ,
    2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
      At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
    3. Find the value of \(T _ { 2 }\)
    Edexcel S1 2024 October Q1
    Easy -1.2
    The back-to-back stem and leaf diagram on page 3 shows information about the running times of 31 Action films and 31 Comedy films.
    The running times are given to the nearest minute.
    1. Write down the modal running time for these Action films. Some of the quartiles for these two distributions are shown in the table below.
      Action filmsComedy films
      Lower quartile121\(a\)
      Median\(b\)117
      Upper quartile138\(c\)
    2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
    3. For these Action films find, to one decimal place,
      1. the mean running time,
      2. the standard deviation of the running times.
        (You may use \(\sum x = 4016\) and \(\sum x ^ { 2 } = 525056\) where \(x\) is the running time, in minutes, of an Action film.) One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
      (d) Evaluate this measure and describe the skewness for the running times of these Action films.

    (e) Comment on one difference between the distribution of the running times of these Action films and the distribution of the running times of these Comedy films. State the values of any statistics you have used to support your comment.
    TotalsAction filmsComedy filmsTotals
    (1)092235(5)
    (0)10356689(6)
    (5)986421102467999(8)
    (10)99876543101212466777789(11)
    (8)87775421131(1)
    (7)776643114(0)
    Key: \(0 | 9 | 2\) means 90 minutes for an Action film and 92 minutes for a Comedy film