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Edexcel S1 2024 October Q2
Moderate -0.8
  1. A biologist records the length, \(y \mathrm {~cm}\), and the weight, \(w \mathrm {~kg}\), of 50 rabbits. The following summary statistics are calculated from these data.
$$\sum y = 2015 \quad \sum y ^ { 2 } = 81938.5 \quad \sum w = 125 \quad \mathrm {~S} _ { w w } = 72.25 \quad \mathrm {~S} _ { y w } = 219.55$$
    1. Show that \(\mathrm { S } _ { y y } = 734\)
    2. Calculate the product moment correlation coefficient for these data. Give your answer to 3 decimal places.
  2. Interpret your value of the product moment correlation coefficient. The biologist believes that a linear regression model may be appropriate to describe these data.
  3. State, with a reason, whether or not your value of the product moment correlation coefficient is consistent with the biologist’s belief.
  4. Find the equation of the regression line of \(w\) on \(y\), giving your answer in the form \(w = a + b y\) Jeff has a pet rabbit of length 45 cm .
  5. Use your regression equation to estimate the weight of Jeff's rabbit.
Edexcel S1 2024 October Q3
Moderate -0.8
  1. A group of 200 adults were asked whether they read cooking magazines, travel magazines or sport magazines.
    Their replies showed that
  • 29 read only cooking magazines
  • 33 read only travel magazines
  • 42 read only sport magazines
  • 17 read cooking magazines and sport magazines but not travel magazines
  • 11 read travel magazines and sport magazines but not cooking magazines
  • 22 read cooking magazines and travel magazines but not sport magazines
  • 32 do not read cooking magazines, travel magazines or sport magazines
    1. Using this information, complete the Venn diagram on page 11
One of these adults was chosen at random.
  • Find the probability that this adult,
    1. reads cooking magazines and travel magazines and sport magazines,
    2. does not read cooking magazines. Given that this adult reads travel magazines,
  • find the probability that this adult also reads sport magazines.
    \includegraphics[max width=\textwidth, alt={}]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-11_851_1086_296_493}
    •
    Edexcel S1 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 distances, \(m\) miles, a motorbike travels on a full tank of petrol can be modelled by a normal distribution with mean 170 miles and standard deviation 16 miles.
    1. Find the probability that, on a randomly selected journey, the motorbike could travel at least 190 miles on a full tank of petrol. The probability that, on a randomly selected journey, the motorbike could travel at least \(d\) miles on a full tank of petrol is 0.9
    2. Find the value of \(d\)
    Edexcel S1 2024 October Q5
    Moderate -0.3
    5.
    \includegraphics[max width=\textwidth, alt={}]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-16_990_1473_246_296}
    The histogram shows the number of hours worked in a given week by a group of 64 freelance photographers.
    1. Give a reason to justify the use of a histogram to represent these data. Given that 16 of these freelance photographers spent between 10 and 20 hours working in this week,
    2. estimate the number that spent between 12 and 24 hours working in this week.
    3. Find an estimate for the median time spent working in this week by these 64 freelance photographers. Charlie decides to model these data using a normal distribution. Charlie calculates an estimate of the mean to be 23.9 hours to one decimal place.
    4. Comment on Charlie's decision to use a normal distribution. Give a justification for your answer.
    Edexcel S1 2024 October Q6
    Moderate -0.3
    1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score which is uppermost. The cumulative distribution function of \(X\) is shown in the table below.
    \(x\)123456
    \(\mathrm {~F} ( x )\)0.10.2\(3 k\)\(5 k\)\(7 k\)\(10 k\)
    1. Find the value of the constant \(k\)
    2. Find the probability distribution of \(X\) A biased die with eight faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The probability distribution of \(Y\) is shown in the table below, where \(a\) and \(b\) are constants.
      \(y\)12345678
      \(\mathrm { P } ( Y = y )\)\(a\)\(a\)\(a\)\(b\)\(b\)\(b\)0.110.05
      Given that \(\mathrm { E } ( Y ) = 4.02\)
    3. form and solve two equations in \(a\) and \(b\) to show that \(a = 0.15\) You must show your working.
      (Solutions relying on calculator technology are not acceptable.)
    4. Show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 20.7\)
    5. Find \(\operatorname { Var } ( 5 - 2 Y )\) These dice are each rolled once. The scores on the two dice are independent.
    6. Find the probability that the sum of these two scores is 3
    Edexcel S1 2024 October Q7
    Moderate -0.3
    1. A box contains only red counters and black counters.
    There are \(n\) red counters and \(n + 1\) black counters.
    Two counters are selected at random, one at a time without replacement, from the box.
    1. Complete the tree diagram for this information. Give your probabilities in terms of \(n\) where necessary. \includegraphics[max width=\textwidth, alt={}, center]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-24_940_1180_591_413}
    2. Show that the probability that the two counters selected are different colours is $$\frac { n + 1 } { 2 n + 1 }$$ The probability that the two counters selected are different colours is \(\frac { 25 } { 49 }\)
    3. Find the total number of counters in the box before any counters were selected. Given that the two counters selected are different colours,
    4. find the probability that the 1st counter is black. You must show your working.
    Edexcel S1 2024 October Q8
    Standard +0.8
    1. An orchard produces apples.
    The weights, \(A\) grams, of its apples are normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams. It is known that $$\mathrm { P } ( A < 162 ) = 0.1 \text { and } \mathrm { P } ( 162 < A < 175 ) = 0.7508$$
    1. Calculate the value of \(\mu\) and the value of \(\sigma\) A second orchard also produces apples.
      The weights, \(B\) grams, of its apples have distribution \(B \sim N \left( 215,10 ^ { 2 } \right)\) An outlier is a value that is
      greater than \(\mathrm { Q } _ { 3 } + 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) or smaller than \(\mathrm { Q } _ { 1 } - 1.5 \times \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) An apple is selected at random from this second orchard.
      Using \(\mathrm { Q } _ { 3 } = 221.74\) grams,
    2. find the probability that this apple is an outlier.
    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
    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.
    Edexcel PURE 2024 October Q6
    Standard +0.3
    1. The functions f and g are defined by
    $$\begin{array} { l l } \mathrm { f } ( x ) = 6 - \frac { 21 } { 2 x + 3 } & x \geqslant 0 \\ \mathrm {~g} ( x ) = x ^ { 2 } + 5 & x \in \mathbb { R } \end{array}$$
    1. Find \(\mathrm { gf } ( 2 )\)
    2. Find \(f ^ { - 1 }\)
    3. Solve the equation $$\operatorname { gg } ( x ) = 126$$
    Edexcel PURE 2024 October Q7
    Standard +0.3
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-20_554_559_264_753} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
    2. Find the coordinates of \(P\).
    3. Hence find the range of the function g defined by $$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
    4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$
    Edexcel PURE 2024 October Q8
    Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-24_472_595_246_735} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The heart rate of a horse is being monitored.
    The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
    Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}
    1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
    2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
    3. Find the value of \(T\), giving your answer to 3 decimal places.
      (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
    4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
      1. find, to 4 decimal places, the value of \(t _ { 2 }\)
      2. find, to 4 decimal places, the value of \(M\)
    Edexcel PURE 2024 October Q9
    Standard +0.8
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-28_753_1111_248_477} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$
    1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
    2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
    3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
    4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
      (Solutions based entirely on calculator technology are not acceptable.)
    Edexcel PURE 2024 October Q1
    Moderate -0.3
    1. Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
    2. Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)
    Edexcel PURE 2024 October Q2
    Standard +0.8
    1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    The curve \(C _ { 1 }\) has equation $$y = x ^ { 4 } + 10 x ^ { 2 } + 8 \quad x \in \mathbb { R }$$ The curve \(C _ { 2 }\) has equation $$y = 2 x ^ { 2 } - 7 \quad x \in \mathbb { R }$$ Use algebra to prove by contradiction that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
    Edexcel PURE 2024 October Q3
    Standard +0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-06_549_750_251_660} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$
    1. Show that $$\frac { d y } { d x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
    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.
    3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leqslant x \leqslant q$$ where \(q\) is a constant to be found.
    Edexcel PURE 2024 October Q4
    Standard +0.8
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-10_634_638_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).
    1. Verify that \(P\) lies on \(C\).
    2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
    3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.
    Edexcel PURE 2024 October Q5
    Standard +0.3
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-14_569_616_242_785} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a container in the shape of a hollow, inverted, right circular cone.
    The height of the container is 30 cm and the radius is 12 cm , as shown in Figure 3.
    The container is initially empty when water starts flowing into it.
    When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\)
    1. Show that $$V = \frac { 4 \pi h ^ { 3 } } { 75 }$$ [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ] Given that water flows into the container at a constant rate of \(2 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
    2. find, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), the rate at which \(h\) is changing, exactly 1.5 minutes after water starts flowing into the container.
    Edexcel PURE 2024 October Q6
    Challenging +1.2
    1. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that
    $$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + c$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
    Edexcel PURE 2024 October Q7
    Standard +0.3
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-18_510_680_251_696} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$
    1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
    2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.
    Edexcel PURE 2024 October Q8
    Standard +0.3
    1. Relative to a fixed origin \(O\)
    • the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
    • the point \(B\) has coordinates \(( - 6,4 , - 1 )\)
    The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
    1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
      Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
    2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
    3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
    4. find the exact length of \(A C\), giving your answer as a surd.
    Edexcel PURE 2024 October Q9
    Challenging +1.2
    1. Express \(\frac { 1 } { x ( 2 x - 1 ) }\) in partial fractions. The height above ground, \(h\) metres, of a carriage on a fairground ride is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 1 } { 50 } h ( 2 h - 1 ) \cos \left( \frac { t } { 10 } \right)$$ where \(t\) seconds is the time after the start of the ride.
      Given that, at the start of the ride, the carriage is 2.5 m above ground,
    2. solve the differential equation to show that, according to the model, $$h = \frac { 5 } { 10 - 8 \mathrm { e } ^ { k \sin \left( \frac { t } { 10 } \right) } }$$ where \(k\) is a constant to be found.
    3. Hence find, according to the model, the time taken for the carriage to reach its maximum height above ground for the 3rd time.
      Give your answer to the nearest second.
      (Solutions relying entirely on calculator technology are not acceptable.)