Questions — Edexcel (10514 questions)

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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
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 \(P\).