Questions — Edexcel S2 (494 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
Edexcel S2 2024 June Q4
4 A bag contains 50 counters, each with one of the numbers 4,7 or 10 written on it in the ratio \(2 : 3 : 5\) respectively. A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as \(D _ { 1 }\) and \(D _ { 2 }\) The random variable \(M\) represents the mean of \(D _ { 1 }\) and \(D _ { 2 }\)
  1. Show that \(\mathrm { P } ( M = 4 ) = \frac { 9 } { 245 }\)
  2. Find the sampling distribution of \(M\) A random sample of \(n\) sets of 2 counters is taken. The random variable \(T\) represents the number of these \(n\) sets of 2 counters that have a mean of 4 Given that each set of 2 counters is replaced after it is drawn,
  3. calculate the minimum value of \(n\) such that \(\mathrm { P } ( T = 0 ) < 0.15\)
Edexcel S2 2024 June Q5
5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.
  1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
  2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
  3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
  4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.
Edexcel S2 2024 June Q6
6 In this question solutions relying entirely on calculator technology are not acceptable.
The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & - 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(4 a + 4 b = 1\) Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 17 } { 5 }\)
    1. find an equation in terms of \(a\) only
    2. hence show that \(b = 0.1\)
  3. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\)
  4. Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.8\)
Edexcel S2 2016 October Q1
  1. A mobile phone company claims that each year \(5 \%\) of its customers have their mobile phone stolen. An insurance company claims this percentage is higher. A random sample of 30 of the mobile phone company's customers is taken and 4 of them have had their mobile phone stolen during the last year.
    1. Test the insurance company's claim at the \(10 \%\) level of significance. State your hypotheses clearly.
    A new random sample of 90 customers is taken. A test is carried out using these 90 customers, to see if the percentage of customers who have had a mobile phone stolen in the last year is more than 5\%
  2. Using a suitable approximation and a \(10 \%\) level of significance, find the critical region for this test.
Edexcel S2 2016 October Q2
  1. The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function
$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 8
\frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12
1 & t > 12 \end{array} \right.$$
  1. Show that \(k = - 432\)
  2. Find the probability density function of \(T\), for all values of \(t\).
  3. Write down the mode of \(T\).
  4. Find the median of \(T\).
  5. Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours,
  6. find the probability that its lifetime is no more than 11 hours.
Edexcel S2 2016 October Q3
  1. A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by \(40 \%\) of the students.
In a random sample of 20 students who sat the examination, \(X\) denotes the number of students who answered the first question correctly.
  1. Write down the distribution of the random variable \(X\)
  2. Find \(\mathrm { P } ( 4 \leqslant X < 9 )\) Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly.
  3. Find the probability that the total number of points scored on the first question by the 20 students is more than 0
  4. Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.
Edexcel S2 2016 October Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ecee051-3a6f-4c12-8c53-926e8c3e241f-14_451_976_233_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1 $$\mathrm { f } ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 5
k & 5 < x \leqslant 10.5
0 & \text { otherwise } \end{cases}$$ where \(m\) and \(k\) are constants.
    1. Show that \(k = \frac { 1 } { 8 }\)
    2. Find the value of \(m\)
  2. Find \(\mathrm { E } ( X )\)
  3. Find the interquartile range of \(X\)
Edexcel S2 2016 October Q5
  1. A string of length 40 cm is cut into 2 pieces at a random point. The continuous random variable \(L\) represents the length of the longer piece of string.
    1. Write down the distribution of \(L\)
    2. Find the probability that the length of the longer piece of string is 28 cm to the nearest cm
    Each piece of string is used to form the perimeter of a square.
  2. Calculate the probability that the area of the larger square is less than \(64 \mathrm {~cm} ^ { 2 }\)
  3. Calculate the probability that the difference in area between the two squares is greater than \(81 \mathrm {~cm} ^ { 2 }\)
Edexcel S2 2016 October Q6
  1. According to an electric company, power failures occur randomly at a rate of \(\lambda\) every 10 weeks, \(1 < \lambda < 10\)
    1. Write down an expression in terms of \(\lambda\) for the probability that there are fewer than 2 power failures in a randomly selected 10 week period.
    2. Write down an expression in terms of \(\lambda\) for the probability that there is exactly 1 power failure in a randomly selected 5 week period.
    Over a 100 week period, the probability, using a normal approximation, that fewer than 15 power failures occur is 0.0179 (to 3 significant figures).
    1. Justify the use of a normal approximation.
    2. Find the value of \(\lambda\). Show each stage of your working clearly.
Edexcel S2 2016 October Q7
  1. An ice cream shop sells a large number of 1 scoop, 2 scoop and 3 scoop ice cream cones to its customers in the ratio \(5 : 2 : 1\)
A random sample of 2 customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. Let \(S\) represent the total number of ice cream scoops ordered by these 2 customers.
  1. Find the sampling distribution of \(S\) A random sample of \(n\) customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. The probability that more than \(n\) scoops of ice cream are ordered by these customers is greater than 0.99
  2. Find the smallest possible value of \(n\)
    \includegraphics[max width=\textwidth, alt={}]{4ecee051-3a6f-4c12-8c53-926e8c3e241f-28_2632_1828_121_121}
Edexcel S2 2017 October Q1
  1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)
The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .
  1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
  2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
  3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
Edexcel S2 2017 October Q2
2. The weekly sales, \(S\), in thousands of pounds, of a small business has probability density function $$\mathrm { f } ( s ) = \left\{ \begin{array} { c c } k ( s - 2 ) ( 10 - s ) & 2 < s < 10
0 & \text { otherwise } \end{array} \right.$$
  1. Use algebraic integration to show that \(k = \frac { 3 } { 256 }\)
  2. Write down the value of \(\mathrm { E } ( S )\)
  3. Use algebraic integration to find the standard deviation of the weekly sales. A week is selected at random.
  4. Showing your working, find the probability that this week's sales exceed \(\pounds 7100\) Give your answer to one decimal place. A quarter is defined as 12 consecutive weeks. The discrete random variable \(X\) is the number of weeks in a quarter in which the weekly sales exceed £7100 The manager earns a bonus at the following rates:
    \(\boldsymbol { X }\)Bonus Earned
    \(X \leqslant 5\)\(\pounds 0\)
    \(X = 6\)\(\pounds 1000\)
    \(X \geqslant 7\)\(\pounds 5000\)
  5. Using your answer to part (d), calculate the manager's expected bonus per quarter.
Edexcel S2 2017 October Q3
3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.
  1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
  2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
  3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
  4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.
Edexcel S2 2017 October Q4
4. In a computer game, a ship appears randomly on a rectangular screen. The continuous random variable \(X \mathrm {~cm}\) is the distance of the centre of the ship from the bottom of the screen. The random variable \(X\) is uniformly distributed over the interval \([ 0 , \alpha ]\) where \(\alpha \mathrm { cm }\) is the height of the screen. Given that \(\mathrm { P } ( X > 6 ) = 0.6\)
  1. find the value of \(\alpha\)
  2. find \(\mathrm { P } ( 4 < X < 10 )\) The continuous random variable \(Y\) cm is the distance of the centre of the ship from the left-hand side of the screen. The random variable \(Y\) is uniformly distributed over the interval [ 0,20 ] where 20 cm is the width of the screen.
  3. Find the mean and the standard deviation of \(Y\).
  4. Find \(\mathrm { P } ( | Y - 4 | < 2 )\)
  5. Given that \(X\) and \(Y\) are independent, find the probability that the centre of the ship appears
    1. in a square of side 4 cm which is at the centre of the screen,
    2. within 5 cm of a side or the top or the bottom of the screen.
Edexcel S2 2017 October Q5
5. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 3
k \left( y ^ { 2 } - 2 y - 3 \right) & 3 \leqslant y \leqslant \alpha
4 k ( 2 y - 7 ) & \alpha < y \leqslant 6
1 & y > 6 \end{array} \right.$$ where \(k\) and \(\alpha\) are constants.
  1. Find \(\mathrm { P } ( 4.5 < Y \leqslant 5.5 )\)
  2. Find the probability density function \(\mathrm { f } ( \mathrm { y } )\)
Edexcel S2 2017 October Q6
6. A fair 6 -sided die is thrown \(n\) times. The number of sixes, \(X\), is recorded. Using a normal approximation, \(\mathrm { P } ( X < 50 ) = 0.0082\) correct to 4 decimal places. Find the value of \(n\).
(10)
END
Edexcel S2 2018 October Q1
  1. Each day a restaurant opens between 11 am and 11 pm . During its opening hours, the restaurant receives calls for reservations at an average rate of 6 per hour.
    1. Find the probability that the restaurant receives exactly 1 call for a reservation between 6 pm and 7 pm .
    The restaurant distributes leaflets to local residents to try and increase the number of calls for reservations. After distributing the leaflets, it records the number of calls for reservations it receives over a 90 minute period. Given that it receives 14 calls for reservations during the 90 minute period,
  2. test, at the \(5 \%\) level of significance, whether the rate of calls for reservations has increased from 6 per hour. State your hypotheses clearly.
Edexcel S2 2018 October Q2
  1. At a cafe, customers ordering hot drinks order either tea or coffee.
Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.
  1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
    1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
    2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
  3. A random sample of 10 customers ordering hot drinks is selected.
    1. Find the probability that exactly 4 of these 10 customers take sugar.
    2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
  4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.
Edexcel S2 2018 October Q3
3. The function \(\mathrm { f } ( x )\) is defined as $$f ( x ) = \begin{cases} \frac { 1 } { 9 } ( x + 5 ) ( 3 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Albert believes that \(\mathrm { f } ( x )\) is a valid probability density function.
  1. Sketch \(\mathrm { f } ( x )\) and comment on Albert's belief. The continuous random variable \(Y\) has probability density function given by $$g ( y ) = \begin{cases} k y \left( 12 - y ^ { 2 } \right) & 1 \leqslant y \leqslant 3
    0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  2. Use calculus to find the mode of \(Y\)
  3. Use algebraic integration to find the value of \(k\)
  4. Find the median of \(Y\) giving your answer to 3 significant figures.
  5. Describe the skewness of the distribution of \(Y\) giving a reason for your answer.
Edexcel S2 2018 October Q4
4. A bag contains a large number of marbles, each of which is blue or red. A random sample of 3 marbles is taken from the bag. The random variable \(D\) represents the number of blue marbles taken minus the number of red marbles taken. Given that 20\% of the marbles in the bag are blue,
  1. show that \(\mathrm { P } ( D = - 1 ) = 0.384\)
  2. find the sampling distribution of \(D\)
  3. write down the mode of \(D\) Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
  4. Find the critical region for this test at the 10\% level of significance.
  5. State the actual significance level of this test. \includegraphics[max width=\textwidth, alt={}, center]{d2f40cdb-917a-4377-88f4-396766a299e2-15_2255_47_314_37}
Edexcel S2 2018 October Q5
5. The random variable \(X\) has cumulative distribution function given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
\frac { 1 } { 100 } \left( a x ^ { 3 } + b x ^ { 2 } + 15 x \right) & 0 \leqslant x \leqslant 5
1 & x > 5 \end{array} \right.$$ Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 6.25\)
  1. show that \(6 a + b = 0\)
  2. find the value of \(a\) and the value of \(b\)
  3. find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\)
Edexcel S2 2018 October Q6
  1. One side of a square is measured to the nearest centimetre and this measurement is multiplied by 4 to estimate the perimeter of the square. The random variable, \(W \mathrm {~cm}\), represents the estimated perimeter of the square minus the true perimeter of the square.
    \(W\) is uniformly distributed over the interval \([ a , b ]\)
    1. Explain why \(a = - 2\) and \(b = 2\)
    The standard deviation of \(W\) is \(\sigma\)
    1. Find \(\sigma\)
    2. Find the probability that the estimated perimeter of the square is within \(\sigma\) of the true perimeter of the square. One side of each of 100 squares are now measured. Using a suitable approximation,
  3. find the probability that \(W\) is greater than 1.9 for at least 5 of these squares.
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Edexcel S2 2018 October Q7
7. Members of a conservation group record the number of sightings of a rare animal. The number of sightings follows a Poisson distribution with a rate of 1 every 2 months.
  1. Find the smallest value of \(n\) such that the probability that there are at least \(n\) sightings in 2 months is less than 0.05
  2. Find the smallest number of months, \(m\), such that the probability of no sightings in \(m\) months is less than 0.05
  3. Find the probability that there is at least 1 sighting per month in each of 3 consecutive months.
  4. Find the probability that the number of sightings in an 8 month period is equal to the expected number of sightings for that period.
  5. Given that there were 4 sightings in a 4 month period, find the probability that there were more sightings in the last 2 months than in the first 2 months.
Edexcel S2 2020 October Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3a781851-e2cc-4379-8b8c-abb3060a6019-02_572_497_299_726} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). For \(1 \leqslant x \leqslant 2 , \mathrm { f } ( x )\) is represented by a curve with equation \(\mathrm { f } ( x ) = k \left( \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + a x + 1 \right)\) where \(k\) and \(a\) are constants. For all other values of \(x , \mathrm { f } ( x ) = 0\)
  1. Use algebraic integration to show that \(k ( 12 a - 33 ) = 8\) Given that \(a = 5\)
  2. calculate the mode of \(X\).
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Edexcel S2 2020 October Q2
  1. In the summer Kylie catches a local steam train to work each day. The published arrival time for the train is 10 am.
The random variable \(W\) is the train's actual arrival time minus the published arrival time, in minutes. When the value of \(W\) is positive, the train is late. The cumulative distribution function \(\mathrm { F } ( w )\) is shown in the sketch below.
\includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-06_583_1235_589_349}
  1. Specify fully the probability density function \(\mathrm { f } ( w )\) of \(W\).
  2. Write down the value of \(\mathrm { E } ( \mathrm { W } )\)
  3. Calculate \(\alpha\) such that \(\mathrm { P } ( \alpha \leqslant W \leqslant 1.6 ) = 0.35\) A day is selected at random.
  4. Calculate the probability that on this day the train arrives between 1.2 minutes late and 2.4 minutes late. Given that on this day the train was between 1.2 minutes late and 2.4 minutes late,
  5. calculate the probability that it was more than 2 minutes late. A random sample of 40 days is taken.
  6. Calculate the probability that for at least 10 of these days the train is between 1.2 minutes late and 2.4 minutes late. DO NOT WRITEIN THIS AREA