Questions — CAIE (7646 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
CAIE S1 2014 November Q4
8 marks Moderate -0.8
Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.
  1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
  2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]
CAIE S1 2014 November Q5
8 marks Moderate -0.8
  1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
    1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
    2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
  2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]
CAIE S1 2014 November Q6
9 marks Easy -1.2
On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.
Height (cm)\(4 - 10\)\(11 - 15\)\(16 - 20\)\(21 - 25\)\(26 - 30\)
Frequency2232784028
  1. Draw a cumulative frequency graph to illustrate the data. [4]
  2. 28\% of these daffodils are of height \(h\) cm or more. Estimate \(h\). [2]
  3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils. [3]
CAIE S1 2014 November Q7
9 marks Standard +0.3
In Marumbo, three quarters of the adults own a cell phone.
  1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
  2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
  3. Justify the use of your approximation in part (ii). [1]
CAIE S2 2021 June Q1
4 marks Standard +0.3
Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\). Find the probability that the total number of accidents in the two factories during a 2-month period is more than 3. [4]
CAIE S2 2021 June Q2
4 marks Moderate -0.8
The time, in minutes, taken by students to complete a test has the distribution \(\text{N}(125, 36)\).
  1. Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. [3]
  2. Explain whether it was necessary to use the Central Limit theorem in the solution to part (a). [1]
CAIE S2 2021 June Q3
2 marks Moderate -0.8
The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\text{P}(X < 5) = \frac{20}{39}\), find \(\text{P}(3 < X < 5)\). [2]
CAIE S2 2021 June Q4
6 marks Moderate -0.3
100 randomly chosen adults each throw a ball once. The length, \(l\) metres, of each throw is recorded. The results are summarised below. $$n = 100 \qquad \sum l = 3820 \qquad \sum l^2 = 182200$$ Calculate a 94% confidence interval for the population mean length of throws by adults. [6]
CAIE S2 2021 June Q5
7 marks Standard +0.3
On average, 1 in 75000 adults has a certain genetic disorder.
  1. Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder. [3]
  2. In a random sample of \(n\) people, where \(n\) is large, the probability that no-one has the genetic disorder is more than 0.9. Find the largest possible value of \(n\). [4]
CAIE S2 2021 June Q6
6 marks Standard +0.3
The probability density function, f, of a random variable \(X\) is given by $$\text{f}(x) = \begin{cases} k(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant. State the value of \(\text{E}(X)\) and show that \(\text{Var}(X) = \frac{9}{5}\). [6]
CAIE S2 2021 June Q7
10 marks Standard +0.3
The masses, in kilograms, of large and small sacks of flour have the distributions \(\text{N}(55, 3^2)\) and \(\text{N}(27, 2.5^2)\) respectively.
  1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]
  2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]
CAIE S2 2021 June Q8
11 marks Standard +0.3
At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.
  1. It is suggested that a random sample of the students in Grade 12 should be used for the test. Give a reason why this would not be an appropriate sample. [1]
  2. State suitable null and alternative hypotheses. [1]
  3. Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully. [5]
  4. In fact 4 students out of the 50 are not wearing correct uniform. State the conclusion of the test, explaining your answer. [2]
  5. State, with a reason, which of the errors, Type I or Type II, may have been made. [2]
A suitable sample of 50 students is selected and the number not wearing correct uniform is noted. This figure is used to carry out a test at the 5% significance level.
CAIE S2 2022 November Q1
7 marks Standard +0.3
Each of a random sample of 80 adults gave an estimate, \(h\) metres, of the height of a particular building. The results were summarised as follows. $$n = 80 \quad \sum h = 2048 \quad \sum h^2 = 52760$$
  1. Calculate unbiased estimates of the population mean and variance. [3]
  2. Using this sample, the upper boundary of an \(\alpha\%\) confidence interval for the population mean is 26.0. Find the value of \(\alpha\). [4]
CAIE S2 2022 November Q2
5 marks Moderate -0.3
In the past, the mean length of a particular variety of worm has been 10.3 cm, with standard deviation 2.6 cm. Following a change in the climate, it is thought that the mean length of this variety of worm has decreased. The lengths of a random sample of 100 worms of this variety are found and the mean of this sample is found to be 9.8 cm. Assuming that the standard deviation remains at 2.6 cm, carry out a test at the 2% significance level of whether the mean length has decreased. [5]
CAIE S2 2022 November Q3
6 marks Moderate -0.8
1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.
  1. Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
  2. Justify your approximating distribution. [2]
CAIE S2 2022 November Q4
8 marks Standard +0.3
The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per 10 m\(^2\). An adjustment is made to the machine. It is required to test at the 5% significance level whether the mean number of faults has decreased. A randomly selected 30 m\(^2\) of cloth is checked and the number of faults is found.
  1. State suitable null and alternative hypotheses for the test. [1]
  2. Find the probability of a Type I error. [3]
Exactly 3 faults are found in the randomly selected 30 m\(^2\) of cloth.
  1. Carry out the test at the 5% significance level. [2]
Later a similar test was carried out at the 5% significance level, using another randomly selected 30 m\(^2\) of cloth.
  1. Given that the number of faults actually has a Poisson distribution with mean 0.5 per 10 m\(^2\), find the probability of a Type II error. [2]
CAIE S2 2022 November Q5
6 marks Moderate -0.8
\(X\) is a random variable with distribution B(10, 0.2). A random sample of 160 values of \(X\) is taken.
  1. Find the approximate distribution of the sample mean, including the values of the parameters. [3]
  2. Hence find the probability that the sample mean is less than 1.8. [3]
CAIE S2 2022 November Q6
10 marks Standard +0.3
The masses, in grams, of small and large bags of flour have the distributions N(510, 100) and N(1015, 324) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.
  1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]
  2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]
CAIE S2 2022 November Q7
8 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the graph of the probability density function, f, of a random variable \(X\) which takes values between \(-3\) and 2 only.
  1. Given that the graph is symmetrical about the line \(x = -0.5\) and that P(\(X < 0\)) = \(p\), find P(\(-1 < X < 0\)) in terms of \(p\). [2]
  2. It is now given that the probability density function shown in the diagram is given by $$\text{f}(x) = \begin{cases} a - b(x^2 + x) & -3 \leq x \leq 2, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are positive constants.
    1. Show that \(30a - 55b = 6\). [3]
    2. By substituting a suitable value of \(x\) into f(\(x\)), find another equation relating \(a\) and \(b\) and hence determine the values of \(a\) and \(b\). [3]
CAIE S2 2023 November Q1
3 marks Moderate -0.5
A random variable \(X\) has the distribution N(410, 400). Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405. [3]
CAIE S2 2023 November Q2
4 marks Standard +0.8
In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an \(\alpha\)% confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be 0.487, correct to 3 significant figures. Find the value of \(\alpha\) correct to the nearest integer. [4]
CAIE S2 2023 November Q3
10 marks Standard +0.3
A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.
  1. Assume that the owner is correct.
    1. Find the probability that there will be at least 4 hits during a 10-minute period. [3]
    2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]
A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).
    1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution. [1]
    2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]
CAIE S2 2023 November Q4
8 marks Standard +0.3
The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim\) N(10.3, 5.76) and \(Y \sim\) N(11.4, 9.61). The income generated by the chemicals is \\(2.50 per kilogram for \)A\( and \\)3.25 per kilogram for \(B\).
  1. Find the mean and variance of the daily income generated by chemical \(A\). [2]
  2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\). [6]
CAIE S2 2023 November Q5
5 marks Standard +0.3
In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution Po(0.31). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the 2.5% significance level. [5]
CAIE S2 2023 November Q6
8 marks Standard +0.8
A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that P\((a < X < b) = p\) and P\((b < X < 3) = \frac{13}{10}p\), where \(p\) is a positive constant.
  1. Show that \(p \leq \frac{5}{23}\). [1]
  2. Find P\((b < X < 6 - a)\) in terms of \(p\). [2]
It is now given that the probability density function of \(X\) is \(f\), where $$f(x) = \begin{cases} \frac{1}{36}(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Given that \(b = 2\) and \(p = \frac{5}{81}\), find the value of \(a\). [5]