Questions — CAIE (7646 questions)

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CAIE S1 2014 November Q1
3 marks Easy -1.8
1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$
CAIE S1 2014 November Q2
6 marks Easy -1.3
2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.
\(x\)01234\(\geqslant 5\)
\(\mathrm { P } ( X = x )\)0.240.35\(2 k\)\(k\)0.050
  1. Find the value of \(k\).
  2. Find the mode of \(X\).
  3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
CAIE S1 2014 November Q3
5 marks Standard +0.3
3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:
  • if the coin shows a head, the two numbers from the dice are added together;
  • if the coin shows a tail, the two numbers from the dice are multiplied together.
Find the probability that the coin shows a head given that Jodie's score is 8 .
CAIE S1 2014 November Q4
7 marks Easy -1.2
4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7)
Type \(A\)Type \(B\)
976643321358
55442223044566667889
998887664322040112368899
655432110525669
973061389
874410757
766653321081244
86555906
Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).
  1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
  2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
  3. Compare the loading times for these two types of smartphone.
CAIE S1 2014 November Q5
9 marks Standard +0.3
5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .
  1. Show that the variance of the number of faulty screws in a packet is 1.104 .
  2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
  3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.
CAIE S1 2014 November Q6
10 marks Moderate -0.3
6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .
  1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
  2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
  3. Find the value of \(\mu\).
CAIE S1 2014 November Q7
10 marks Standard +0.3
7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
  1. if there are more women than men on the committee,
  2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
  3. In how many different ways can the committee members be arranged in a line if the man is not at either end?
CAIE S1 2014 November Q1
3 marks Moderate -0.5
1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]
CAIE S1 2014 November Q2
5 marks Easy -1.2
2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$
  1. Find \(\Sigma ( x - 62 )\).
  2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
  3. Find the mean and variance of the speeds of the 8 cars.
CAIE S1 2014 November Q3
5 marks Easy -1.2
3 The number of books read by members of a book club each year has the binomial distribution \(B ( 12,0.7 )\).
  1. State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.
  2. Find the probability that a member reads fewer than 10 books in a particular year.
CAIE S1 2014 November Q4
8 marks Easy -1.3
4 A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.
2319321425
2226364542
4728173815
4618262241
1921282430
  1. Draw a stem-and-leaf diagram to represent the data.
  2. On graph paper draw a box-and-whisker plot to represent the data.
CAIE S1 2014 November Q5
9 marks Standard +0.3
5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g
  1. Find the proportion of gem stones in each of these three categories.
  2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).
CAIE S1 2014 November Q6
9 marks Standard +0.3
6
  1. Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
  2. Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.
CAIE S1 2014 November Q7
11 marks Standard +0.3
7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
  2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
  3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.
CAIE S1 2015 November Q1
4 marks Moderate -0.8
1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.
CAIE S1 2015 November Q2
4 marks Moderate -0.3
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 2015 November Q3
6 marks Easy -1.8
3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).
Time \(( t\) minutes \()\)\(60 - 62\)\(63 - 64\)\(65 - 67\)\(68 - 71\)
Frequency (number of days)396\(b\)
Frequency density1\(a\)21.5
  1. Find the values of \(a\) and \(b\).
  2. On graph paper, draw a histogram to represent Robert's times.
CAIE S1 2015 November Q4
7 marks Moderate -0.8
4
  1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
  2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.
CAIE S1 2015 November Q5
9 marks Moderate -0.8
5
  1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
    1. if all the letters E are together,
    2. if the T is at one end and there is an S at the other end.
  2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.
  1. Draw a fully labelled tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of \(X\).
    \(x\)0123
    \(\mathrm { P } ( X = x )\)0.24
  3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.
CAIE S1 2015 November Q7
11 marks Standard +0.3
7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.
  1. Find the probability of throwing a 3 .
  2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
  3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.
CAIE S1 2015 November Q1
3 marks Easy -1.2
1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).
CAIE S1 2015 November Q2
3 marks Moderate -0.8
2 A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.
CAIE S1 2015 November Q3
6 marks Easy -1.2
3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that
  1. he gets a green robot on opening his first packet,
  2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
  3. Find the probability that the first four packets Amos opens all contain different coloured robots.
CAIE S1 2015 November Q4
7 marks Standard +0.3
4 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.
  1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_284_467_1491_495} \includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_286_471_1489_1183} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
  2. Find the number of different seating arrangements that are now possible for the 8 friends.