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CAIE P3 2021 November Q9
11 marks Standard +0.3
9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.
  1. Show that \(l\) and \(m\) are perpendicular.
  2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
  3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.
CAIE P3 2020 Specimen Q2
4 marks Moderate -0.8
2
  1. Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 Specimen Q3
4 marks Easy -1.3
3
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Solve the inequality \(3 x - 1 > | 2 x - 3 |\).
CAIE P3 2020 Specimen Q4
9 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t - 3 } , \quad y = 4 \ln t$$ where \(t > 0\). When \(t = a\) the gradient of the curve is 2 .
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2020 Specimen Q5
7 marks Standard +0.8
5
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
8 marks Moderate -0.5
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the argument of \(\frac { u } { v }\).
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2020 Specimen Q7
9 marks Standard +0.3
7
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2020 Specimen Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-14_485_716_262_676} In the diagram, \(O A B C\) is a pyramid in which \(O A = 2\) units, \(O B = 4\) units and \(O C = 2\) units. The edge \(O C\) is vertical, the base \(O A B\) is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O B\) and \(O C\) respectively. The midpoints of \(A B\) and \(B C\) are \(M\) and \(N\) respectively.
  1. Express the vectors \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Calculate the angle between the directions of \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\).
  3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2020 Specimen Q10
11 marks Standard +0.5
10 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\).
The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x ) .$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE S1 2020 Specimen Q1
5 marks Easy -1.2
1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
FemalesMales
(4)5200203(1)
(9)98876400021007(3)
(8)8753310022004566(6)
(6)64210023002335677(9)
(6)754000240112556889(10)
(4)9500253457789(7)
(2)5026046(3)
Key: 2 | 20 | 3 means \\(20200for females and \\)20300 for males.
  1. Find the median and the quartiles of the females' salaries.
    You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
  2. Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}
CAIE S1 2020 Specimen Q2
4 marks Easy -1.2
2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \text { and } \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).
CAIE S1 2020 Specimen Q3
7 marks Moderate -0.5
3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.
  1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
  2. Draw up the probability distribution table for \(X\).
  3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).
CAIE S1 2020 Specimen Q4
10 marks Moderate -0.5
4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
  1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
    The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
  2. Find the value of \(m\).
  3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
CAIE S1 2020 Specimen Q5
7 marks Moderate -0.5
5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.
  1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
  2. Justify your use of the approximation in part (a).
    On another occasion, the same die is thrown repeatedly until a 3 is obtained.
  3. Find the probability that obtaining a 3 requires fewer than 7 throws.
CAIE S1 2020 Specimen Q6
7 marks Standard +0.3
6 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]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} 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.
CAIE S1 2020 Specimen Q7
10 marks Standard +0.3
7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.
  • Event \(X\) is 'exactly two of the selected balls have the same number'.
  • Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
    1. Find \(\mathrm { P } ( X )\).
    2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
    3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.
CAIE Further Paper 4 2020 Specimen Q1
7 marks Moderate -0.5
1
  1. State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test. The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.
    5.625.736.556.816.105.755.876.475.866.266.995.91
  2. Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a \(5\%\) significance level.
    [0pt] [6]
CAIE Further Paper 4 2020 Specimen Q2
7 marks Challenging +1.2
2 Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws needed is recorded and the results are summarised in the following table.
Number of throws123456\(\geqslant 7\)
Frequency12643223510
Carry out a goodness of fit test, at the \(5\%\) significance level, to test whether \(\operatorname{Geo}(0.6)\) is a satisfactory model for the data.
CAIE Further Paper 4 2020 Specimen Q3
8 marks Standard +0.3
3 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm. To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm. For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Before4235967420578451460
After34321007231261351400
Test, at the \(10\%\) significance level, whether the population mean number of hours of absence has decreased following the introduction of flexi-time, stating any assumption that you make.
CAIE Further Paper 4 2020 Specimen Q4
7 marks Standard +0.8
4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).
CAIE Further Paper 4 2020 Specimen Q5
8 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} 0 & x < 0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x > 1 \end{cases}$$
  1. Find \(\mathrm{P}(X > 1)\).
  2. Find the median value of \(X\).
  3. Given that \(\mathrm{E}(X) = 1\), find the variance of \(X\).
  4. Find \(\mathrm{E}(\sqrt{X})\).
CAIE Further Paper 4 2020 Specimen Q6
13 marks Standard +0.3
6 Aisha has a bag containing 3 red balls and 3 white balls. She selects a ball at random, notes its colour and returns it to the bag; the same process is repeated twice more. The number of red balls selected by Aisha is denoted by \(X\).
  1. Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).
Basant also has a bag containing 3 red balls and 3 white balls. He selects three balls at random, without replacement, from his bag. The number of red balls selected by Basant is denoted by \(Y\).
  1. Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).
The random variable \(Z\) is the total number of red balls selected by Aisha and Basant.
  1. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  2. Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).
CAIE FP1 2008 June Q1
4 marks Standard +0.8
1 The finite region enclosed by the line \(y = k x\), where \(k\) is a positive constant, the \(x\)-axis for \(0 \leqslant x \leqslant h\), and the line \(x = h\) is rotated through 1 complete revolution about the \(x\)-axis. Prove by integration that the centroid of the resulting cone is at a distance \(\frac { 3 } { 4 } h\) from the origin \(O\).
[0pt] [The volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]