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CAIE S2 2018 November Q6
6 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { - 1 } & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { \ln 3 }\).
  2. Show that \(\mathrm { E } ( X ) = 3.64\), correct to 3 significant figures.
  3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < \mathrm { E } ( X ) )\).
CAIE S2 2018 November Q7
7 A mill owner claims that the mean mass of sacks of flour produced at his mill is 51 kg . A quality control officer suspects that the mean mass is actually less than 51 kg . In order to test the owner's claim she finds the mass, \(x \mathrm {~kg}\), of each of a random sample of 150 sacks and her results are summarised as follows. $$n = 150 \quad \Sigma x = 7480 \quad \Sigma x ^ { 2 } = 380000$$
  1. Carry out the test at the \(2.5 \%\) significance level.
    You may now assume that the population standard deviation of the masses of sacks of flour is 6.856 kg . The quality control officer weighs another random sample of 150 sacks and carries out another test at the 2.5\% significance level.
  2. Given that the population mean mass is 49 kg , find the probability of a Type II error.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 2 2022 November Q1
1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2022 November Q2
2
  1. Show that the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x - y - 3 z = a \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) is a constant, does not have a unique solution.
  2. Given that \(a = - 5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
  3. Given instead that \(a \neq - 5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
CAIE Further Paper 2 2022 November Q3
3 The curve \(C\) has parametric equations $$\mathrm { x } = \mathrm { e } ^ { \mathrm { t } } - \frac { 1 } { 3 } \mathrm { t } ^ { 3 } , \quad \mathrm { y } = 4 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { t } } ( \mathrm { t } - 2 ) , \quad \text { for } 0 \leqslant t \leqslant 2$$ Find, in terms of e , the length of \(C\).
CAIE Further Paper 2 2022 November Q4
4
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1 .$$
  2. Show that \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \tan ^ { - 1 } ( \sinh x ) \right) = \operatorname { sech } x\).
  3. Sketch the graph of \(y = \operatorname { sechx }\), stating the equation of the asymptote.
  4. By considering a suitable set of \(n\) rectangles of unit width, use your sketch to show that $$\sum _ { r = 1 } ^ { n } \operatorname { sechr } < \tan ^ { - 1 } ( \operatorname { sinhn } )$$
  5. Hence state an upper bound, in terms of \(\pi\), for \(\sum _ { r = 1 } ^ { \infty }\) sech \(r\).
CAIE Further Paper 2 2022 November Q5
5 Find the particular solution of the differential equation $$2 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + y = 4 x ^ { 2 } + 3 x + 3$$ given that, when \(x = 0 , y = \frac { d y } { d x } = 0\).
CAIE Further Paper 2 2022 November Q6
6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7 \\ 0 & 5 & 7 \\ 0 & 0 & - 2 \end{array} \right) .$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.
CAIE Further Paper 2 2022 November Q7
7
  1. State the sum of the series \(1 + \mathrm { w } + \mathrm { w } ^ { 2 } + \mathrm { w } ^ { 3 } + \ldots + \mathrm { w } ^ { \mathrm { n } - 1 }\), for \(w \neq 1\).
  2. Show that \(( 1 + i \tan \theta ) ^ { k } = \sec ^ { k } \theta ( \cos k \theta + i \sin k \theta )\), where \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
  3. By considering \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { \mathrm { k } }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \sec ^ { k } \theta \sin k \theta = \cot \theta \left( 1 - \sec ^ { n } \theta \cos n \theta \right)$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
  4. Hence find \(\sum _ { k = 0 } ^ { 6 m - 1 } 2 ^ { k } \sin \left( \frac { 1 } { 3 } k \pi \right)\) in terms of \(m\).
CAIE Further Paper 2 2022 November Q8
8
  1. Use the substitution \(u = 1 - ( \theta - 1 ) ^ { 2 }\) to find $$\int \frac { \theta - 1 } { \sqrt { 1 - ( \theta - 1 ) ^ { 2 } } } \mathrm {~d} \theta$$
  2. Find the solution of the differential equation $$\theta \frac { d y } { d \theta } - y = \theta ^ { 2 } \sin ^ { - 1 } ( \theta - 1 ) ,$$ where \(0 < \theta < 2\), given that \(y = 1\) when \(\theta = 1\). Give your answer in the form \(y = \mathrm { f } ( \theta )\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2020 June Q1
1 Two randomly selected groups of students, with similar ranges of abilities, take the same examination in different rooms. One group of 140 students takes the examination with background music playing. The other group of 210 students takes the examination in silence. Each student is awarded a grade for their performance in the examination and the numbers from each group gaining each grade are shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Grade awarded
\cline { 2 - 4 } \multicolumn{1}{c|}{}ABC
Background music495140
Silence936849
Test at the 10\% significance level whether grades awarded are independent of whether background music is playing during the examination.
CAIE Further Paper 4 2020 June Q2
2 The times, in milliseconds, taken by a computer to perform a certain task were recorded on 10 randomly chosen occasions. The times were as follows. $$\begin{array} { l l l l l l l l l l } 6.44 & 6.16 & 5.62 & 5.82 & 6.51 & 6.62 & 6.19 & 6.42 & 6.34 & 6.28 \end{array}$$ It is claimed that the median time to complete the task is 6.4 milliseconds.
  1. Carry out a Wilcoxon signed-rank test at the \(5 \%\) significance level to test this claim.
  2. State an underlying assumption that is made when using a Wilcoxon signed-rank test.
CAIE Further Paper 4 2020 June Q3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( 2 - \sqrt { x } ) & 0 \leqslant x < 1 \\ \frac { 3 } { 16 \sqrt { x } } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( X )\).
    The random variable \(Y\) is such that \(Y = \sqrt { X }\).
  2. Find the probability density function of \(Y\).
CAIE Further Paper 4 2020 June Q4
4 A company has two different machines, \(X\) and \(Y\), each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, \(x\) and \(y\), measured in appropriate units, in the cups filled by machines \(X\) and \(Y\) respectively. She chooses a random sample of 50 cups filled by machine \(X\) and a random sample of 40 cups filled by machine \(Y\). The volumes are summarised as follows. $$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$ The manager claims that there is no difference between the mean volume of coffee in cups filled by machine \(X\) and the mean volume of coffee in cups filled by machine \(Y\). Test the manager's claim at the \(10 \%\) significance level.
CAIE Further Paper 4 2020 June Q5
5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows. \(\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}\)
  1. Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the population mean distance thrown is more than 22.0 metres.
  2. Find a 95\% confidence interval for the population mean distance thrown.
CAIE Further Paper 4 2020 June Q6
6 A bag contains 4 red balls and 6 blue balls. Rassa selects two balls at random, without replacement, from the bag. The number of red balls selected by Rassa is denoted by \(X\).
  1. Find the probability generating function, \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\), of \(X\).
    Rassa also tosses two coins. One coin is biased so that the probability of a head is \(\frac { 2 } { 3 }\). The other coin is biased so that the probability of a head is \(p\). The probability generating function of \(Y\), the number of heads obtained by Rassa, is \(\mathrm { G } _ { Y } ( \mathrm { t } )\). The coefficient of \(t\) in \(\mathrm { G } _ { Y } ( \mathrm { t } )\) is \(\frac { 7 } { 12 }\).
  2. Find \(\mathrm { G } _ { Y } ( \mathrm { t } )\).
    The random variable \(Z\) is the sum of the number of red balls selected and the number of heads obtained by Rassa.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 4 2021 June Q1
1 A random sample of 7 observations of a variable \(X\) are as follows. $$\begin{array} { l l l l l l l } 8.26 & 7.78 & 7.92 & 8.04 & 8.27 & 7.95 & 8.34 \end{array}$$ The population mean of \(X\) is \(\mu\).
  1. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu = 8.22\) against the alternative hypothesis \(\mu < 8.22\).
  2. State an assumption necessary for the test in part (a) to be valid.
CAIE Further Paper 4 2021 June Q2
2 A driving school employs four instructors to prepare people for their driving test. The allocation of people to instructors is random. For each of the instructors, the following table gives the number of people who passed and the number who failed their driving test last year.
Instructor \(A\)Instructor \(B\)Instructor \(C\)Instructor \(D\)Total
Pass72425268234
Fail33344158166
Total1057693126400
Test at the 10\% significance level whether success in the driving test is independent of the instructor.
CAIE Further Paper 4 2021 June Q3
3 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 81 } x ^ { 2 } & 0 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{cases}$$
  1. Find \(\mathrm { E } ( \sqrt { X } )\).
  2. Find \(\operatorname { Var } ( \sqrt { X } )\).
  3. The random variable \(Y\) is given by \(Y ^ { 3 } = X\). Find the probability density function of \(Y\).
CAIE Further Paper 4 2021 June Q4
4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random sample of 50 leaves from birch trees in region \(A\) and a random sample of 60 leaves from birch trees in region \(B\). He records their lengths in \(\mathrm { cm } , x\) and \(y\), respectively. His results are summarised as follows. $$\sum x = 282 \quad \sum x ^ { 2 } = 1596 \quad \sum y = 328 \quad \sum y ^ { 2 } = 1808$$ The population mean lengths of leaves from birch trees in regions \(A\) and \(B\) are \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively. Carry out a test at the \(5 \%\) significance level to test the null hypothesis \(\mu _ { \mathrm { A } } = \mu _ { \mathrm { B } }\) against the alternative hypothesis \(\mu _ { \mathrm { A } } \neq \mu _ { \mathrm { B } }\).
CAIE Further Paper 4 2021 June Q5
5 Georgio has designed two new uniforms \(X\) and \(Y\) for the employees of an airline company. A random sample of 11 employees are each asked to assess each of the two uniforms for practicality and appearance, and to give a total score out of 100. The scores are given in the table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)
Uniform \(X\)8274425960739498623650
Uniform \(Y\)7875635667829990724861
  1. Give a reason why a Wilcoxon signed-rank test may be more appropriate than a \(t\)-test for investigating whether there is any evidence of a preference for one of the uniforms.
  2. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level.
CAIE Further Paper 4 2021 June Q6
6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).
  1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
    Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\).
  2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
    The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained.
  3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
  4. Use the probability generating function of \(Z\) to find \(E ( Z )\) and \(\operatorname { Var } ( Z )\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 4 2022 June Q1
1 A manager is investigating the times taken by employees to complete a particular task as a result of the introduction of new technology. He claims that the mean time taken to complete the task is reduced by more than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes, before and after the introduction of the new technology are recorded in the table.
Employee\(A\)\(B\)\(C\)D\(E\)\(F\)G\(H\)IJ
Time before new technology10.29.812.411.610.811.214.610.612.311.0
Time after new technology9.68.512.410.910.210.612.810.812.510.6
  1. Test at the 10\% significance level whether the manager's claim is justified.
  2. State an assumption that is necessary for this test to be valid.
CAIE Further Paper 4 2022 June Q2
2 The probability generating function, \(\mathrm { G } _ { Y } ( t )\), of the random variable \(Y\) is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$
  1. Find \(\operatorname { Var } ( Y )\).
    The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
  2. Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).
CAIE Further Paper 4 2022 June Q3
3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 3 } { 40 }\).
  2. Given that \(\mathrm { E } ( X ) = 2.5\), find \(\operatorname { Var } ( X )\).
  3. Find the median value of \(X\).