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CAIE FP2 2013 November Q9
Standard +0.3
9 For a random sample of 10 observations of pairs of values \(( x , y )\), the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are $$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36 ,$$ respectively.
  1. Find the value of the product moment correlation coefficient for the sample.
  2. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
  3. Find the mean values of \(x\) and \(y\) for this sample.
  4. Estimate the value of \(x\) when \(y = 2.3\) and comment on the reliability of your answer.
CAIE FP2 2013 November Q11
Challenging +1.8
11 Answer only one of the following two alternatives.
EITHER
A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.
OR
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).
CAIE Further Paper 2 2020 Specimen Q0
Standard +0.3
0 & 2 & 2
- 1 & 1 & 3 \end{array} \right) .$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
CAIE M1 2014 June Q4
Standard +0.3
4 A particle \(P\) moves on a straight line, starting from rest at a point \(O\) of the line. The time after \(P\) starts to move is \(t \mathrm {~s}\), and the particle moves along the line with constant acceleration \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes through a point \(A\) at time \(t = 8\). After passing through \(A\) the velocity of \(P\) is \(\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the acceleration of \(P\) immediately after it passes through \(A\). Hence show that the acceleration of \(P\) decreases by \(\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it passes through \(A\).
  2. Find the distance moved by \(P\) from \(t = 0\) to \(t = 27\).
CAIE M1 2014 June Q5
Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block \(A\) of mass 5 kg attached at one end, and a block \(B\) of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Block \(A\) is held at rest at the bottom of the plane and block \(B\) hangs below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { 3 } }\). Block \(A\) is released from rest and the system starts to move. When each of the blocks has moved a distance of \(x \mathrm {~m}\) each has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the gain in kinetic energy of the system in terms of \(v\).
  2. Find, in terms of \(x\),
    1. the loss of gravitational potential energy of the system,
    2. the work done against the frictional force.
    3. Show that \(21 v ^ { 2 } = 220 x\).
CAIE FP2 2014 June Q10
Standard +0.3
10 The lengths of a random sample of eight fish of a certain species are measured, in cm, as follows. $$\begin{array} { l l l l l l l l } 17.3 & 15.8 & 18.2 & 15.6 & 16.0 & 18.8 & 15.3 & 15.0 \end{array}$$ Assuming that lengths are normally distributed,
  1. test, at the \(10 \%\) significance level, whether the population mean length of fish of this species is greater than 15.8 cm ,
  2. calculate a \(95 \%\) confidence interval for the population mean length of fish of this species.
CAIE FP2 2014 June Q11
Challenging +1.2
11 Answer only one of the following two alternatives.
EITHER
A particle \(P\) of mass \(m\) is suspended from a fixed point by a light elastic string of natural length \(l\), and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is \(\frac { 13 } { 7 } l\). The particle is released from rest in this position and reaches its greatest height when the length of the string is \(\frac { 11 } { 7 } l\).
  1. Show that the modulus of elasticity of the string is \(\frac { 7 } { 5 } \mathrm { mg }\).
  2. Show that \(P\) moves in simple harmonic motion about the equilibrium position and state the period of the motion.
  3. Find the time after release when the speed of \(P\) is first equal to half of its maximum value.
    OR
    For a random sample of 12 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) and the equation of the regression line of \(x\) on \(y\) are $$y = b x + 4.5 \quad \text { and } \quad x = a y + c$$ respectively, where \(a , b\) and \(c\) are constants. The product moment correlation coefficient for the sample is 0.6 .
  1. Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
  2. Given that \(b - a = 0.5\), find the values of \(a\) and \(b\).
  3. Given that the sum of the \(x\)-values in the sample data is 66, find the value of \(c\) and sketch the two regression lines on the same diagram. For each of the 12 pairs of values of \(( x , y )\) in the sample, another variable \(z\) is considered, where \(z = 5 y\).
  4. State the coefficient of \(x\) in the equation of the regression line of \(z\) on \(x\) and find the value of the product moment correlation coefficient between \(x\) and \(z\), justifying your answer.
CAIE FP1 2013 November Q22
Moderate -0.5
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r }
CAIE FP1 2013 November Q7
Standard +0.3
7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l }
CAIE FP1 2013 November Q9
Challenging +1.8
9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\).
CAIE FP1 2013 November Q11
Challenging +1.3
11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\). \end{document}
Edexcel D1 2022 January Q17
Moderate -0.8
17 & 9 & 15 & 8 & 20 & 13 & 28 & 4 & 12 & 5 \end{array}$$ The numbers in the list shown above are the weights, in kilograms, of ten boxes. The boxes are to be transported in containers that will each hold a maximum weight of 40 kilograms.
  1. Calculate a lower bound for the number of containers that will be needed to transport the boxes. You must show your working.
  2. Use the first-fit bin packing algorithm to allocate the boxes to the containers.
  3. Using the list provided, carry out a quick sort to produce a list of the weights in ascending order. You must make your pivots clear.
  4. Use the binary search algorithm to try to locate the weight of 9 in the sorted list. Clearly indicate how you choose your pivots and which part of the list is being rejected at each stage.
Edexcel D1 2022 January Q0
Easy -1.8
0 \leqslant x & \leqslant 27
Edexcel D1 2023 January Q10
Moderate -0.8
10 x + 7 y & \leqslant 140
& x + y \leqslant 15
& 2 x + 3 y \geqslant 36
& x \geqslant 0 , \quad y \geqslant 0 \end{aligned} \end{array}$$ (c) Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, \(R\).
(d) Use the objective line method to find the optimal number of each type of cake that Martin should make, and the amount of sugar used.
(e) Determine how much flour and how many eggs Martin will have left over after making the optimal number of cakes. BLANK PAGE \end{document}
OCR MEI FP2 2013 June Q1
Standard +0.3
1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x
y
z \end{array} \right) = \left( \begin{array} { c } p
Edexcel D1 2022 January Q7
Moderate -0.8
7. \section*{Question 7 continued} \section*{Question 7 continued} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{765ea64e-d4b8-4f0f-9a43-2619f9db0c18-19_2109_1335_299_372} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure} \section*{Question 7 continued} \section*{Pearson Edexcel International Advanced Level} Time 1 hour 30 minutes \section*{Paper reference WDM11/01} \section*{Mathematics} \section*{You must have:} Decision Mathematics Answer Book (enclosed), calculator Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
  • Use black ink or ball-point pen.
  • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
  • Write your answers for this paper in the Decision Mathematics answer book provided.
  • Fill in the boxes at the top of the answer book with your name, centre number and candidate number.
  • Do not return the question paper with the answer book.
  • Answer all questions and ensure that your answers to parts of questions are clearly labelled.
  • Answer the questions in the spaces provided
  • there may be more space than you need.
  • You should show sufficient working to make your methods clear. Answers without working may not gain full credit.
  • Inexact answers should be given to three significant figures unless otherwise stated.
\section*{Information}
  • There are 7 questions in this question paper. The total mark for this paper is 75.
  • The marks for each question are shown in brackets
  • use this as a guide as to how much time to spend on each question.
\section*{Advice}
  • Read each question carefully before you start to answer it.
  • Try to answer every question.
  • Check your answers if you have time at the end.
  • If you change your mind about an answer, cross it out and put your new answer and any working underneath.
\section*{Write your answers in the D1 answer book for this paper.}
Edexcel D1 2023 January Q6
Easy -1.3
6. \section*{Question 6 continued}
\includegraphics[max width=\textwidth, alt={}]{ed8418c4-cdc9-480f-aa09-a16e16933acb-17_1845_1463_296_303}
\section*{Diagram 1} \section*{Question 6 continued} \section*{Question 6 continued} \section*{Question 6 continued} \section*{Pearson Edexcel International Advanced Level} Time 1 hour 30 minutes \section*{Paper reference} \section*{Mathematics} \section*{You must have:} Decision Mathematics Answer Book (enclosed), calculator Candidates may use any calculator allowed by Pearson regulations. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
  • Use black ink or ball-point pen.
  • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used.
  • Fill in the boxes on the top of the answer book with your name, centre number and candidate number.
  • Answer all questions and ensure that your answers to parts of questions are clearly labelled.
  • Answer the questions in the D1 answer book provided - there may be more space than you need.
  • You should show sufficient working to make your methods clear. Answers without working may not gain full credit.
  • When a calculator is used, the answer should be given to an appropriate degree of accuracy.
  • Do not return the question paper with the answer book.
\section*{Information}
  • The total mark for this paper is 75.
  • The marks for each question are shown in brackets
  • use this as a guide as to how much time to spend on each question.
\section*{Advice}
  • Read each question carefully before you start to answer it.
  • Try to answer every question.
  • Check your answers if you have time at the end.
\section*{Write your answers in the D1 answer book for this paper.}
OCR MEI S4 2016 June Q4
4 The cardiovascular unit of a hospital is studying the effect on patients' heart rates of three different light exercises, \(\mathrm { A } , \mathrm { B }\) and C . Patients are given an exercise to do and the increases in their pulse rates are measured after 5 minutes. There are 16 patients in the study: 5 are chosen randomly and allocated to exercise A, 6 to exercise B, and 5 to exercise C. The data obtained are as follows.
ABC
636956
417244
425265
516448
475453
ABC
Sum of data244368266
Sum of squares122242291014410
  1. State the usual one-way analysis of variance model. Explain what the terms in the model mean in this context.
    State the distributional assumptions required for the standard test.
    Carry out the test at the \(5 \%\) level of significance and report your conclusions.
  2. Someone unfamiliar with analysis of variance analysed these data. They used three \(t\) tests to compare A with \(\mathrm { B } , \mathrm { B }\) with C , and C with A . The test comparing A with B was significant at the \(5 \%\) level; the other two tests were not significant at the \(5 \%\) level. Comment on this analysis, explaining whether it is better than, worse than or equivalent to the analysis carried out in part (i). Your comments should include consideration of the independence of the \(t\) tests and the overall level of significance of the procedure.