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CAIE FP2 2011 June Q11 EITHER
Challenging +1.8
\includegraphics[max width=\textwidth, alt={}]{020ebd88-b920-40ce-84cf-5c26d45e2935-5_511_508_392_817}
A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.
CAIE FP2 2011 June Q11 OR
Standard +0.8
\includegraphics[max width=\textwidth, alt={}]{020ebd88-b920-40ce-84cf-5c26d45e2935-5_383_839_1635_651}
The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find
  1. the probability density function of \(Y\),
  2. the median value of \(Y\).
CAIE FP2 2011 June Q1
8 marks Standard +0.8
1 Three small spheres, \(A , B\) and \(C\), of masses \(m , k m\) and \(6 m\) respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\) and the coefficient of restitution between \(B\) and \(C\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u\) and is brought to rest by the subsequent collision. Show that \(k = 2\). Given that there are no further collisions after \(B\) has collided with \(C\), show that \(e \leqslant \frac { 1 } { 3 }\).
CAIE FP2 2011 June Q2
11 marks Challenging +1.8
2 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_510_755_667_696} A uniform circular disc with centre \(A\) has mass \(M\) and radius \(3 a\). A second uniform circular disc with centre \(B\) has mass \(\frac { 1 } { 9 } M\) and radius \(a\). The two discs are rigidly joined together so that they lie in the same plane with their circumferences touching. The line of centres meets the circumference of the larger disc at \(P\) and the circumference of the smaller disc at \(O\). A particle of mass \(\frac { 1 } { 3 } M\) is attached at \(P\) (see diagram). Show that the moment of inertia of the system about an axis through \(O\), perpendicular to the plane of the discs, is \(51 M a ^ { 2 }\). The system is free to rotate about a fixed horizontal axis through \(O\), perpendicular to the plane of the discs. The system is held with \(O P\) horizontal and is then released from rest. Given that \(a = 0.5 \mathrm {~m}\), find the greatest speed of \(P\) in the subsequent motion, giving your answer correct to 2 significant figures.
[0pt] [5]
CAIE FP2 2011 June Q3
12 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_419_1102_1859_520} The diagram shows two uniform rods \(B A\) and \(A C\), smoothly hinged at \(A\). The rod \(B A\) has length \(8 a\) and weight \(W\); the rod \(A C\) has length \(6 a\) and weight \(2 W\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) resting on a rough horizontal floor and angle \(C A B\) equal to \(90 ^ { \circ }\). Show that the normal contact force at \(B\) is \(\frac { 26 } { 25 } W\). The coefficient of friction between each rod and the floor is \(\mu\). Find the least possible value of \(\mu\).
CAIE FP2 2011 June Q4
12 marks Challenging +1.8
4 A particle \(P\) of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). When hanging at rest under gravity, \(P\) is given a horizontal velocity of magnitude \(\sqrt { } ( 3 a g )\) and subsequently moves freely in a vertical circle. Show that the tension \(T\) in the string when \(O P\) makes an angle \(\theta\) with the downward vertical is given by $$T = m g ( 1 + 3 \cos \theta )$$ When the string is horizontal, it comes into contact with a small smooth peg \(Q\) which is at the same horizontal level as \(O\) and at a distance \(x\) from \(O\), where \(x < a\). Given that \(P\) completes a vertical circle about \(Q\), find the least possible value of \(x\).
CAIE FP2 2011 June Q5
6 marks Moderate -0.8
5 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.01 \mathrm { e } ^ { - 0.01 x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find the median value of \(X\).
  3. Find the probability that \(X\) lies between the median and the mean.
CAIE FP2 2011 June Q6
7 marks Standard +0.8
6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).
CAIE FP2 2011 June Q7
8 marks Moderate -0.3
7 A fair die is thrown until a 6 appears for the first time. Assuming that the throws are independent, find
  1. the probability that exactly 5 throws are needed,
  2. the probability that fewer than 8 throws are needed,
  3. the least integer \(n\) such that the probability of obtaining a 6 before the \(n\)th throw is at least 0.99 .
CAIE FP2 2011 June Q8
11 marks Standard +0.3
8 A company decides that its employees should follow an exercise programme for 30 minutes each day, with the aim that they lose weight and increase productivity. The weights, in kg , of a random sample of 8 employees at the start of the programme and after following the programme for 6 weeks are shown in the table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Weight before \(( \mathrm { kg } )\)98.687.390.485.2100.592.489.991.3
Weight after \(( \mathrm { kg } )\)93.585.288.284.695.489.386.087.6
Assuming that loss in weight is normally distributed, find a 95\% confidence interval for the mean loss in weight of the company's employees. Test at the \(5 \%\) significance level whether, after the exercise programme, there is a reduction of more than 2.5 kg in the population mean weight.
CAIE FP2 2011 June Q9
11 marks Standard +0.3
9 The marks achieved by a random sample of 15 college students in a Physics examination ( \(x\) ) and in a General Studies examination (y) are summarised as follows. $$\Sigma x = 752 \quad \Sigma x ^ { 2 } = 38814 \quad \Sigma y = 773 \quad \Sigma y ^ { 2 } = 45351 \quad \Sigma x y = 40236$$
  1. Find the mean values, \(\bar { x }\) and \(\bar { y }\).
  2. Another college student achieved a mark of 56 in the General Studies examination, but was unable to take the Physics examination. Use the equation of a suitable regression line to estimate the mark that the student would have obtained in the Physics examination.
  3. Find the product moment correlation coefficient for the given data.
  4. Stating your hypotheses, test at the \(5 \%\) level of significance whether there is a non-zero product moment correlation coefficient between examination marks in Physics and in General Studies achieved by college students.
CAIE FP2 2011 June Q10 EITHER
Challenging +1.8
One end of a light elastic string is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and hangs freely under gravity. In the equilibrium position, the extension of the string is \(d\). Show that the period of small vertical oscillations about the equilibrium position is \(2 \pi \sqrt { } \left( \frac { d } { g } \right)\). The particle is now pulled down and released from rest at a distance \(2 d\) below the equilibrium position. Given that the particle does not reach \(O\) in the subsequent motion, show that the time taken until the particle first comes to instantaneous rest is \(\left( \sqrt { } 3 + \frac { 2 } { 3 } \pi \right) \sqrt { } \left( \frac { d } { g } \right)\).
CAIE FP2 2011 June Q10 OR
Standard +0.3
A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table.
Number of letters012345\(\geqslant 6\)
Number of days57605325410
It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.
Number of letters012345\(\geqslant 6\)
Expected number of days53.96470.693\(p\)\(q\)6.6221.7350.463
  1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
  2. Carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2012 June Q1
7 marks Standard +0.3
1 Two smooth spheres \(A\) and \(B\), of equal radii and of masses \(3 m\) and \(6 m\) respectively, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Show that the kinetic energy lost in the collision between \(A\) and \(B\) is \(m u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).
CAIE FP2 2012 June Q2
7 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length \(a\) and modulus of elasticity \(2 m g\), are attached to a particle \(P\) of mass \(m\). The strings join the particle to the points \(A\) and \(B\) which are fixed and at a distance \(4 a\) apart on a smooth horizontal surface. The particle is at rest at the mid-point \(O\) of \(A B\). The particle is now displaced a small distance in a direction perpendicular to \(A B\), on the surface, and released from rest. At time \(t\), the displacement of \(P\) from \(O\) is \(x\) (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that \(\frac { x } { a }\) is so small that \(\left( \frac { x } { a } \right) ^ { 2 }\) and higher powers may be neglected, show that the motion of \(P\) is approximately simple harmonic and state the period of the motion.
CAIE FP2 2012 June Q3
8 marks Challenging +1.2
3 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 6 \mathrm {~m}\) and \(O B = 8 \mathrm {~m}\), with \(O\) between \(A\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). When \(P\) is at \(A\) its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when \(P\) is at \(B\) its speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is 10 m and find the period of the motion. Find the time taken by \(P\) to travel directly from \(A\) to \(B\), through \(O\).
CAIE FP2 2012 June Q4
11 marks Standard +0.8
4 A smooth sphere, with centre \(O\) and radius \(a\), has its lowest point fixed on a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the highest point on the outer surface of the sphere. In the subsequent motion, \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\). Show that, while \(P\) remains in contact with the sphere, the magnitude of the reaction of the sphere on \(P\) is \(m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the surface of the sphere when \(\theta = \alpha\). Given that \(u = \frac { 1 } { 2 } \sqrt { } ( g a )\), find
  1. \(\cos \alpha\),
  2. the vertical component of the velocity of \(P\) as it strikes the horizontal plane.
CAIE FP2 2012 June Q5
11 marks Challenging +1.8
5 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-3_319_794_255_678} A uniform rod \(A B\), of mass \(m\) and length \(6 a\), is rigidly attached at \(B\) to a point on the circumference of a uniform circular lamina of mass \(m\), radius \(2 a\) and centre \(O\). The lamina and the rod are in the same vertical plane, and \(A B O\) is a straight line (see diagram). Show that the moment of inertia of the system about an axis \(l\) through \(A\) perpendicular to the plane of the lamina is \(78 m a ^ { 2 }\). A particle of mass \(2 m\) is now attached at \(B\) and the system is free to rotate in a vertical plane about the fixed axis \(l\) which is horizontal. Initially \(A B\) is horizontal, with \(O\) moving downwards and the system having angular velocity \(\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)\). At time \(t , A B\) makes an angle \(\theta\) with the downward vertical through \(A\).
  1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }\).
  2. Find the angular velocity of the system when \(B\) is vertically below \(A\).
CAIE FP2 2012 June Q6
7 marks Standard +0.3
6 A random sample of 10 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 8.254 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.912 .$$ Using a \(5 \%\) significance level, test whether the mean of \(X\) is greater than 8.05.
CAIE FP2 2012 June Q7
7 marks Standard +0.3
7 The waiting time, \(T\) minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & t < 0 \end{cases}$$ where \(\lambda\) is a positive constant. The standard deviation of \(T\) is 8 . Find
  1. the value of \(\lambda\),
  2. the probability that a customer has to wait between 5 and 10 minutes before being served,
  3. the median value of \(T\).
CAIE FP2 2012 June Q8
9 marks Standard +0.3
8 Residents of three towns \(A , B\) and \(C\) were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table.
GoodSatisfactoryPoor
Town \(A\)243414
Town \(B\)586026
Town \(C\)203430
Test, at the 2.5\% significance level, whether grade of reliability is independent of town. Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question.
CAIE FP2 2012 June Q9
9 marks Standard +0.3
9 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find the distribution function of \(Y\). Given that \(a = 4\), find the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = 0.25\).
CAIE FP2 2012 June Q10
12 marks Standard +0.3
10 Engineers are investigating the speed of the internet connection received by households in two towns \(P\) and \(Q\). The speeds, in suitable units, in \(P\) and \(Q\) are denoted by \(x\) and \(y\) respectively. For a random sample of 50 houses in town \(P\) and a random sample of 40 houses in town \(Q\) the results are summarised as follows. $$\Sigma x = 240 \quad \Sigma x ^ { 2 } = 1224 \quad \Sigma y = 168 \quad \Sigma y ^ { 2 } = 754$$ Calculate a \(95 \%\) confidence interval for \(\mu _ { P } - \mu _ { Q }\), where \(\mu _ { P }\) and \(\mu _ { Q }\) are the population mean speeds for \(P\) and \(Q\). Test, at the \(1 \%\) significance level, whether \(\mu _ { P }\) is greater than \(\mu _ { Q }\).
CAIE FP2 2012 June Q11 EITHER
Challenging +1.8
\includegraphics[max width=\textwidth, alt={}]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-5_474_796_479_676}
The diagram shows a uniform rod \(A B\), of length \(4 a\) and weight \(W\), resting in equilibrium with its end \(A\) on rough horizontal ground. The rod rests at \(C\) on the surface of a smooth cylinder whose axis is horizontal. The cylinder rests on the ground and is fixed to it. The rod is in a vertical plane perpendicular to the axis of the cylinder and is inclined at an angle \(\theta\) to the horizontal, where \(\cos \theta = \frac { 3 } { 5 }\). A particle of weight \(k W\) is attached to the rod at \(B\). Given that \(A C = 3 a\), show that the least possible value of the coefficient of friction \(\mu\) between the rod and the ground is \(\frac { 8 ( 2 k + 1 ) } { 13 k + 19 }\). Given that \(\mu = \frac { 9 } { 10 }\), find the set of values of \(k\) for which equilibrium is possible.
CAIE FP2 2012 June Q11 OR
Challenging +1.2
For a random sample of 5 pairs of values of \(x\) and \(y\), the equations of the regression lines of \(y\) on \(x\) and \(x\) on \(y\) are respectively $$y = - 0.5 x + 5 \quad \text { and } \quad x = - 1.2 y + 7.6$$ Find the value of the product moment correlation coefficient for this sample. Test, at the \(5 \%\) significance level, whether the population product moment correlation coefficient differs from zero. The following table shows the sample data.
\(x\)1255\(p\)
\(y\)5342\(q\)
Find the values of \(p\) and \(q\).