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CAIE FP2 2011 November Q6
6 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 1
\frac { 1 } { 2 } & 1 \leqslant x \leqslant 3
0 & x > 3 \end{cases}$$ Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find
  1. the probability density function of \(Y\),
  2. the expected value and variance of \(Y\).
CAIE FP2 2011 November Q7
7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable \(T\). The probability density function f of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0
\lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where \(\lambda\) is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .
CAIE FP2 2011 November Q8
8 A sample of 216 observations of the continuous random variable \(X\) was obtained and the results are summarised in the following table.
Interval\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)\(5 \leqslant x < 6\)
Observed frequency13153159107
It is suggested that these results are consistent with a distribution having probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x < 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant. The relevant expected frequencies are given in the following table.
Interval\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)\(5 \leqslant x < 6\)
Expected frequency17\(a\)\(b\)\(c\)91
  1. Show that \(a = 19\) and find the values of \(b\) and \(c\).
  2. Carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2011 November Q9
9 A random sample of five metal rods produced by a machine is taken. Each rod is tested for hardness. The results, in suitable units, are as follows. $$\begin{array} { l l l l l } 524 & 526 & 520 & 523 & 530 \end{array}$$ Assuming a normal distribution, calculate a \(95 \%\) confidence interval for the population mean. Some adjustments are made to the machine. Assume that a normal distribution is still appropriate and that the population variance remains unchanged. A second random sample, this time of ten metal rods, is now taken. The results for hardness are as follows. $$\begin{array} { l l l l l l l l l l } 525 & 520 & 522 & 524 & 518 & 520 & 519 & 525 & 527 & 516 \end{array}$$ Stating suitable hypotheses, test at the \(10 \%\) significance level whether there is any difference between the population means before and after the adjustments.
CAIE FP2 2011 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{96b6c92d-6d13-452f-84ec-37c45651b232-5_606_787_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).
CAIE FP2 2011 November Q10 OR
The regression line of \(y\) on \(x\) obtained from a random sample of five pairs of values of \(x\) and \(y\) is $$y = 2.5 x - 1.5$$ The data is given in the following table.
\(x\)12426
\(y\)236\(p\)\(q\)
  1. Show that \(p + q = 19\).
  2. Find the values of \(p\) and \(q\).
  3. Determine the value of the product moment correlation coefficient for this sample.
  4. It is later discovered that the values of \(x\) given in the table have each been divided by 10 (that is, the actual values are \(10,20,40,20,60\) ). Without any further calculation, state
    (a) the equation of the actual regression line of \(y\) on \(x\),
    (b) the value of the actual product moment correlation coefficient.
CAIE FP2 2011 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find
  1. the period of small oscillations,
  2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.
CAIE FP2 2011 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-5_606_787_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).
CAIE FP2 2011 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find
  1. the period of small oscillations,
  2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.
CAIE FP2 2011 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{0d4a352c-4eda-45b4-9284-60c6fc680f02-5_606_789_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).
CAIE FP2 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-2_216_1205_253_470} A rigid body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), the centres of which are rigidly attached to the ends \(A\) and \(B\) of a uniform rod of mass \(3 m\) and length \(10 a\). The discs and the rod are in the same plane and \(O\) is the point on the rod such that \(A O = 4 a\) (see diagram). Show that the moment of inertia of the body about an axis through \(O\) perpendicular to the plane of the discs is \(81 m a ^ { 2 }\).
CAIE FP2 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find
  1. the angular acceleration of the disc,
  2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).
CAIE FP2 2012 November Q3
3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and horizontal and is then released. When the string is vertical, it comes into contact with a small smooth peg \(A\) which is vertically below \(O\) and at a distance \(x ( < a )\) from \(O\). In the subsequent motion, when \(A P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = m g \left( 3 \cos \theta + \frac { 2 x } { a - x } \right)$$ Given that \(P\) completes a vertical circle about \(A\), find the least possible value of \(\frac { x } { a }\).
CAIE FP2 2012 November Q4
4 A particle \(P\) of mass \(2 m\), moving on a smooth horizontal plane with speed \(u\), strikes a fixed smooth vertical barrier. Immediately before the collision the angle between the direction of motion of \(P\) and the barrier is \(60 ^ { \circ }\). The coefficient of restitution between \(P\) and the barrier is \(\frac { 1 } { 3 }\). Show that \(P\) loses two-thirds of its kinetic energy in the collision. Subsequently \(P\) collides directly with a particle \(Q\) of mass \(m\) which is moving on the plane with speed \(u\) towards \(P\). The magnitude of the impulse acting on each particle in the collision is \(\frac { 2 } { 3 } m u ( 1 + \sqrt { 3 } )\).
  1. Show that the speed of \(P\) after this collision is \(\frac { 1 } { 3 } u\).
  2. Find the exact value of the coefficient of restitution between \(P\) and \(Q\).
CAIE FP2 2012 November Q5
5 A particle \(P\) of mass \(m\) lies on a smooth horizontal surface. \(A\) and \(B\) are fixed points on the surface, where \(A B = 10 a\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(8 m g\), joins \(P\) to \(A\). Another light elastic string, of natural length \(4 a\) and modulus of elasticity \(16 m g\), joins \(P\) to \(B\). Show that when \(P\) is in equilibrium, \(A P = 4 a\). The particle is held at rest at the point \(C\) between \(A\) and \(B\) on the line \(A B\) where \(A C = 3 a\). The particle is now released.
  1. Show that the subsequent motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { a } { 2 g } \right)\).
  2. Find the maximum speed of \(P\).
CAIE FP2 2012 November Q6
6 In a skiing resort, for each day during the winter season, the probability that snow will fall on that day is 0.2 , independently of any other day. The first day of the winter season is 1 December. Find, for the winter season,
  1. the probability that the first snow falls on 20 December,
  2. the probability that the first snow falls before 5 December,
  3. the earliest date in December such that the probability that the first snow falls on or before that date is at least 0.95 .
CAIE FP2 2012 November Q7
7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 2 } { 15 } x & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Show that the distribution function G of \(Y\) is given by $$\mathrm { G } ( y ) = \begin{cases} 0 & y < 1
\frac { 1 } { 15 } \left( y ^ { \frac { 2 } { 3 } } - 1 \right) & 1 \leqslant y \leqslant 64
1 & y > 64 \end{cases}$$ Find
  1. the median value of \(Y\),
  2. \(\mathrm { E } ( Y )\).
CAIE FP2 2012 November Q8
8 The yield of a particular crop on a farm is thought to depend principally on the amount of sunshine during the growing season. For a random sample of 8 years, the average yield, \(y\) kilograms per square metre, and the average amount of sunshine per day, \(x\) hours, are recorded. The results are given in the following table.
\(x\)12.210.45.26.311.810.014.22.3
\(y\)159107811126
$$\left[ \Sigma x = 72.4 , \Sigma x ^ { 2 } = 769.9 , \Sigma y = 78 , \Sigma y ^ { 2 } = 820 , \Sigma x y = 761.3 . \right]$$
  1. Find the equation of the regression line of \(y\) on \(x\).
  2. Find the product moment correlation coefficient.
  3. Test, at the \(5 \%\) significance level, whether there is positive correlation between the average yield and the average amount of sunshine per day.
CAIE FP2 2012 November Q9
9 marks
9 The leaves from oak trees growing in two different areas \(A\) and \(B\) are being measured. The lengths, in cm , of a random sample of 7 oak leaves from area \(A\) are $$6.2 , \quad 8.3 , \quad 7.8 , \quad 9.3 , \quad 10.2 , \quad 8.4 , \quad 7.2$$ Assuming that the distribution is normal, find a 95\% confidence interval for the mean length of oak leaves from area \(A\). The lengths, in cm, of a random sample of 5 oak leaves from area \(B\) are $$5.9 , \quad 7.4 , \quad 6.8 , \quad 8.2 , \quad 8.7$$ Making suitable assumptions, which should be stated, test, at the \(5 \%\) significance level, whether the mean length of oak leaves from area \(A\) is greater than the mean length of oak leaves from area \(B\). [9]
CAIE FP2 2012 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{34024618-0ff9-44a1-ac57-d4d7e8a3655e-5_389_702_484_719}
Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$
CAIE FP2 2012 November Q10 OR
A continuous random variable \(X\) is believed to have the probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 10 } \left( 5 x - x ^ { 2 } - 4 \right) & 2 \leqslant x < 4
0 & \text { otherwise } \end{cases}$$ A random sample of 60 observations was taken and these values are summarised in the following grouped frequency table.
Interval\(2 \leqslant x < 2.4\)\(2.4 \leqslant x < 2.8\)\(2.8 \leqslant x < 3.2\)\(3.2 \leqslant x < 3.6\)\(3.6 \leqslant x < 4\)
Observed frequency19171680
The estimated mean, based on the grouped data in the table above, is 2.69 , correct to 2 decimal places. It is decided that a goodness of fit test will only be conducted if the mean predicted from the probability density function is within \(10 \%\) of the estimated mean. Show that this condition is satisfied. The relevant expected frequencies are as follows.
Interval\(2 \leqslant x < 2.4\)\(2.4 \leqslant x < 2.8\)\(2.8 \leqslant x < 3.2\)\(3.2 \leqslant x < 3.6\)\(3.6 \leqslant x < 4\)
Expected frequency15.45616.03214.30410.2723.936
Show how the expected frequency for the interval \(3.2 \leqslant x < 3.6\) is obtained. Carry out the goodness of fit test at the 10\% significance level.
CAIE FP2 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_216_1205_253_470} A rigid body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), the centres of which are rigidly attached to the ends \(A\) and \(B\) of a uniform rod of mass \(3 m\) and length \(10 a\). The discs and the rod are in the same plane and \(O\) is the point on the rod such that \(A O = 4 a\) (see diagram). Show that the moment of inertia of the body about an axis through \(O\) perpendicular to the plane of the discs is \(81 m a ^ { 2 }\).
CAIE FP2 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find
  1. the angular acceleration of the disc,
  2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).
CAIE FP2 2012 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-5_389_702_484_719}
Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$
CAIE FP2 2012 November Q1
1 A particle \(P\) is moving in a circle of radius 1.5 m . At time \(t \mathrm {~s}\) its velocity is \(\left( k - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a positive constant. When \(t = 3\), the magnitudes of the radial and transverse components of the acceleration of \(P\) are equal. Find the possible values of \(k\).