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CAIE FP2 2013 November Q4
4 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\), where \(O E = 5 a\). The particle is pulled down a vertical distance \(\frac { 1 } { 2 } a\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic and state the period of the motion. Find the two possible values of the distance \(O P\) when the speed of \(P\) is equal to one half of its maximum speed.
CAIE FP2 2013 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{38694ab3-44cd-48d1-922a-d5eb09b62826-3_650_698_248_721} Two parallel vertical smooth walls \(E F\) and \(C D\) meet a horizontal plane at \(E\) and \(C\) respectively. A uniform smooth rod \(A B\), of weight \(2 W\) and length \(3 a\), is freely hinged to the horizontal plane at the point \(A\), between \(E\) and \(C\). The end \(B\) rests against \(C D\). A uniform smooth circular disc of weight \(W\) is in contact with the wall \(E F\) at the point \(P\) and with the rod at the point \(Q\). It is given that angle \(B A C\) is \(60 ^ { \circ }\) and that \(A Q = a\) (see diagram). The rod and the disc are in equilibrium in the same vertical plane, which is perpendicular to both walls. Show that
  1. the magnitude of the reaction at \(P\) is \(\sqrt { } 3 W\),
  2. the magnitude of the reaction at \(B\) is \(\frac { 7 \sqrt { } 3 } { 9 } W\). Find, in the form \(k W\), the magnitude of the reaction on \(A B\) at \(A\), giving the value of \(k\) correct to 3 significant figures.
CAIE FP2 2013 November Q6
6 The random variable \(T\) is the time, in suitable units, between two successive arrivals in a hospital casualty department. The probability density function of \(T\) is f , where $$\mathrm { f } ( t ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ State the expected value of \(T\). Write down the distribution function of \(T\) and find \(\mathrm { P } ( T > 10 )\).
CAIE FP2 2013 November Q7
7 Two independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of \(n\) observations of \(X\) and \(2 n\) observations of \(Y\) are taken and the results are summarised by $$\Sigma x = 10.0 , \quad \Sigma x ^ { 2 } = 25.0 , \quad \Sigma y = 15.0 , \quad \Sigma y ^ { 2 } = 43.5 .$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 2 , find the value of \(n\).
CAIE FP2 2013 November Q8
8 A factory produces china mugs. Random samples of size 6 are selected at regular intervals, and the mugs are inspected for defects. During one week, 100 samples are selected and the numbers of defective mugs found are summarised in the following table.
Number of defective mugs0123456
Number of samples1143358210
Fit a binomial distribution to the data and carry out a goodness of fit test at the 5\% significance level.
CAIE FP2 2013 November Q9
9 A random sample of 9 observations of a normally distributed random variable \(X\) gave the following summarised data. $$\Sigma x = 94.5 \quad \Sigma x ^ { 2 } = 993.6$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is 10.2 . Calculate a \(90 \%\) confidence interval for the population mean of \(X\).
CAIE FP2 2013 November Q10
10 The lengths, \(x \mathrm {~m}\), and masses, \(y \mathrm {~kg}\), of 12 randomly chosen babies born at a particular hospital last year are summarised as follows. $$\Sigma x = 7.50 \quad \Sigma x ^ { 2 } = 4.73 \quad \Sigma y = 38.6 \quad \Sigma y ^ { 2 } = 124.84 \quad \Sigma x y = 24.25$$ Find the value of the product moment correlation coefficient for this sample. Obtain an estimate for the mass of a baby, born last year at the hospital, whose length is 0.64 m . Test, at the \(2 \%\) significance level, whether there is non-zero correlation between the two variables.
CAIE FP2 2013 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{38694ab3-44cd-48d1-922a-d5eb09b62826-5_320_831_459_657}
A rigid body consists of a thin uniform rod \(A B\), of mass \(4 m\) and length \(6 a\), joined at \(B\) to a point on the circumference of a uniform circular disc, with centre \(O\), mass \(8 m\) and radius \(2 a\). The point \(C\) on the circumference of the disc is such that \(B C\) is a diameter and \(A B C\) is a straight line (see diagram). The body rotates about a smooth fixed horizontal axis through \(C\), perpendicular to the plane of the disc. The angle between \(C A\) and the downward vertical at time \(t\) is denoted by \(\theta\).
  1. Given that the body is performing small oscillations about the downward vertical, show that the period of these oscillations is approximately \(16 \pi \sqrt { } \left( \frac { a } { 11 g } \right)\).
  2. Given instead that the body is released from rest in the position given by \(\cos \theta = 0.6\), find the maximum speed of \(A\).
CAIE FP2 2013 November Q11 OR
Guided tours of a museum begin every 60 minutes. A randomly chosen tourist arrives \(X\) minutes after the start of a tour. The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { ( x - 20 ) ^ { 2 } } { 24000 } & 0 < x < 60
0 & \text { otherwise } \end{cases}$$ The random variable \(T\) is the time that the tourist has to wait for the next tour to begin. Show that the distribution function G of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 0 & t \leqslant 0
\frac { 8 } { 9 } - \frac { ( 40 - t ) ^ { 3 } } { 72000 } & 0 < t < 60
1 & t \geqslant 60 \end{cases}$$ Find the median and the mean of \(T\).
CAIE FP2 2014 November Q1
1 Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2 m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4 u\) and \(3 u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision.
CAIE FP2 2014 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).
CAIE FP2 2014 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_413_414_1155_863} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(A O B\) is a diameter of this circular cross-section and the radius \(O A\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R _ { A }\) when \(P\) is at \(A\), and is \(R _ { B }\) when \(P\) is at \(B\). It is given that \(R _ { B } = 10 R _ { A }\). Show that \(u ^ { 2 } = a g\). The particle loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\).
CAIE FP2 2014 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-3_513_643_260_749} A uniform rod \(A B\), of length \(l\) and mass \(m\), rests in equilibrium with its lower end \(A\) on a rough horizontal floor and the end \(B\) against a smooth vertical wall. The rod is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring \(C D\) which is in compression in a vertical line with its lower end \(D\) fixed on the floor. The upper end \(C\) is attached to the rod at a distance \(\frac { 1 } { 4 } l\) from \(B\) (see diagram). The coefficient of friction at \(A\) between the rod and the floor is \(\frac { 1 } { 3 }\) and the system is in limiting equilibrium.
  1. Show that the normal reaction of the floor at \(A\) has magnitude \(\frac { 1 } { 2 } m g\) and find the force in the spring.
  2. Given that the modulus of elasticity of the spring is \(2 m g\), find the natural length of the spring.
CAIE FP2 2014 November Q5
5 The points \(A\) and \(B\) are on a smooth horizontal table at a distance \(8 a\) apart. A particle \(P\) of mass \(m\) lies on the table on the line \(A B\), between \(A\) and \(B\). The particle is attached to \(A\) by a light elastic string of natural length \(3 a\) and modulus of elasticity 6 mg , and to \(B\) by a light elastic string of natural length \(2 a\) and modulus of elasticity \(m g\). In equilibrium, \(P\) is at the point \(O\) on \(A B\).
  1. Show that \(A O = 3.6 a\). The particle is released from rest at the point \(C\) on \(A B\), between \(A\) and \(B\), where \(A C = 3.4 a\).
  2. Show that \(P\) moves in simple harmonic motion and state the period.
  3. Find the greatest speed of \(P\).
CAIE FP2 2014 November Q6
6 A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates for the population variances, \(s _ { x } ^ { 2 }\) and \(s _ { y } ^ { 2 }\), respectively, are as follows. $$\bar { x } = 25.4 \quad \bar { y } = 23.6 \quad s _ { x } ^ { 2 } = 23.2 \quad s _ { y } ^ { 2 } = 27.8$$ A test, at the \(\alpha \%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\).
CAIE FP2 2014 November Q7
7 The time, \(T\) seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 100 } \mathrm { e } ^ { - 0.01 t } & t \geqslant 0
0 & \text { otherwise } . \end{cases}$$
  1. State the expected value of \(T\).
  2. Find the median value of \(T\). Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes.
CAIE FP2 2014 November Q8
8 The numbers of a particular type of laptop computer sold by a store on each of 100 consecutive Saturdays are summarised in the following table.
Number sold01234567\(\geqslant 8\)
Number of Saturdays7203916142110
Fit a Poisson distribution to the data and carry out a goodness of fit test at the \(2.5 \%\) significance level.
CAIE FP2 2014 November Q9
9 A random sample of 10 pairs of values of \(x\) and \(y\) is given in the following table.
\(x\)466827121495
\(y\)24686109865
  1. Find the equation of the regression line of \(y\) on \(x\).
  2. Find the product moment correlation coefficient for the sample.
  3. Find the estimated value of \(y\) when \(x = 10\), and comment on the reliability of this estimate.
  4. Another sample of \(N\) pairs of data from the same population has the same product moment correlation coefficient as the first sample given. A test, at the \(1 \%\) significance level, on this second sample indicates that there is sufficient evidence to conclude that there is positive correlation. Find the set of possible values of \(N\).
CAIE FP2 2014 November Q10
10 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find the distribution function of \(Y\). Sketch the graph of the probability density function of \(Y\). Find the probability that \(Y\) lies between its median value and its mean value.
CAIE FP2 2014 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-5_595_522_477_810}
A uniform plane object consists of three identical circular rings, \(X , Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3 m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2 r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(A B\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left( 1 + \frac { 2 } { 3 } \sqrt { } 3 \right) r\).
  1. Show that the moment of inertia of the object about \(A B\) is \(( 7 + 2 \sqrt { } 3 ) m r ^ { 2 }\). The line \(C D\) is the diameter of \(W\) that is perpendicular to \(A B\). A particle of mass \(9 m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(A B\).
  2. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60 ^ { \circ }\).
CAIE FP2 2014 November Q11 OR
Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$\begin{array} { l l l l l l l l } 15.3 & 12.0 & 15.1 & 11.2 & 14.4 & 13.8 & 12.4 & 11.8 \end{array}$$ Assuming a normal distribution, find a \(95 \%\) confidence interval for the mean length of fish in \(A\). The lengths of the 6 fish from \(B\), in the same units, are as follows. $$\begin{array} { l l l l l l } 15.0 & 10.7 & 13.6 & 12.4 & 11.6 & 12.6 \end{array}$$ Stating any assumptions that you make, test at the \(5 \%\) significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). Calculate a 95\% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\).
CAIE FP2 2014 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{699490ab-a01a-46e2-aa7c-3fd48c962c0c-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).
CAIE FP2 2014 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{699490ab-a01a-46e2-aa7c-3fd48c962c0c-2_413_414_1155_863} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(A O B\) is a diameter of this circular cross-section and the radius \(O A\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R _ { A }\) when \(P\) is at \(A\), and is \(R _ { B }\) when \(P\) is at \(B\). It is given that \(R _ { B } = 10 R _ { A }\). Show that \(u ^ { 2 } = a g\). The particle loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\).
CAIE FP2 2014 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{699490ab-a01a-46e2-aa7c-3fd48c962c0c-3_513_643_260_749} A uniform rod \(A B\), of length \(l\) and mass \(m\), rests in equilibrium with its lower end \(A\) on a rough horizontal floor and the end \(B\) against a smooth vertical wall. The rod is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring \(C D\) which is in compression in a vertical line with its lower end \(D\) fixed on the floor. The upper end \(C\) is attached to the rod at a distance \(\frac { 1 } { 4 } l\) from \(B\) (see diagram). The coefficient of friction at \(A\) between the rod and the floor is \(\frac { 1 } { 3 }\) and the system is in limiting equilibrium.
  1. Show that the normal reaction of the floor at \(A\) has magnitude \(\frac { 1 } { 2 } m g\) and find the force in the spring.
  2. Given that the modulus of elasticity of the spring is \(2 m g\), find the natural length of the spring.
CAIE FP2 2014 November Q7
7 The time, \(T\) seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 100 } \mathrm { e } ^ { - 0.01 t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$
  1. State the expected value of \(T\).
  2. Find the median value of \(T\). Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes.