Questions FP2 (1157 questions)

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CAIE FP2 2013 November Q7
7 A random sample of 10 observations of a normally distributed random variable \(X\) gave the following summarised data, where \(\bar { x }\) denotes the sample mean. $$\Sigma x = 70.4 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 8.48$$ Test, at the \(10 \%\) significance level, whether the population mean of \(X\) is less than 7.5.
CAIE FP2 2013 November Q8
8 The lifetime, in years, of an electrical component is the random variable \(T\), with probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} A \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ where \(A\) and \(\lambda\) are positive constants.
  1. Show that \(A = \lambda\). It is known that out of 100 randomly chosen components, 16 failed within the first year.
  2. Find an estimate for the value of \(\lambda\), and hence find an estimate for the median value of \(T\).
CAIE FP2 2013 November Q9
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 Q10
10 Customers were asked which of three brands of coffee, \(A , B\) and \(C\), they prefer. For a random sample of 80 male customers and 60 female customers, the numbers preferring each brand are shown in the following table.
\(A\)\(B\)\(C\)
Male323612
Female183012
Test, at the \(5 \%\) significance level, whether there is a difference between coffee preferences of male and female customers. A larger random sample is now taken. It consists of \(80 n\) male customers and \(60 n\) female customers, where \(n\) is a positive integer. It is found that the proportions choosing each brand are identical to those in the smaller sample. Find the least value of \(n\) that would lead to a different conclusion for the 5\% significance level hypothesis test.
CAIE FP2 2013 November Q11 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.
CAIE FP2 2013 November Q11 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 FP2 2013 November Q2
2 Three uniform small smooth spheres \(A , B\) and \(C\), of equal radii and of masses \(4 m , \lambda m\) and \(m\) respectively, are at rest in a straight line on a smooth horizontal plane, with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(\frac { 1 } { 2 }\). Show that the speed of \(B\) after it is struck by \(A\) is \(\frac { 6 u } { \lambda + 4 }\). Given that the speed of \(C\) after it is struck by \(B\) is \(u\), find the value of \(\lambda\).
CAIE FP2 2013 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 path of the particle is a complete vertical circle with centre \(O\). When \(P\) is at its lowest point, its speed is \(u\). When \(P\) is at the point \(A\), the tension in the string is \(T\) and the string makes an angle \(\theta\) with the downward vertical, where \(\cos \theta = \frac { 3 } { 5 }\). When \(P\) is at the point \(B\), above the level of \(O\), the tension in the string is \(\frac { 1 } { 8 } T\) and angle \(B O A = 90 ^ { \circ }\). Find \(u\) in terms of \(a\) and \(g\).
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.