Questions — CAIE FP2 (474 questions)

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CAIE FP2 2015 November Q6
6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\). Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\). Find
  1. \(\mathrm { P } ( X = 4 )\),
  2. \(\mathrm { P } ( X > 4 )\),
  3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).
CAIE FP2 2015 November Q7
7 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\). Show that \(Y\) has probability density function given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16
0 & \text { otherwise } \end{cases}$$ Find
  1. the median value of \(Y\),
  2. the expected value of \(Y\).
CAIE FP2 2015 November Q8
8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the \(5 \%\) significance level.
CAIE FP2 2015 November Q9
9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \quad \Sigma x ^ { 2 } = 29950 \quad \Sigma y = 400 \quad \Sigma y ^ { 2 } = 21226 \quad \Sigma x y = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination. Estimate the mark that this student would have obtained in the French examination. Test, at the \(5 \%\) significance level, whether there is non-zero correlation between marks in Mathematics and marks in French.
CAIE FP2 2015 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{13457d19-ee13-4f91-a22f-240c85068f48-5_604_609_434_769}
An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).
CAIE FP2 2015 November Q10 OR
A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$ Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer B's Crown plants. Test, at the \(5 \%\) significance level, whether Farmer A's claim is justified.
CAIE FP2 2015 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{27d3ee31-7c6e-4451-9c3d-aa4cfc0fdb22-2_744_504_255_824} A uniform ladder \(A B\), of length \(3 a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(A C = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(D C\) is in the same vertical plane as the ladder \(A B\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \theta = 2 \tan \alpha\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\).
CAIE FP2 2015 November Q8
8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.
CAIE FP2 2015 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{27d3ee31-7c6e-4451-9c3d-aa4cfc0fdb22-5_604_609_434_769}
An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).
CAIE FP2 2015 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{a8e37fb1-14c7-4004-b186-d607878e200d-2_744_504_255_824} A uniform ladder \(A B\), of length \(3 a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(A C = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(D C\) is in the same vertical plane as the ladder \(A B\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \theta = 2 \tan \alpha\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\).
CAIE FP2 2015 November Q7
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\). Show that \(Y\) has probability density function given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16
0 & \text { otherwise } \end{cases}$$ Find
  1. the median value of \(Y\),
  2. the expected value of \(Y\).
CAIE FP2 2015 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{a8e37fb1-14c7-4004-b186-d607878e200d-5_604_609_434_769}
An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\). Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\). Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).
CAIE FP2 2016 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the magnitude of the maximum acceleration of \(P\),
  2. the number of complete oscillations made by \(P\) in one minute,
  3. the time that \(P\) takes to travel directly from \(A\) to \(C\).
CAIE FP2 2016 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).
  1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
  2. Find the value of \(\alpha\) and the value of \(e\).
CAIE FP2 2016 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.
  1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
  2. Find the numerical value of \(\mu\).
  3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).
CAIE FP2 2016 November Q4
4 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 vertically above \(O\) with the string taut and then projected horizontally with speed \(\sqrt { } \left( \frac { 13 } { 3 } a g \right)\). It begins to move in a vertical circle with centre \(O\). When \(P\) is at its lowest point, it collides with a stationary particle of mass \(\lambda m\). The two particles coalesce.
  1. Show that the speed of the combined particle immediately after the impact is \(\frac { 5 } { \lambda + 1 } \sqrt { } \left( \frac { 1 } { 3 } a g \right)\). In the subsequent motion, the string becomes slack when the combined particle is at a height of \(\frac { 1 } { 3 } a\) above the level of \(O\).
  2. Find the value of \(\lambda\).
  3. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.
CAIE FP2 2016 November Q5
5 The distance, \(X \mathrm {~km}\), completed by a new car before any mechanical fault occurs has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - a x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. The mean value of \(X\) is 10000 . Find
  1. the value of \(a\),
  2. the probability that a new car completes less than 15000 km before any mechanical fault occurs. The probability that a new car completes at least \(d \mathrm {~km}\) before any mechanical fault occurs is 0.75 .
  3. Find the value of \(d\).
CAIE FP2 2016 November Q6
6 A random sample of 8 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is less than 6.44.
CAIE FP2 2016 November Q7
7 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find
  2. the probability density function of \(Y\),
  3. the value of \(k\) for which \(\mathrm { P } ( Y \geqslant k ) = \frac { 7 } { 12 }\).
CAIE FP2 2016 November Q8
8 The amounts spent on the weekly food shopping by families in the big city \(P\) and the small town \(Q\) are to be compared. The amounts spent, in dollars, in \(P\) and \(Q\) are denoted by \(x\) and \(y\) respectively. For a random sample of 60 families in \(P\) and a random sample of 50 families in \(Q\), the amounts are summarised as follows. $$\Sigma x = 9600 \quad \Sigma x ^ { 2 } = 1560000 \quad \Sigma y = 7200 \quad \Sigma y ^ { 2 } = 1052500$$ Assuming a common population variance, find
  1. a pooled estimate for the population variance,
  2. a \(95 \%\) confidence interval for the difference in the population means in \(P\) and \(Q\).
CAIE FP2 2016 November Q9
9 The number of visitors arriving at an art exhibition is recorded for each 10 -minute period of time during the ten hours that it is open on a particular day. The results are as follows.
Number of visitors in a 10 -minute period012345678\(\geqslant 9\)
Number of 10 -minute periods2212811134710
  1. Calculate the mean and variance for this sample and explain whether your answers support a suggestion that a Poisson distribution might be a suitable model for the number of visitors in a 10-minute period.
  2. Use an appropriate Poisson distribution to find the two expected frequencies missing from the following table.
    Number of visitors in
    a 10-minute period
    		012345678\(\geqslant 9\)
    Expected number of
    10 -minute periods
    		1.108.7911.729.386.253.571.791.28
  3. Test, at the \(10 \%\) significance level, the goodness of fit of this Poisson distribution to the data.
CAIE FP2 2016 November Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-6_515_625_411_758}
A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.
  1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
  2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
  3. Show that \(\cos \theta = \frac { 8 } { 21 }\).
CAIE FP2 2016 November Q10 OR
For a random sample, \(A\), 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 \(y = 4.5 + 0.3 x\) and \(x = 3 y - 13\). Four of the five pairs of data are given in the following table.
\(x\)1579
\(y\)5677
Find
  1. the fifth pair of values of \(x\) and \(y\),
  2. the value of the product moment correlation coefficient. A second random sample, \(B\), of 5 pairs of values of \(x\) and \(y\) is summarised as follows. $$\Sigma x = 20 \quad \Sigma x ^ { 2 } = 100 \quad \Sigma y = 17 \quad \Sigma y ^ { 2 } = 69 \quad \Sigma x y = 75$$ The two samples, \(A\) and \(B\), are combined to form a single random sample of size 10 .
  3. Use this combined sample to test, at the \(5 \%\) significance level, whether the population product moment correlation coefficient is different from zero.
CAIE FP2 2016 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the magnitude of the maximum acceleration of \(P\),
  2. the number of complete oscillations made by \(P\) in one minute,
  3. the time that \(P\) takes to travel directly from \(A\) to \(C\).
CAIE FP2 2016 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).
  1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
  2. Find the value of \(\alpha\) and the value of \(e\).