Questions — CAIE (7646 questions)

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CAIE FP2 2014 November Q4
11 marks Challenging +1.2
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
12 marks Challenging +1.2
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
5 marks Challenging +1.2
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
6 marks Moderate -0.5
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
9 marks Standard +0.8
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
11 marks Standard +0.8
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
12 marks Challenging +1.2
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
Challenging +1.8
\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
Standard +0.8
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
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-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
10 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-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
11 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-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 Q11 EITHER
Challenging +1.8
\includegraphics[max width=\textwidth, alt={}]{5d40f5b4-e3d4-482c-8d8d-05a01bd3b43f-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 2015 November Q1
9 marks Standard +0.8
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
10 marks Standard +0.8
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
Challenging +1.2
\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
9 marks Standard +0.8
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
9 marks Standard +0.8
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
Challenging +1.2
\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
8 marks Standard +0.3
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
10 marks Challenging +1.8
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
11 marks Challenging +1.8
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
14 marks Challenging +1.2
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
6 marks Moderate -0.5
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
7 marks Standard +0.3
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.