Questions — CAIE FP2 (474 questions)

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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 \(k\).
CAIE FP2 2012 November Q2
2 A small bead of mass \(m\) is threaded on a thin smooth wire which forms a circle of radius \(a\). The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass \(4 m\) which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed \(\sqrt { } ( k g a )\). Show that, when the angle between the two parts of the string is \(\theta\), the normal force exerted on the bead by the wire is \(m g ( 3 \cos \theta + k - 6 )\), towards the centre. Given that the bead reaches the highest point of the wire, find an inequality which must be satisfied by \(k\).
CAIE FP2 2012 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{bcd7ee99-e382-4cb6-aa39-d8b385b01319-2_506_623_977_760} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\) and mass \(m\), are smoothly hinged at \(B\). They rest in equilibrium with \(C\) in contact with a smooth vertical wall and \(A\) in contact with a rough horizontal floor. The rods are in a vertical plane perpendicular to the wall. The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) respectively with the horizontal (see diagram). Show that
  1. the reaction at \(C\) has magnitude \(\frac { 1 } { 2 } m g \cot \beta\),
  2. \(\tan \alpha = 3 \tan \beta\). The coefficient of friction at \(A\) is \(\mu\). Given that \(\alpha = 60 ^ { \circ }\), find the least possible value of \(\mu\).
CAIE FP2 2012 November Q4
4 Three particles \(A , B\) and \(C\) have masses \(m , 2 m\) and \(m\) respectively. The particles are able to move on a smooth horizontal surface in a straight line, and \(B\) is between \(A\) and \(C\). Initially \(A\) is moving towards \(B\) with speed \(2 u\) and \(C\) is moving towards \(B\) with speed \(u\). The particle \(B\) is at rest. The coefficient of restitution between any pair of particles is \(e\). The first collision is between \(A\) and \(B\).
  1. Show that the speed of \(B\) immediately before its collision with \(C\) is \(\frac { 2 } { 3 } u ( 1 + e )\).
  2. Find the velocity of \(B\) immediately after its collision with \(C\).
  3. Given that \(e > \frac { 1 } { 2 }\), show that there are no further collisions between the particles.
CAIE FP2 2012 November Q5
5 Four identical uniform rods, each of mass \(m\) and length \(2 a\), are rigidly joined to form a square frame \(A B C D\). Show that the moment of inertia of the frame about an axis through \(A\) perpendicular to the plane of the frame is \(\frac { 40 } { 3 } m a ^ { 2 }\). The frame is suspended from \(A\) and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through \(A\). When the frame is at rest with \(C\) vertically below \(A\), it is given an angular velocity \(\sqrt { } \left( \frac { 6 g } { 5 a } \right)\). Find the angular velocity of the frame when \(A C\) makes an angle \(\theta\) with the downward vertical through \(A\). When \(A C\) is horizontal, the speed of \(C\) is \(k \sqrt { } ( g a )\). Find the value of \(k\) correct to 3 significant figures.
CAIE FP2 2012 November Q6
6 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 6 } \mathrm { e } ^ { - \frac { 1 } { 6 } x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$ Find
  1. the distribution function of \(X\),
  2. the probability that \(X\) lies between the median and the mean.
CAIE FP2 2012 November Q7
7 The speed \(v\) at which a javelin is thrown by an athlete is measured in \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results for 10 randomly chosen throws are summarised by $$\Sigma v = 1110.8 , \quad \Sigma ( v - \bar { v } ) ^ { 2 } = 333.9$$ where \(\bar { v }\) is the sample mean.
  1. Stating any necessary assumption, calculate a \(99 \%\) confidence interval for the mean speed of a throw. The results for a further 5 randomly chosen throws are now combined with the above results. It is found that the sample variance is smaller than that used in part (i).
  2. State, with reasons, whether a \(95 \%\) confidence interval calculated from the combined 15 results will be wider or less wide than that found in part (i).
CAIE FP2 2012 November Q8
8 Drinking glasses are sold in packs of 4. The manufacturer conducts a survey to assess the quality of the glasses. The results from a sample of 50 randomly chosen packs are summarised in the following table.
Number of perfect glasses01234
Number of packs13101719
Fit a binomial distribution to the data and carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2012 November Q9
9 Experiments are conducted to test the breaking strength of each of two types of rope, \(P\) and \(Q\). A random sample of 50 ropes of type \(P\) and a random sample of 70 ropes of type \(Q\) are selected. The breaking strengths, \(p\) and \(q\), measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$ Test, at the \(10 \%\) significance level, whether the mean breaking strengths of type \(P\) and type \(Q\) ropes are the same.
CAIE FP2 2012 November Q10
10 Delegates who travelled to a conference were asked to report the distance, \(y \mathrm {~km}\), that they had travelled and the time taken, \(x\) minutes. The values reported by a random sample of 8 delegates are given in the following table.
Delegate\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(x\)90467298526510582
\(y\)90556985455011074
$$\left[ \Sigma x = 610 , \Sigma x ^ { 2 } = 49682 , \Sigma y = 578 , \Sigma y ^ { 2 } = 45212 , \Sigma x y = 47136 . \right]$$ Find the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\). Estimate the time taken by a delegate who travelled 100 km to the conference. Calculate the product moment correlation coefficient for this sample.