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CAIE FP1 2017 Specimen Q11 OR
Standard +0.8
The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\).
  1. Sketch \(C\).
  2. Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\).
  3. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).
CAIE FP2 2009 June Q1
5 marks Challenging +1.2
1 A line \(O P\) of fixed length \(l\) rotates in a plane about the fixed point \(O\). At time \(t = 0\), the line is at the position \(O A\). At time \(t\), angle \(A O P = \theta\) radians and \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta\). Show that, for all \(t\), the magnitude of the acceleration of \(P\) is equal to the magnitude of its velocity.
CAIE FP2 2009 June Q2
7 marks Moderate -0.5
2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is \(2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
CAIE FP2 2009 June Q3
8 marks Challenging +1.8
3 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-2_513_711_890_717} A uniform lamina of mass \(m\) is bounded by concentric circles with centre \(O\) and radii \(a\) and \(2 a\). The lamina is free to rotate about a fixed smooth horizontal axis \(T\) which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about \(T\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). When hanging at rest, with \(O\) vertically below \(T\), the lamina is given an angular speed \(\omega\) about \(T\). The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of \(\omega\).
CAIE FP2 2009 June Q4
11 marks Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre \(O\), radius 0.6 m and weight 36 N . A uniform rod \(A B\), of weight 14 N and length 1 m , rests with \(A\) in contact with the plane and \(B\) in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is \(C\), and \(A , B , C\) and \(O\) lie in the same vertical plane (see diagram). The contacts at \(A , B\) and \(C\) are rough and the system is in equilibrium. By taking moments about \(C\) for the system, show that the magnitude of the normal contact force at \(A\) is 10 N . Show that the magnitudes of the frictional forces at \(A , B\) and \(C\) are equal. The coefficients of friction at \(A , B\) and \(C\) are all equal to \(\mu\). Find the smallest possible value of \(\mu\).
CAIE FP2 2009 June Q5
12 marks Challenging +1.2
5 Two spheres \(A\) and \(B\), of equal radius, have masses \(m _ { 1 }\) and \(m _ { 2 }\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards sphere \(B\) with speed \(u\) and, as a result of the collision, \(A\) is brought to rest. Show that
  1. the speed of \(B\) immediately after the collision cannot exceed \(u\),
  2. \(m _ { 1 } \leqslant m _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, \(B\) hits a smooth vertical wall which is at an angle of \(60 ^ { \circ }\) to the direction of motion of \(B\) (see diagram). In the impact with the wall \(B\) loses \(\frac { 2 } { 3 }\) of its kinetic energy. Find the coefficient of restitution between \(B\) and the wall and show that the direction of motion of \(B\) turns through \(90 ^ { \circ }\).
CAIE FP2 2009 June Q6
6 marks Standard +0.8
6 The times taken by employees in a factory to complete a certain task have a normal distribution with mean \(\mu\) seconds and standard deviation \(\sigma\) seconds, both of which are unknown. Based on a random sample of 20 employees, the symmetric \(95 \%\) confidence interval for \(\mu\) is \(( 481,509 )\). Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
[0pt] [6]
CAIE FP2 2009 June Q7
8 marks Standard +0.3
7 An experiment was carried out to determine how much weedkiller to apply per \(100 \mathrm {~m} ^ { 2 }\) in a large field. Ten \(100 \mathrm {~m} ^ { 2 }\) areas of the field were randomly chosen and sprayed with predetermined volumes of the weedkiller. The volume of the weedkiller is denoted by \(x\) litres and the number of weeds that survived is denoted by \(y\). The results are given in the table.
\(x\)0.100.150.200.250.300.350.400.450.500.55
\(y\)484044353924101396
$$\left[ \Sigma x = 3.25 , \Sigma x ^ { 2 } = 1.2625 , \Sigma y = 268 , \Sigma y ^ { 2 } = 9548 , \Sigma x y = 66.10 . \right]$$ It is given that the product moment correlation coefficient for the data is - 0.951 , correct to 3 decimal places.
  1. Calculate the equation of a suitable regression line, giving a reason for your choice of line.
  2. Estimate the best volume of weedkiller to apply, and comment on the reliability of your estimate.
CAIE FP2 2009 June Q8
8 marks Standard +0.3
8 Part of a research study of identical twins who had been separated at birth involved a random sample of 9 pairs, in which one twin had been raised by the natural parents and the other by adoptive parents. The IQ scores of these twins were measured, with the following results.
Twin pair123456789
IQ of twin raised by natural parents8292115132889511283123
IQ of twin raised by adoptive parents9288115134979410788130
It may be assumed that the difference in IQ scores has a normal distribution. The mean IQ scores of separated twins raised by natural parents and by adoptive parents are denoted by \(\mu _ { N }\) and \(\mu _ { A }\) respectively. Obtain a \(90 \%\) confidence interval for \(\mu _ { N } - \mu _ { A }\). One of the researchers claimed that there was no evidence of a difference between the two population means. State, giving a reason, whether the confidence interval supports this claim.
CAIE FP2 2009 June Q9
9 marks Standard +0.3
9 The proportions of blood types \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) and O in the Australian population are \(38 \% , 10 \% , 3 \%\) and \(49 \%\) respectively. In order to test whether the population in Sydney conforms to these figures, a random sample of 200 residents is selected. The table shows the observed frequencies of these types in the sample.
Blood TypeABABO
Frequency57249110
Carry out a suitable test at the 5\% significance level. Find the smallest sample size that could be used for the test.
CAIE FP2 2009 June Q10
12 marks Challenging +1.2
10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by \(T\) minutes. Show that \(\mathrm { P } ( T > t ) = \mathrm { e } ^ { - 0.8 t }\) and hence show that \(T\) has a negative exponential distribution. Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.
CAIE FP2 2009 June Q11 EITHER
Challenging +1.2
\includegraphics[max width=\textwidth, alt={}]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-5_250_878_808_632}
Two particles \(A\) and \(B\), of equal mass \(m\), are connected by a light elastic string of natural length \(a\) and modulus of elasticity \(4 m g\). Particle \(A\) rests on a rough horizontal table at a distance \(a\) from the edge of the table. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. At time \(t = 0 , B\) is released from rest at \(P\) and falls vertically. At time \(t , B\) has fallen a distance \(x\), without \(A\) slipping (see diagram). Show that $$\ddot { x } = - \frac { g } { a } ( 4 x - a ) .$$ Deduce that, while \(A\) does not slip, \(B\) moves in simple harmonic motion and identify the centre of the motion. Given that the coefficient of friction between \(A\) and the table is \(\frac { 1 } { 3 }\), find the value of \(x\) when \(A\) starts to slip, and the corresponding value of \(t\), expressing this answer in the form \(k \sqrt { } \left( \frac { a } { g } \right)\). Give the value of \(k\) correct to 3 decimal places.
CAIE FP2 2009 June Q11 OR
Standard +0.8
A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by \(x _ { E }\) and \(x _ { W }\) respectively. The sample means \(\bar { x } _ { E }\) and \(\bar { x } _ { W }\), and unbiased estimates of the two population variances, \(s _ { E } ^ { 2 }\) and \(s _ { W } ^ { 2 }\), are as follows. $$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$ The population means on the eastern and western sides are denoted by \(\mu _ { E }\) and \(\mu _ { W }\) respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island. Stating any required assumptions, obtain the rejection region for a test at the \(5 \%\) significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test. Find the largest value of \(a\) for which the samples above provide evidence at the \(5 \%\) significance level that \(\mu _ { E } - \mu _ { W } > a\).
CAIE FP2 2010 June Q1
5 marks Standard +0.3
1 A particle \(P\), of mass 0.2 kg , moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude \(F \mathrm {~N}\). The distance between the end-points of the motion is 0.6 m , and the period of the motion is 0.5 s . Find the greatest value of \(F\) during the motion.
CAIE FP2 2010 June Q2
7 marks Standard +0.8
2 A uniform \(\operatorname { rod } A B\) of weight \(W\) rests in equilibrium with \(A\) in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude \(P\) acting at \(B\) in this vertical plane. The rod makes an angle of \(60 ^ { \circ }\) with the wall, and the force makes an angle of \(30 ^ { \circ }\) with the rod (see diagram). Find the value of \(P\). Find also the set of possible values of the coefficient of friction between the rod and the wall.
CAIE FP2 2010 June Q3
9 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-2_159_707_1443_721} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.
CAIE FP2 2010 June Q4
9 marks Standard +0.3
4
[diagram]
Two coplanar discs, of radii 0.5 m and 0.3 m , rotate about their centres \(A\) and \(B\) respectively, where \(A B = 0.8 \mathrm {~m}\). At time \(t\) seconds the angular speed of the larger disc is \(\frac { 1 } { 2 } t \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find
  1. the angular speed of the smaller disc,
  2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(P A\).
CAIE FP2 2010 June Q5
13 marks Challenging +1.8
5 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-3_378_625_1272_758} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m , g\) and \(\beta\), and hence show that \(\beta = \frac { 1 } { 4 } \pi\). The particle is given a velocity of magnitude \(\sqrt { } ( a g )\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.
CAIE FP2 2010 June Q6
5 marks Moderate -0.3
6 The lifetime, \(X\) days, of a particular insect is such that \(\log _ { 10 } X\) has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime. Find also \(\mathrm { P } ( X \geq 50 )\).
CAIE FP2 2010 June Q7
7 marks Challenging +1.2
7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of \(X\), find the probability that 2 lies between \(X\) and \(4 X\). Find also the expected value of the width of the interval ( \(X , 4 X\) ).
CAIE FP2 2010 June Q8
9 marks Standard +0.8
8 An examination involved writing an essay. In order to compare the time taken to write the essay by students in two large colleges, a sample of 12 students from college \(A\) and a sample of 8 students from college \(B\) were randomly selected. The times, \(t _ { A }\) and \(t _ { B }\), taken for these students to write the essay were measured, correct to the nearest minute, and are summarised by $$n _ { A } = 12 , \quad \Sigma t _ { A } = 257 , \quad \Sigma t _ { A } ^ { 2 } = 5629 , \quad n _ { B } = 8 , \quad \Sigma t _ { B } = 206 , \quad \Sigma t _ { B } ^ { 2 } = 5359$$ Stating any required assumptions, calculate a \(95 \%\) confidence interval for the difference in the population means. State, giving a reason, whether your confidence interval supports the statement that the population means, for the two colleges, are equal.
CAIE FP2 2010 June Q9
9 marks Standard +0.3
9 A set of 20 pairs of bivariate data \(( x , y )\) is summarised by $$\Sigma x = 200 , \quad \Sigma x ^ { 2 } = 2125 , \quad \Sigma y = 240 , \quad \Sigma y ^ { 2 } = 8245 .$$ The product moment correlation coefficient is - 0.992 .
  1. What does the value of the product moment correlation coefficient indicate about a scatter diagram of the data points?
  2. Find the equation of the regression line of \(y\) on \(x\).
  3. The equation of the regression line of \(x\) on \(y\) is \(x = a ^ { \prime } + b ^ { \prime } y\). Find the value of \(b ^ { \prime }\).
CAIE FP2 2010 June Q10
13 marks Standard +0.3
10 Three new flu vaccines, \(A , B\) and \(C\), were tested on 500 volunteers. The vaccines were assigned randomly to the volunteers and 178 received \(A , 149\) received \(B\) and 173 received \(C\). During the following year, 30 of the volunteers given \(A\) caught flu, 29 of the volunteers given \(B\) caught flu, and 16 of the volunteers given \(C\) caught flu. Carry out a suitable test for independence at the 5\% significance level. Without using a statistical test, decide which of the vaccines appears to be most effective.
CAIE FP2 2010 June Q11 EITHER
Challenging +1.8
\includegraphics[max width=\textwidth, alt={}]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-5_538_572_456_788}
A uniform disc, of mass \(4 m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2 a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(O A , O B\) and \(O C\), each of mass \(m\) and length \(2 a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42 m a ^ { 2 }\). The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(A O\) making an angle of \(30 ^ { \circ }\) with the horizontal. Find the angular speed of the wheel when \(A O\) is horizontal. When \(A O\) is horizontal the disc becomes detached from the wheel. Find the angle that \(A O\) makes with the horizontal when the wheel first comes to instantaneous rest.
CAIE FP2 2010 June Q11 OR
Challenging +1.2
The continuous random variable \(T\) has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2 \\ \frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$
  1. Find the distribution function of \(T\), and find also \(\mathrm { P } ( T > 5 )\).
  2. Consecutive independent observations of \(T\) are made until the first observation that exceeds 5 is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding 5. Find \(\mathrm { P } ( N > \mathrm { E } ( N ) )\).
  3. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { T - 1 }\).