Questions — CAIE FP2 (515 questions)

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CAIE FP2 2012 June Q9
10 marks Standard +0.3
A random sample of 8 observations of a normal random variable \(X\) gave the following summarised data, where \(\overline{x}\) denotes the sample mean. $$\Sigma x = 42.5 \quad \Sigma(x - \overline{x})^2 = 15.519$$ Test, at the 5% significance level, whether the population mean of \(X\) is greater than 4.5. [7] Calculate a 95% confidence interval for the population mean of \(X\). [3]
CAIE FP2 2012 June Q10
11 marks Standard +0.8
Random samples of employees are taken from two companies, \(A\) and \(B\). Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.
CappuccinoLatteGround
Company \(A\)605232
Company \(B\)354031
Test, at the 5% significance level, whether coffee preferences of employees are independent of their company. [7] Larger random samples, consisting of \(N\) times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of \(N\) that would lead to the conclusion, at the 1% significance level, that coffee preferences of employees are not independent of their company. [4]
CAIE FP2 2012 June Q11
24 marks Standard +0.3
Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{3l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.
Day12345678
\(x\)1.21.40.91.10.81.00.61.5
\(y\)0.30.40.60.60.250.750.60.35
  1. Calculate the product moment correlation coefficient for this sample. [4]
  2. Stating your hypotheses, test, at the 2.5% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]
Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.
  1. Find the range of possible values of \(N\). [3]
CAIE FP2 2012 June Q1
4 marks Standard +0.3
A circular flywheel of radius \(0.3\) m, with moment of inertia about its axis \(18\) kg m\(^2\), is rotating freely with angular speed \(6\) rad s\(^{-1}\). A tangential force of constant magnitude \(48\) N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to \(2\) rad s\(^{-1}\). [4]
CAIE FP2 2012 June Q2
6 marks Standard +0.8
Two particles, of masses \(3m\) and \(m\), are moving in the same straight line towards each other with speeds \(2u\) and \(u\) respectively. When they collide, the impulse acting on each particle has magnitude \(4mu\). Show that the total loss in kinetic energy is \(\frac{5}{2}mu^2\). [6]
CAIE FP2 2012 June Q3
10 marks Challenging +1.8
A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos\theta)\). [4] Find the speed of \(P\)
  1. when it loses contact with the sphere, [3]
  2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]
CAIE FP2 2012 June Q4
12 marks Challenging +1.8
\(AB\) is a diameter of a uniform circular disc \(D\) of mass \(9m\), radius \(3a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112ma^2\). [5] A particle of mass \(3m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k\sqrt{(ga)}\). Find the value of \(k\), correct to 3 significant figures. [7]
CAIE FP2 2012 June Q5
12 marks Challenging +1.8
\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{14}{5}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]
CAIE FP2 2012 June Q6
6 marks Moderate -0.8
The probability that a particular type of light bulb is defective is \(0.01\). A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E(\(N\)). [1] Find the least value of \(n\) such that P(\(N \leqslant n\)) is greater than \(0.9\). [3]
CAIE FP2 2012 June Q7
8 marks Standard +0.3
A random sample of 8 swimmers from a swimming club were timed over a distance of 100 metres, once in an outdoor pool and once in an indoor pool. Their times, in seconds, are given in the following table.
Swimmer\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Outdoor time66.262.460.865.468.864.365.267.2
Indoor time66.160.360.965.266.463.862.469.8
Assuming a normal distribution, test, at the 5\% significance level, whether there is a non-zero difference between mean time in the outdoor pool and mean time in the indoor pool. [8]
CAIE FP2 2012 June Q8
9 marks Standard +0.3
The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\text{F}(x) = \begin{cases} 1 - e^{-0.016x} & x \geqslant 0, \\ 0 & x < 0, \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. [4] Find
  1. the median distance between two successive flaws, [3]
  2. the probability that there is a distance of at least 50 metres between two successive flaws. [2]
CAIE FP2 2012 June Q9
10 marks Standard +0.3
A random sample of 8 observations of a normal random variable \(X\) gave the following summarised data, where \(\bar{x}\) denotes the sample mean. $$\Sigma x = 42.5 \quad \Sigma(x - \bar{x})^2 = 15.519$$ Test, at the 5\% significance level, whether the population mean of \(X\) is greater than 4.5. [7] Calculate a 95\% confidence interval for the population mean of \(X\). [3]
CAIE FP2 2012 June Q10
11 marks Standard +0.8
Random samples of employees are taken from two companies, \(A\) and \(B\). Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.
CappuccinoLatteGround
Company \(A\)605232
Company \(B\)354031
Test, at the 5\% significance level, whether coffee preferences of employees are independent of their company. [7] Larger random samples, consisting of \(N\) times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of \(N\) that would lead to the conclusion, at the 1\% significance level, that coffee preferences of employees are not independent of their company. [4]
CAIE FP2 2012 June Q11
24 marks Standard +0.3
Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.
Day12345678
\(x\)1.21.40.91.10.81.00.61.5
\(y\)0.30.40.60.60.250.750.60.35
  1. Calculate the product moment correlation coefficient for this sample. [4]
  2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]
Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.
  1. Find the range of possible values of \(N\). [3]
CAIE FP2 2017 June Q1
3 marks Moderate -0.5
A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed 300 m s\(^{-1}\) and emerges horizontally after 0.02 s. There is a constant horizontal resisting force of magnitude 1000 N. Find the speed with which the bullet emerges from the barrier. [3]
CAIE FP2 2017 June Q2
8 marks Challenging +1.8
\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{5}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]
CAIE FP2 2017 June Q3
10 marks Standard +0.8
Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).
  1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision. [3]
Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac{3}{4}\). When the spheres subsequently collide, \(A\) is brought to rest.
  1. Find the value of \(e\). [7]
CAIE FP2 2017 June Q4
10 marks Challenging +1.2
\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{4}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).
  1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]
The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\).
  1. Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]
CAIE FP2 2017 June Q5
12 marks Challenging +1.8
\includegraphics{figure_5} A particle 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 point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{(ag)}\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{5}{4}a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).
  1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt{(ag(1 + 2\cos\alpha))}\). [2]
The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(CBO = 150°\), the tension in the string is the same as it was when the particle was at the point \(A\).
  1. Find the value of \(\cos\alpha\). [10]
CAIE FP2 2017 June Q6
5 marks Moderate -0.3
A fair die is thrown repeatedly until a 6 is obtained.
  1. Find the probability that obtaining a 6 takes no more than four throws. [2]
  2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\)th throw is more than 0.95. [3]
CAIE FP2 2017 June Q7
7 marks Standard +0.8
A farmer grows a particular type of fruit tree. On average, the mass of fruit produced per tree has been 6.2 kg. He has developed a new kind of soil and claims that the mean mass of fruit produced per tree when growing in this new soil has increased. A random sample of 10 trees grown in the new soil is chosen. The masses, \(x\) kg, of fruit produced are summarised as follows. $$\Sigma x = 72.0 \qquad \Sigma x^2 = 542.0$$ Test at the 5% significance level whether the farmer's claim is justified, assuming a normal distribution. [7]
CAIE FP2 2017 June Q8
10 marks Standard +0.8
The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} \frac{1}{4}(x - 1) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise.} \end{cases}$$
  1. Find the distribution function of \(X\). [3]
The random variable \(Y\) is defined by \(Y = (X - 1)^3\).
  1. Find the probability density function of \(Y\). [4]
  2. Find the median value of \(Y\). [3]
CAIE FP2 2017 June Q9
10 marks Challenging +1.2
Two fish farmers \(X\) and \(Y\) produce a particular type of fish. Farmer \(X\) chooses a random sample of 8 of his fish and records the masses, \(x\) kg, as follows. 1.2 \quad 1.4 \quad 0.8 \quad 2.1 \quad 1.8 \quad 2.6 \quad 1.5 \quad 2.0 Farmer \(Y\) chooses a random sample of 10 of his fish and summarises the masses, \(y\) kg, as follows. $$\Sigma y = 20.2 \qquad \Sigma y^2 = 44.6$$ You should assume that both distributions are normal with equal variances. Test at the 10% significance level whether the mean mass of fish produced by farmer \(X\) differs from the mean mass of fish produced by farmer \(Y\). [10]
CAIE FP2 2017 June Q10
11 marks Standard +0.3
A random sample of 5 pairs of values \((x, y)\) is given in the following table.
\(x\)12458
\(y\)75864
  1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
  2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
  3. Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]
CAIE FP2 2017 June Q11
28 marks Standard +0.8
Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(3m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac{3}{5}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4}a\).
  1. Find the value of \(k\). [3]
The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\).
  1. Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
  2. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion. [5]
OR A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.
Number of damaged pots per pack (\(x\))0123456
Frequency486978322210
  1. Show that the mean number of damaged pots per pack in this sample is 1.656. [1]
The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.
Number of damaged pots per pack (\(x\))0123456
Expected frequency36.0182.36\(a\)39.89\(b\)1.740.11
  1. Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
  2. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]