Questions — CAIE FP2 (515 questions)

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CAIE FP2 2018 November Q1
3 marks Moderate -0.5
A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(AB = 6\) m. The period of the motion is \(4\pi\) s. Find the speed of \(P\) when it is 2 m from \(B\). [3]
CAIE FP2 2018 November Q2
9 marks Standard +0.3
Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5m\) and \(2m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2u\). The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) after the collision is \(\frac{1}{7}u(1 + 15e)\) and find an expression for the speed of \(A\). [4]
In the collision, the speed of \(A\) is halved and its direction of motion is reversed.
  1. Find the value of \(e\). [2]
  2. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). [3]
CAIE FP2 2018 November Q3
9 marks Challenging +1.8
\includegraphics{figure_3} A uniform disc, of radius \(a\) and mass \(2M\), is attached to a thin uniform rod \(AB\) of length \(6a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
  1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc. [4]
The object is free to rotate about the axis \(l\). The object is held with \(AB\) horizontal and is released from rest. When \(AB\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac{3}{5}\), the angular speed of the object is \(\sqrt{\left(\frac{2g}{5a}\right)}\).
  1. Find the possible values of \(x\). [5]
CAIE FP2 2018 November Q4
11 marks Challenging +1.8
A uniform rod \(AB\) of length \(4a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(AC = 3a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(ACD = 2\theta\). A particle of weight \(\frac{1}{4}W\) is attached to the rod at \(B\). It is given that \(\tan \theta = \frac{5}{12}\).
  1. Show that the tension in the string is \(\frac{17}{12}W\). [4]
  2. Find the magnitude and direction of the reaction at the hinge. [5]
  3. Given that the natural length of the string is \(2a\), find its modulus of elasticity. [2]
CAIE FP2 2018 November Q5
12 marks Standard +0.8
The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(AB = 2.6\) m. One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(0.6\) N, is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg. One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.62\) N, is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).
  1. Show that \(AE = 1.4\) m. [4]
The particle \(P\) is now displaced slightly from \(E\), along the line \(AB\).
  1. Show that, in the subsequent motion, \(P\) performs simple harmonic motion. [5]
  2. Given that the period of the motion is \(\frac{4}{\pi}\) s, find the value of \(\lambda\). [3]
CAIE FP2 2018 November Q6
6 marks Standard +0.8
The continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{1}{80}\left(3\sqrt{x} - \frac{8}{\sqrt{x}}\right) & 4 \leqslant x \leqslant 16, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
The random variable \(Y\) is defined by \(Y = \sqrt{X}\).
  1. Find the probability density function of \(Y\). [3]
CAIE FP2 2018 November Q7
7 marks Standard +0.3
The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.
  1. Write down the probability density function of \(T\). [1]
  2. Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours. [3]
  3. Find the median value of \(T\). [3]
CAIE FP2 2018 November Q8
9 marks Standard +0.3
The weekly salaries of employees at two large electronics companies, \(A\) and \(B\), are being compared. The weekly salary of an employee from company \(A\) and an employee from company \(B\) are denoted by \(\\)x\( and \)\\(y\) respectively. A random sample of 50 employees from company \(A\) and a random sample of 40 employees from company \(B\) give the following summarised data. $$\Sigma x = 5120 \quad \Sigma x^2 = 531000 \quad \Sigma y = 3760 \quad \Sigma y^2 = 375135$$
  1. The population mean salaries of employees from companies \(A\) and \(B\) are denoted by \(\\)\mu_A\( and \)\\(\mu_B\) respectively. Using a 5\% significance level, test the null hypothesis \(\mu_A = \mu_B\) against the alternative hypothesis \(\mu_A \neq \mu_B\). [8]
  2. State, with a reason, whether any assumptions about the distributions of employees' salaries are needed for the test in part (i). [1]
CAIE FP2 2018 November Q9
10 marks Standard +0.3
There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows. $$1.75 \quad 1.72 \quad 1.62 \quad 1.70 \quad 1.82 \quad 1.75 \quad 1.68 \quad 1.84$$ You may assume that heights of students are normally distributed.
  1. Test, at the 5\% significance level, whether the population mean height of students at this college is greater than 1.70 metres. [7]
  2. Find a 95\% confidence interval for the population mean height of students at this college. [3]
CAIE FP2 2018 November Q10
12 marks Standard +0.8
For a random sample of 10 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = 1.1664 + 0.4604x\). It is given that $$\Sigma x^2 = 1419.98 \quad \text{and} \quad \Sigma y^2 = 439.68.$$ The mean value of \(y\) is 6.24.
  1. Find the equation of the regression line of \(x\) on \(y\). [6]
  2. Find the product moment correlation coefficient. [2]
  3. Test at the 5\% significance level whether there is evidence of positive correlation between the two variables. [4]
CAIE FP2 2018 November Q11
24 marks Challenging +1.8
Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(AOB\) makes an acute angle \(\alpha\) with the vertical, where \(\cos \alpha = \frac{4}{5}\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8 : 9\).
  1. Show that \(u^2 = 4ag\). [6]
  2. Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere. [6]
OR A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x\) cm, of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm. The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.
Interval\(146 \leqslant x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
Observed frequency122352
\(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x < 154\)
6936152
As a first check, the sample is used to calculate an estimate for the mean.
  1. Show that an estimate for the mean from this sample is close to 150 cm. [2]
As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm. The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.
Interval\(x < 147\)\(147 \leqslant x < 148\)\(148 \leqslant x < 149\)\(149 \leqslant x < 150\)
Observed frequency122352
Expected frequency1.248.3230.9459.50
\(150 \leqslant x < 151\)\(151 \leqslant x < 152\)\(152 \leqslant x < 153\)\(153 \leqslant x\)
6936152
59.5030.948.321.24
  1. Show how the expected frequency for \(151 \leqslant x < 152\) is obtained. [3]
  2. Test, at the 5\% significance level, the goodness of fit of the normal distribution to the results. [7]
CAIE FP2 2018 November Q1
3 marks Standard +0.8
The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(OA = 0.1\) m and \(OB = 0.5\) m, with \(B\) between \(O\) and \(A\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion. [3]
CAIE FP2 2018 November Q2
9 marks Standard +0.3
Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) 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 \(\frac{2}{3}\).
  1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
  2. Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]
CAIE FP2 2018 November Q3
9 marks Challenging +1.8
\includegraphics{figure_3} 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, where \(\tan \alpha = \frac{15}{8}\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270°\) from its initial position, with the particle now at the point \(B\). \begin{enumerate}[label=(\roman*)] \item Show that \(u^2 = 2ag\). [5] \item Find the maximum tension in the string as the particle moves from \(A\) to \(B\). [4] \end{enumerate]
CAIE FP2 2018 November Q4
11 marks Challenging +1.2
\includegraphics{figure_4} A uniform rod \(AB\) of length \(2a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(AC = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45°\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\). \begin{enumerate}[label=(\roman*)] \item Find an expression for \(x\) in terms of \(a\) and \(\mu\). [5] \item Hence show that \(\mu \geqslant \frac{1}{3}\). [2] \item Given that \(x = \frac{5}{3}a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\). [4] \end{enumerate]
CAIE FP2 2018 November Q5
11 marks Challenging +1.8
An object is formed from a uniform circular disc, of radius \(2a\) and mass \(3M\), and a uniform rod \(AB\), of length \(4a\) and mass \(kM\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(OBA\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(AB\).
  1. Show that the moment of inertia of the object about the axis \(l\) is \(3Ma^2(26 + k)\). [5]
  2. The object is free to rotate about \(l\). Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4\pi\sqrt{\frac{a}{g}}\), find the value of \(k\). [6]
CAIE FP2 2018 November Q6
6 marks Moderate -0.3
The heights, in metres, of a random sample of 8 trees of a particular type are as follows. 14.2 11.3 10.8 8.4 12.8 11.5 12.1 9.2 Assuming that heights of trees of this type are normally distributed, calculate a 95% confidence interval for the mean height of trees of this type. [6]
CAIE FP2 2018 November Q7
6 marks Challenging +1.2
The continuous random variable \(X\) has distribution function given by $$\text{F}(x) = \begin{cases} 0 & x < 0, \\ \frac{1}{90}(x^2 + x^4) & 0 \leqslant x \leqslant 3, \\ 1 & x > 3. \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^2\).
  1. Find the probability density function of \(Y\). [4]
  2. Find the mean value of \(Y\). [2]
CAIE FP2 2018 November Q8
8 marks Standard +0.3
Lan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value \(p\). The random variable \(X\) denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096.
  1. Show that \(p = 0.2\). [2]
  2. Find the probability that Lan first gets a seat on Monday of the second week in his new job. [2]
  3. Find the least integer \(N\) such that \(\text{P}(X \leqslant N) > 0.9\), and identify the day and the week that corresponds to this value of \(N\). [4]
CAIE FP2 2018 November Q9
11 marks Standard +0.8
For a random sample of 5 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = -4.2 + c\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + dy\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is \(-0.7214\) and the mean value of \(x\) is 7.018. \begin{enumerate}[label=(\roman*)] \item Test at the 5% significance level whether there is evidence of non-zero correlation between the variables. [4] \item Find the values of \(c\) and \(d\). [5] \item Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate. [2] \end{enumerate]
CAIE FP2 2018 November Q10
12 marks Standard +0.8
The number of accidents, \(x\), that occur each day on a motorway are recorded over a period of 40 days. The results are shown in the following table.
Number of accidents0123456\(\geqslant 7\)
Observed frequency358105720
\begin{enumerate}[label=(\roman*)] \item Show that the mean number of accidents each day is 2.95 and calculate the variance for this sample. Explain why these values suggest that a Poisson distribution might fit the data. [3] \item A Poisson distribution with mean 2.95, as found from the data, is used to calculate the expected frequencies, correct to 2 decimal places. The results are shown in the following table.
Number of accidents0123456\(\geqslant 7\)
Observed frequency358105720
Expected frequency2.096.189.118.966.613.901.921.23
Show how the expected frequency of 6.61 for \(x = 4\) is obtained. [2] \item Test at the 5% significance level the goodness of fit of this Poisson distribution to the data. [7] \end{enumerate]
CAIE FP2 2018 November Q11
28 marks Moderate -0.5
Answer only one of the following two alternatives. **EITHER** One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards. \begin{enumerate}[label=(\roman*)] \item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations. [7] \item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m. Find the amplitude of the motion. [3] \item Deduce the value of \(M\). [4] \end{enumerate] **OR** In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. \(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\) A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\). \begin{enumerate}[label=(\roman*)] \item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal. [9] \item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as \(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\), where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim. Find the greatest possible value of \(p\). [5] \end{enumerate]
CAIE FP2 2019 November Q1
5 marks Standard +0.3
A particle \(P\) is moving in a circle of radius 2 m. At time \(t\) seconds, its velocity is \((t - 1)^2\) m s\(^{-1}\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is 8 m s\(^{-2}\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant. [5]
CAIE FP2 2019 November Q2
8 marks Challenging +1.2
\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
  2. Given that the lamina is about to slip, find the value of \(\mu\). [3]
CAIE FP2 2019 November Q3
9 marks Standard +0.8
Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(5m\), \(5m\) and \(3m\) respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).
  1. Show that the speed of \(A\) after its collision with \(B\) is \(\frac{1}{2}u(1 - e)\) and find the speed of \(B\). [3]
Sphere \(B\) now collides with sphere \(C\). Subsequently there are no further collisions between any of the spheres.
  1. Find the set of possible values of \(e\). [6]