Questions — CAIE FP2 (515 questions)

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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 \(ED = \frac{3}{4}a\). 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]
  2. 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. 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 \(\frac{1}{3}m\) 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]
  2. 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 \(A\). 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 \(\frac{3}{4}a\) 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]
  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 \(A\). 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. $$\Sigma x = 72.0 \quad \Sigma x^2 = 542.0$$ 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 +0.3
The continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{1}{4}(x - 1) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
  2. The random variable \(Y\) is defined by \(Y = (X - 1)^3\). Find the probability density function of \(Y\). [4]
  3. 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. $$\Sigma y = 20.2 \quad \Sigma y^2 = 44.6$$ 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]
  2. The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\). Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
  3. 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]
  2. 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
    Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
  3. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]
CAIE FP2 2017 June Q1
5 marks Challenging +1.2
\includegraphics{figure_1} A uniform disc with centre \(O\), mass \(m\) and radius \(a\) is free to rotate without resistance in a vertical plane about a horizontal axis through \(O\). One end of a light inextensible string is attached to the rim of the disc and wrapped around the rim. The other end of the string is attached to a block of mass \(3m\) (see diagram). The system is released from rest with the block hanging vertically. While the block is in motion, it experiences a constant vertical resisting force of magnitude \(0.9mg\). Find the tension in the string in terms of \(m\) and \(g\). [5]
CAIE FP2 2017 June Q2
9 marks Challenging +1.2
A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is \(2.5\) m. The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(OL = 1.5\) m. The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio \(3 : 4\).
  1. Find the distance \(OM\). [2]
The time taken by \(P\) to travel directly from \(L\) to \(M\) is \(2\) s.
  1. Find the period of the motion. [5]
  2. Find the speed of \(P\) when it passes through \(L\). [2]
CAIE FP2 2017 June Q3
10 marks Challenging +1.2
Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). 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 \(\frac{2}{3}\). 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 \(\frac{1}{3}\).
  1. Show that the speed of \(B\) after its collision with the wall is \(\frac{5}{18}u\). [4]
  2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time. [6]
CAIE FP2 2017 June Q4
10 marks Challenging +1.8
\includegraphics{figure_4} A uniform rod \(AB\) of length \(3a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length \(4a\). The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5a\). The string and the rod make angles \(\theta\) and \(2\theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac{3}{5}\) and you may use the fact that \(\cos 2\theta = \frac{7}{25}\).
  1. Find the tension in the string in terms of \(W\). [3]
  2. Find the modulus of elasticity of the string in terms of \(W\). [4]
  3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal. [3]
CAIE FP2 2017 June Q5
10 marks Standard +0.8
\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 particle is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(OP\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(QOP = 90°\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac{4}{5}\).
  1. Show that \(v^2 = u^2 + \frac{14}{5}ag\). [2]
The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).
  1. Obtain another equation relating \(u^2\), \(v^2\), \(a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\). [5]
  2. Find the least tension in the string during the motion. [3]
CAIE FP2 2017 June Q6
5 marks Standard +0.8
The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma^2\). Random samples of \(N\) observations of \(X\) and \(2N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4, \quad \Sigma x^2 = 10, \quad \Sigma y = 8, \quad \Sigma y^2 = 102.$$ These data give a pooled estimate of \(10\) for \(\sigma^2\). Find \(N\). [5]
CAIE FP2 2017 June Q7
6 marks Standard +0.3
A random sample of twelve pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are respectively $$y = 0.46x + 1.62 \quad \text{and} \quad x = 0.93y + 8.24.$$
  1. Find the value of the product moment correlation coefficient for this sample. [2]
  2. Using a \(5\%\) significance level, test whether there is non-zero correlation between the variables. [4]
CAIE FP2 2017 June Q8
9 marks Standard +0.8
The number, \(x\), of beech trees was counted in each of \(50\) randomly chosen regions of equal size in beech forests in country \(A\). The number, \(y\), of beech trees was counted in each of \(40\) randomly chosen regions of the same equal size in beech forests in country \(B\). The results are summarised as follows. $$\Sigma x = 1416 \quad \Sigma x^2 = 41100 \quad \Sigma y = 888 \quad \Sigma y^2 = 20140$$ Find a \(95\%\) confidence interval for the difference between the mean number of beech trees in regions of this size in country \(A\) and in country \(B\). [9]
CAIE FP2 2017 June Q9
12 marks Standard +0.8
The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$ where \(a\) is a positive constant.
  1. Find the value of \(a\). [2]
  2. State the value of E\((X)\). [1]
  3. Find the interquartile range of \(X\). [4]
The variable \(Y\) is related to \(X\) by \(Y = 2^X\).
  1. Find the probability density function of \(Y\). [5]
CAIE FP2 2017 June Q10
12 marks Standard +0.3
Roberto owns a small hotel and offers accommodation to guests. Over a period of \(100\) nights, the numbers of rooms, \(x\), that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.
Number of rooms occupied \((x)\)0123456\(\geqslant 7\)
Number of nights491826201670
  1. Show that the mean number of rooms that are occupied each night is \(3.25\). [1]
The following table shows most of the corresponding expected frequencies, correct to \(2\) decimal places, using a Poisson distribution with mean \(3.25\).
Number of rooms occupied \((x)\)0123456\(\geqslant 7\)
Observed frequency491826201670
Expected frequency3.8812.6020.4822.1818.0211.72
  1. Show how the expected value of \(22.18\), for \(x = 3\), is obtained and find the expected values for \(x = 6\) and for \(x \geqslant 7\). [4]
  2. Use a goodness-of-fit test at the \(5\%\) significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel. [7]
CAIE FP2 2017 June Q11
24 marks Moderate -0.5
Answer only one of the following two alternatives. EITHER \includegraphics{figure_11a} The diagram shows a uniform thin rod \(AB\) of length \(3a\) and mass \(8m\). The end \(A\) is rigidly attached to the surface of a sphere with centre \(O\) and radius \(a\). The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass \(m\) and radius \(a\) surrounded by a thin uniform spherical shell of mass \(m\) and also of radius \(a\). The horizontal axis \(l\) is perpendicular to the rod and passes through the point \(C\) on the rod where \(AC = a\).
  1. Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis \(l\) is \(\frac{289}{15}ma^2\). [6]
The object is free to rotate about the axis \(l\). The object is held so that \(CA\) makes an angle \(\alpha\) with the downward vertical and is released from rest.
  1. Given that \(\cos \alpha = \frac{1}{6}\), find the greatest speed achieved by the centre of the sphere in the subsequent motion. [6]
OR The times taken to run \(200\) metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by \(x\) and the time taken, in seconds, at the end of the year is denoted by \(y\). For a random sample of \(8\) members, the results are shown in the following table.
MemberABCDEFGH
\(x\)24.223.822.825.124.524.023.822.8
\(y\)23.923.622.824.524.223.523.622.7
\([\Sigma x = 191, \quad \Sigma x^2 = 4564.46, \quad \Sigma y = 188.8, \quad \Sigma y^2 = 4458.4, \quad \Sigma xy = 4510.99.]\)
  1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
The athletics coach believes that, on average, the time taken by an athlete to run \(200\) metres decreases between the beginning and the end of the year by more than \(0.2\) seconds.
  1. Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10\%\) significance level. [8]
CAIE FP2 2019 June Q1
4 marks Moderate -0.3
A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m. At time \(t\) seconds, the angle \(POA\) is \(\theta\), where \(\theta = 1 - \cos 2t\), and \(A\) is a fixed point on the arc of the circle.
  1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\) is 6 m s\(^{-2}\). [2]
  2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac{1}{6}\pi\). [2]
CAIE FP2 2019 June Q2
8 marks Challenging +1.2
A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(OA = 3.5\) m and \(OB = 1\) m. The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is 1 m s\(^{-2}\).
  1. Find the speed of \(P\) when it is at \(O\). [4]
  2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\). [4]
CAIE FP2 2019 June Q3
10 marks Standard +0.3
Three uniform small spheres \(A\), \(B\) and \(C\) have equal radii and masses \(2m\), \(4m\) and \(m\) respectively. The spheres are moving in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The coefficient of restitution between each pair of spheres is \(e\). Spheres \(A\) and \(B\) are moving towards each other with speeds \(2u\) and \(u\) respectively. The first collision is between \(A\) and \(B\).
  1. Find the velocities of \(A\) and \(B\) after this collision. [3] Sphere \(C\) is moving towards \(B\) with speed \(\frac{1}{2}u\) and now collides with it. As a result of this collision, \(B\) is brought to rest.
  2. Find the value of \(e\). [4]
  3. Find the total kinetic energy lost by the three spheres as a result of the two collisions. [3]
CAIE FP2 2019 June Q4
10 marks Challenging +1.8
\includegraphics{figure_4} A uniform rod \(AB\) of length \(4a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac{5}{2}a\) has one end attached to the point \(C\) on the rod, where \(AC = \frac{3}{2}a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the rod \(AB\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\). [10]