Questions — CAIE FP2 (515 questions)

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CAIE FP2 2019 June Q5
12 marks Challenging +1.8
\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(kM\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(kM\) and radius \(2a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C\), \(B\), \(A\), \(O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\frac{5}{2}(8k + 3)Ma^2\). [6] The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
  2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\). [6]
CAIE FP2 2019 June Q6
7 marks Moderate -0.3
A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. State the mean value of \(X\). [1]
  2. Find the probability that obtaining a 3 or a 4 takes exactly 6 throws. [1]
  3. Find the probability that obtaining a 3 or a 4 takes more than 4 throws. [2]
  4. Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95. [3]
CAIE FP2 2019 June Q7
8 marks Standard +0.3
The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{3}{4x^2} + \frac{1}{4} & 1 \leqslant x \leqslant 3, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
  2. Find the exact value of the interquartile range of \(X\). [5]
CAIE FP2 2019 June Q8
8 marks Standard +0.3
A large number of runners are attending a summer training camp. A random sample of 6 runners is chosen and their times to run 1500 m at the beginning of the camp and at the end of the camp are recorded. Their times, in minutes, are shown in the following table.
Runner\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
Time at beginning of camp3.823.623.553.713.753.92
Time at end of camp3.723.553.523.683.543.73
The organiser of the training camp claims that a runner's time will improve by more than 0.05 minutes between the beginning and end of the camp. Assuming that differences in time over the two runs are normally distributed, test at the 10% significance level whether the organiser's claim is justified. [8]
CAIE FP2 2019 June Q9
10 marks Standard +0.3
A random sample of 50 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
Interval\(0 \leqslant x < 0.8\)\(0.8 \leqslant x < 1.6\)\(1.6 \leqslant x < 2.4\)\(2.4 \leqslant x < 3.2\)\(3.2 \leqslant x < 4\)
Observed frequency1816862
It is required to test the goodness of fit of the distribution with probability density function f given by $$f(x) = \begin{cases} \frac{3}{16}(4 - x)^{\frac{1}{2}} & 0 \leqslant x < 4, \\ 0 & \text{otherwise}. \end{cases}$$ The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
Interval\(0 \leqslant x < 0.8\)\(0.8 \leqslant x < 1.6\)\(1.6 \leqslant x < 2.4\)\(2.4 \leqslant x < 3.2\)\(3.2 \leqslant x < 4\)
Expected frequency14.2212.5410.598.184.47
  1. Show how the expected frequency for \(1.6 \leqslant x < 2.4\) is obtained. [3]
  2. Carry out a goodness of fit test at the 5% significance level. [7]
CAIE FP2 2019 June Q10
11 marks Standard +0.3
The values from a random sample of five pairs \((x, y)\) taken from a bivariate distribution are shown below.
\(x\)34468
\(y\)57\(q\)67
The equation of the regression line of \(x\) on \(y\) is given by \(x = \frac{5}{4}y + c\).
  1. Given that \(q\) is an integer, find its value. [5]
  2. Find the value of \(c\). [3]
  3. Find the value of the product moment correlation coefficient. [3]
CAIE FP2 2019 June Q11
24 marks Challenging +1.8
Answer only one of the following two alternatives. **EITHER** A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(AOB\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt{\left(\frac{21}{2}ag\right)}\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).
  1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision. [7] In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(OD\) and the upward vertical through \(O\) is \(\theta\).
  2. Find \(\cos \theta\). [5] **OR** A farmer grows two different types of cherries, Type A and Type B. He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type A. He finds that the sample mean mass is 15.1 g and that a 95% confidence interval for the population mean mass, \(\mu\) g, is \(13.5 \leqslant \mu \leqslant 16.7\).
  3. Find an unbiased estimate for the population variance of the masses of cherries of Type A. [3] The farmer now chooses a random sample of 6 cherries of Type B and records their masses as follows. $$12.2 \quad 13.3 \quad 16.4 \quad 14.0 \quad 13.9 \quad 15.4$$
  4. Test at the 5% significance level whether the mean mass of cherries of Type B is less than the mean mass of cherries of Type A. You should assume that the population variances for the two types of cherry are equal. [9]
CAIE FP2 2009 November Q1
5 marks Challenging +1.2
A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120°\). Show that the speed of the particle is then \(\sqrt{\left(\frac{4}{3}ga\right)}\) and find the initial speed of projection. [5]
CAIE FP2 2009 November Q2
7 marks Standard +0.3
A circular wheel is modelled as a uniform disc of mass \(6\) kg and radius \(0.25\) m. It is rotating with angular speed \(2\) rad s\(^{-1}\) about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest \(5\) s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts. [7]
CAIE FP2 2009 November Q3
8 marks Challenging +1.2
Two small smooth spheres \(A\) and \(B\) of equal radius have masses \(m\) and \(3m\) respectively. They lie at rest on a smooth horizontal plane with their line of centres perpendicular to a smooth fixed vertical barrier wall \(9\) feet away from the barrier. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and the barrier, is \(e\), where \(e > \frac{1}{4}\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that after colliding with \(B\) the direction of motion of \(A\) is reversed. [5] After the impact, \(B\) hits the barrier and rebounds. Show that \(B\) will subsequently collide with \(A\) again unless \(e = 1\). [3]
CAIE FP2 2009 November Q4
11 marks Challenging +1.8
A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]
CAIE FP2 2009 November Q5
12 marks Challenging +1.8
\includegraphics{figure_5} Two uniform rods, \(AB\) and \(BC\), each have length \(2a\) and weight \(W\). They are smoothly jointed at \(B\), and \(A\) is attached to a smooth fixed pivot. A light inextensible string of length \((2\sqrt{2})a\) joins \(A\) to \(C\) so that angle \(ABC = 90°\). The system hangs in equilibrium, with \(AB\) making an angle \(\alpha\) with the vertical (see diagram). By taking moments about \(A\) for the system, or otherwise, show that \(\alpha = 18.4°\), correct to the nearest \(0.1°\). [3] Find the tension in the string in the form \(kW\), giving the value of \(k\) correct to 3 significant figures. [3] Find, in terms of \(W\), the magnitude of the force acting on the rod \(BC\) at \(B\). [6]
CAIE FP2 2009 November Q6
6 marks Standard +0.8
A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be \(10\) cm. Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows. 10.03 \quad 10.02 \quad 9.98 \quad 10.06 \quad 10.08 \quad 10.01 Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. [5] Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine. [1]
CAIE FP2 2009 November Q7
8 marks Standard +0.3
A continuous random variable \(X\) has cumulative distribution function F given by $$\mathrm{F}(x) = \begin{cases} 0 & x < -1, \\ \frac{1}{4}(x^3 + 1) & -1 \leqslant x \leqslant 1, \\ 1 & x > 1. \end{cases}$$ Find \(\mathrm{P}\left(X \geqslant \frac{3}{4}\right)\), and state what can be deduced about the upper quartile of \(X\). [3] Obtain the cumulative distribution function of \(Y\), where \(Y = X^2\). [5]
CAIE FP2 2009 November Q8
9 marks Challenging +1.2
150 sheep, chosen from a large flock of sheep, were divided into two groups of 75. Over a fixed period, one group had their grazing controlled and the other group grazed freely. The gains in weight, in kg, were recorded for each animal and the table below shows the sample means and the unbiased estimates of the population variances for the two samples.
Sample meanUnbiased estimate of population variance
Controlled grazing19.1420.54
Free grazing15.369.84
It is required to test whether the population mean for sheep having their grazing controlled exceeds the population mean for sheep grazing freely by less than 5 kg. State, giving a reason, if it is necessary for the validity of the test to assume that the two population variances are equal. [1] Stating any other assumption, carry out the test at the 5\% significance level. [8]
CAIE FP2 2009 November Q9
10 marks Standard +0.3
It has been found that 60\% of the computer chips produced in a factory are faulty. As part of quality control, 100 samples of 4 chips are selected at random, and each chip is tested. The number of faulty chips in each sample is recorded, with the results given in the following table.
Number of faulty chips01234
Number of samples212274910
The expected values for a binomial distribution with parameters \(n = 4\) and \(p = 0.6\) are given in the following table.
Number of faulty chips01234
Expected value2.5615.3634.5634.5612.96
Show how the expected value 34.56 corresponding to 2 faulty chips is obtained. [2] Carry out a goodness of fit test at the 5\% significance level, and state what can be deduced from the outcome of the test. [8]
CAIE FP2 2009 November Q10
10 marks Challenging +1.2
An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20. Find the probability that she requires at least 6 shots to hit the bull's-eye. [3] When she hits the bull's-eye for the third time her total number of shots is \(Y\). Show that $$\mathrm{P}(Y = r) = \frac{1}{2}(r - 1)(r - 2)\left(\frac{3}{20}\right)^3\left(\frac{17}{20}\right)^{r-3}.$$ [3] Simplify \(\frac{\mathrm{P}(Y = r + 1)}{\mathrm{P}(Y = r)}\), and hence find the set of values of \(r\) for which \(\mathrm{P}(Y = r + 1) < \mathrm{P}(Y = r)\). Deduce the most probable value of \(Y\). [4]
CAIE FP2 2009 November Q11
28 marks Standard +0.3
Answer only one of the following two alternatives. EITHER A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt{(gl)}\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot{x} = -\frac{4g}{l}x.$$ [4] Find the speed of \(P\) when the length of the string is \(l\). [4] Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\sqrt{\left(\frac{l}{g}\right)}.$$ [6] OR \includegraphics{figure_11} The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of \(y\) on \(x\). State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. [2] State, giving a reason, whether, for the data shown, the regression line of \(y\) on \(x\) is the same as the regression line of \(x\) on \(y\). [1] A car is travelling along a stretch of road with speed \(v\) km h\(^{-1}\) when the brakes are applied. The car comes to rest after travelling a further distance of \(z\) m. The values of \(z\) (and \(\sqrt{z}\)) for 8 different values of \(v\) are given in the table, correct to 2 decimal places.
\(v\)2530354045505560
\(z\)2.834.634.845.299.7310.3014.8215.21
\(\sqrt{z}\)1.682.152.202.303.123.213.853.90
[\(\sum v = 340\), \(\sum v^2 = 15500\), \(\sum \sqrt{z} = 22.41\), \(\sum z = 67.65\), \(\sum v\sqrt{z} = 1022.15\).]
  1. Calculate the product moment correlation coefficient between \(v\) and \(\sqrt{z}\). What does this indicate about the scatter diagram of the points \((v, \sqrt{z})\)? [4]
  2. Given that the product moment correlation coefficient between \(v\) and \(z\) is 0.965, correct to 3 decimal places, state why the regression line of \(\sqrt{z}\) on \(v\) is more suitable than the regression line of \(z\) on \(v\), and find the equation of the regression line of \(\sqrt{z}\) on \(v\). [5]
  3. Comment, in the context of the question, on the value of the constant term in the equation of the regression line of \(\sqrt{z}\) on \(v\). [2]
CAIE FP2 2010 November Q1
6 marks Standard +0.3
A particle \(P\) is describing simple harmonic motion of amplitude 5 m. Its speed is 6 m s\(^{-1}\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion. [2] Find also
  1. the maximum speed of \(P\), [2]
  2. the magnitude of the maximum acceleration of \(P\). [2]
CAIE FP2 2010 November Q2
6 marks Challenging +1.2
A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac{1}{2}u\) the angle \(\theta\) between \(OP\) and the downward vertical satisfies the equation $$8ga(1 - \cos\theta) = 3u^2.$$ [2] Find, in terms of \(m\), \(u\), \(a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position. [4]
CAIE FP2 2010 November Q3
8 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac{1}{4}u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). [6] Deduce that \(\alpha \geqslant 2\). [2]
CAIE FP2 2010 November Q4
9 marks Challenging +1.8
\includegraphics{figure_4} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(AC\) is inclined at \(2\theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that
  1. the contact force acting on the rod at \(A\) has magnitude \(W\tan\theta\),
  2. the contact force acting on the rod at \(B\) has magnitude \(\frac{W\cos 2\theta}{\cos\theta}\),
  3. \(2r\cos 2\theta = a\cos\theta\).
[9]
CAIE FP2 2010 November Q5
14 marks Challenging +1.8
A uniform circular disc has diameter \(AB\), mass \(2m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac{5}{2}ma^2\). [4] The disc is held with \(AB\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). [5] The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(AB\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position. [5]
CAIE FP2 2010 November Q6
6 marks Standard +0.3
The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at School A is 109. The mean IQ of a random sample of 20 pupils at School B is 112. You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a 90% confidence interval for \(\mu_B - \mu_A\), where \(\mu_A\) and \(\mu_B\) are the population mean IQs. [6]
CAIE FP2 2010 November Q7
7 marks Standard +0.3
The discrete random variable \(X\) has a geometric distribution with mean 4. Find
  1. P\((X = 5)\), [3]
  2. P\((X > 5)\), [2]
  3. the least integer \(N\) such that P\((X \leqslant N) > 0.9995\). [2]