Questions FP2 (1157 questions)

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CAIE FP2 2019 June Q9
9 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.
  2. Carry out a goodness of fit test at the \(5 \%\) significance level.
CAIE FP2 2019 June Q10
10 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.
  2. Find the value of \(c\).
  3. Find the value of the product moment correlation coefficient.
CAIE FP2 2019 June Q11 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 \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) 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 \(4 m\), 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.
    In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(O D\) and the upward vertical through \(O\) is \(\theta\).
  2. Find \(\cos \theta\).
CAIE FP2 2019 June Q11 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, \(\mu \mathrm { g }\), is \(13.5 \leqslant \mu \leqslant 16.7\).
  1. Find an unbiased estimate for the population variance of the masses of cherries of Type \(A\).
    The farmer now chooses a random sample of 6 cherries of Type \(B\) and records their masses as follows.
    12.2
    13.3
    13.9
    14.0
    15.4
    16.4
  2. 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.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2019 June Q1
1 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 \(\theta = 1 - \cos 2 t\), 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 \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-03_65_1573_488_324}
  2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\).
CAIE FP2 2019 June Q2
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 \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the speed of \(P\) when it is at \(O\).
    \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-04_64_1566_492_328}
  2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).
CAIE FP2 2019 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-08_677_812_258_664} A uniform rod \(A B\) of length \(4 a\) 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 \(A C = \frac { 5 } { 2 } a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the \(\operatorname { rod } A B\) 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 \(\mu\).
CAIE FP2 2019 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). 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 \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
    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 \(k\).
CAIE FP2 2019 June Q1
1 A bullet of mass 0.2 kg is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after a time \(T\) seconds with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1200 N . Find \(T\).
CAIE FP2 2019 June Q2
2 A particle \(P\) 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 \(P\) is moving in a complete vertical circle about \(O\). The points \(A\) and \(B\) are on the circle, at opposite ends of a diameter, and such that \(O A\) makes an acute angle \(\alpha\) with the upward vertical through \(O\). The speed of \(P\) as it passes through \(A\) is \(\frac { 3 } { 2 } \sqrt { } ( a g )\). The tension in the string when \(P\) is at \(B\) is four times the tension in the string when \(P\) is at \(A\).
  1. Show that \(\cos \alpha = \frac { 3 } { 4 }\).
  2. Find the tension in the string when \(P\) is at \(B\).
CAIE FP2 2019 June Q3
3 marks
3 Three uniform small spheres \(A , B\) and \(C\) have equal radii and masses \(3 m , m\) and \(m\) respectively. The spheres are at rest 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 \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\).
  1. Find, in terms of \(u\) and \(e\), expressions for the speeds of \(A , B\) and \(C\) after the first two collisions.
  2. Given that \(A\) and \(C\) are moving with equal speeds after these two collisions, find the value of \(e\). [3]
    \includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-08_812_520_260_808} An object consists of two hollow spheres which touch each other, together with a thin uniform \(\operatorname { rod } A B\). The rod passes through small holes in the surfaces of the spheres. The rod is fixed to the spheres so that it passes through the centre of the smaller sphere. The end \(B\) of the rod is at the centre of the larger sphere. The larger sphere has radius \(2 a\) and mass \(M\), the smaller sphere has radius \(a\) and mass \(k M\), and the rod has length \(7 a\) and mass \(5 M\). A fixed horizontal axis \(L\) passes through \(A\) and is perpendicular to \(A B\) (see diagram).
CAIE FP2 2019 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-10_456_684_264_731} A uniform \(\operatorname { rod } A B\) of length \(2 x\) and weight \(W\) rests on the smooth rim of a fixed hemispherical bowl of radius \(a\). The end \(B\) of the rod is in contact with the rough inner surface of the bowl. The coefficient of friction between the rod and the bowl at \(B\) is \(\frac { 1 } { 3 }\). A particle of weight \(\frac { 1 } { 4 } W\) is attached to the end \(A\) of the rod. The end \(B\) is about to slip upwards when \(A B\) is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram).
  1. By resolving parallel to the rod, show that the normal component of the reaction of the bowl on the rod at \(B\) is \(\frac { 3 } { 4 } W\).
  2. Find, in terms of \(W\), the reaction between the rod and the smooth rim of the bowl.
  3. Find \(x\) in terms of \(a\).
CAIE FP2 2019 June Q6
6 The random variable \(T\) is the lifetime, in hours, of a randomly chosen battery of a particular type. It is given that \(T\) has a negative exponential distribution with mean 400 hours.
  1. Write down the probability density function of \(T\).
  2. Find the probability that a battery of this type has a lifetime that is less than 500 hours.
  3. Find the median of the distribution.
CAIE FP2 2019 June Q7
7 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. State the expected value of \(X\).
  2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
  3. Find the probability that fewer than 4 throws are required to obtain a pair of tails.
  4. Find the least integer \(N\) such that the probability of obtaining a pair of tails in fewer than \(N\) throws is more than 0.95 .
CAIE FP2 2019 June Q8
8 Two salesmen, \(A\) and \(B\), work at a company that arranges different types of holidays: self-catering, hotel and cruise. The table shows, for a random sample of 150 holidays, the number of each type arranged by each salesman.
Type of holiday
\cline { 3 - 5 } \multicolumn{2}{|c|}{}Self-cateringHotelCruise
\multirow{2}{*}{Salesman}\(A\)253821
\cline { 2 - 5 }\(B\)282117
Test at the 10\% significance level whether the type of holiday arranged is independent of the salesman.
CAIE FP2 2019 June Q9
9 A farmer grows large amounts of a certain crop. On average, the yield per plant has been 0.75 kg . The farmer has improved the soil in which the crop grows, and she claims that the yield per plant has increased. A random sample of 10 plants grown in the improved soil is chosen. The yields, \(x \mathrm {~kg}\), are summarised as follows. $$\Sigma x = 7.85 \quad \Sigma x ^ { 2 } = 6.19$$
  1. Test at the \(5 \%\) significance level whether the farmer's claim is justified, assuming a normal distribution.
  2. Find a 95\% confidence interval for the population mean yield for plants grown in the new soil.
CAIE FP2 2019 June Q10
4 marks
10 The means and variances for a random sample of 8 pairs of values of \(x\) and \(y\) taken from a bivariate distribution are given in the following table.
MeanVariance
\(x\)3.31253.3086
\(y\)6.73757.9473
The product moment correlation coefficient for the sample is 0.5815 , correct to 4 decimal places.
  1. Find the equation of the regression line of \(y\) on \(x\).
  2. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between \(x\) and \(y\). [4]
  3. Calculate an estimate of \(y\) when \(x = 6.0\) and comment on the reliability of your estimate.
CAIE FP2 2019 June Q11 EITHER
A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P = a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P = \frac { 3 } { 4 } a\).
  1. Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
  2. Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
  3. Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.
CAIE FP2 2019 June Q11 OR
A company produces packets of sweets. Two different machines, \(A\) and \(B\), are used to fill the packets. The manager decides to assess the performance of the two machines. He selects a random sample of 50 packets filled by machine \(A\) and a random sample of 60 packets filled by machine \(B\). The masses of sweets, \(x \mathrm {~kg}\), in packets filled by machine \(A\) and the masses of sweets, \(y \mathrm {~kg}\), in packets filled by machine \(B\) are summarised as follows. $$\Sigma x = 22.4 \quad \Sigma x ^ { 2 } = 10.1 \quad \Sigma y = 28.8 \quad \Sigma y ^ { 2 } = 16.3$$ A test at the \(\alpha \%\) significance level provides evidence that the mean mass of sweets in packets filled by machine \(A\) is less than the mean mass of sweets in packets filled by machine \(B\). Find the set of possible values of \(\alpha\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP2 2008 November Q1
1 A uniform wire, of length \(24 a\) and mass \(m\), is bent into the form of a triangle \(A B C\) with angle \(A B C = 90 ^ { \circ }\), \(A B = 6 a\) and \(B C = 8 a\) (see diagram). Find the moment of inertia of the wire about an axis through \(A\) perpendicular to the plane of the wire.
CAIE FP2 2008 November Q2
2 A small bead \(B\) of mass \(m\) is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius \(a\) and centre \(O\). The highest point of the circle is \(A\). The bead is slightly displaced from rest at \(A\). When angle \(A O B = \theta\), where \(\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\), the force exerted on the bead by the wire has magnitude \(R _ { 1 }\). When angle \(A O B = \pi + \theta\), the force exerted on the bead by the wire has magnitude \(R _ { 2 }\). Show that \(R _ { 2 } - R _ { 1 } = 4 m g\).
CAIE FP2 2008 November Q3
9 marks
3
\includegraphics[max width=\textwidth, alt={}, center]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-2_582_513_1292_815} A uniform disc, of mass \(m\) and radius \(a\), is free to rotate without resistance in a vertical plane about a horizontal axis through its centre. A light inextensible string has one end fixed to the rim of the disc, and is wrapped round the rim. A block of mass \(2 m\) is attached to the other end of the string (see diagram). The system is released from rest with the block hanging vertically. While the block moves it experiences a constant resistance to motion of magnitude \(\frac { 1 } { 10 } m g\). Find the angular acceleration of the disc, and find also the angular speed of the disc when it has turned through one complete revolution.
[0pt] [9]
CAIE FP2 2008 November Q4
4 Two smooth spheres \(A\) and \(B\), of equal radii, have masses 0.1 kg and \(m \mathrm {~kg}\) respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Assume that in the collision \(A\) does not change direction. The speeds of \(A\) and \(B\) after the collision are \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Express \(m\) in terms of \(v _ { A }\) and \(v _ { B }\), and hence show that \(m < 0.25\).
  2. Assume instead that \(m = 0.2\) and that the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find the magnitude of the impulse acting on \(A\) in the collision.
CAIE FP2 2008 November Q5
5 A particle of mass \(m\) moves in a straight line \(A B\) of length \(2 a\). When the particle is at a general point \(P\) there are two forces acting, one in the direction \(\overrightarrow { P A }\) with magnitude \(m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }\) and the other in the direction \(\overrightarrow { P B }\) with magnitude \(m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }\). At time \(t = 0\) the particle is released from rest at the point \(C\), where \(A C = 1.04 a\). At time \(t\) the distance \(A P\) is \(a + x\). Show that the particle moves in approximate simple harmonic motion. Using the approximate simple harmonic motion, find the speed of \(P\) when it first reaches the mid-point of \(A B\) and the time taken for \(P\) to first reach half of this speed.
CAIE FP2 2008 November Q6
6 The independent random variables \(X\) and \(Y\) have normal distributions with the same variance \(\sigma ^ { 2 }\). Samples of 5 observations of \(X\) and 10 observations of \(Y\) are made, and the results are summarised by \(\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36\) and \(\Sigma y ^ { 2 } = 980\). Find a pooled estimate of \(\sigma ^ { 2 }\).