Questions — CAIE (7279 questions)

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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 \(\sigma ^ { 2 }\).
CAIE FP2 2008 November Q7
7 The pulse rate of each member of a random sample of 25 adult UK males who exercise for a given period each week is measured in beats per minute. A \(98 \%\) confidence interval for the mean pulse rate, \(\mu\) beats per minute, for all such UK males was calculated as \(61.21 < \mu < 64.39\), based on a \(t\)-distribution.
  1. Calculate the sample mean pulse rate and the standard deviation used in the calculation.
  2. State an assumption necessary for the validity of the confidence interval.
  3. The mean pulse rate for all UK males is 72 beats per minute. State, giving a reason, if it can be concluded that, on average, UK males who exercise have a reduced pulse rate.
CAIE FP2 2008 November Q8
8 The equations of the regression lines for a random sample of 25 pairs of data \(( x , y )\) from a bivariate population are $$\begin{array} { c c } y \text { on } x : & y = 1.28 - 0.425 x ,
x \text { on } y : & x = 1.05 - 0.516 y . \end{array}$$
  1. Find the sample means, \(\bar { x }\) and \(\bar { y }\).
  2. Find the product moment correlation coefficient for the sample.
  3. Test at the \(5 \%\) significance level whether the population correlation coefficient differs from zero.
CAIE FP2 2008 November Q9
9 A sample of 100 observations of the continuous random variable \(T\) was obtained and the values are summarised in the following table.
Interval\(1 \leqslant t < 1.5\)\(1.5 \leqslant t < 2\)\(2 \leqslant t < 2.5\)\(2.5 \leqslant t < 3\)
Frequency6417163
It is required to test the goodness of fit of the distribution with probability density function given by $$f ( t ) = \begin{cases} \frac { 9 } { 4 t ^ { 3 } } & 1 \leqslant t < 3
0 & \text { otherwise } \end{cases}$$ The relevant expected values are as follows.
Interval\(1 \leqslant t < 1.5\)\(1.5 \leqslant t < 2\)\(2 \leqslant t < 2.5\)\(2.5 \leqslant t < 3\)
Expected frequency62.521.87510.1255.5
Show how the expected value 10.125 is obtained. Carry out the test, at the \(10 \%\) significance level.
CAIE FP2 2008 November Q10
10 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
\frac { a } { 2 ^ { x } } & x \geqslant 0 , \end{cases}$$ where \(a\) is a positive constant. By expressing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where \(k\) is a constant, show that \(X\) has a negative exponential distribution, and find the value of \(a\). State the value of \(\mathrm { E } ( X )\). The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\). Find the distribution function of \(Y\) and hence find its probability density function.
CAIE FP2 2008 November Q11 EITHER
\includegraphics[max width=\textwidth, alt={}]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-5_976_1043_434_550}
The diagram shows a central cross-section \(C D E F\) of a uniform solid cube of weight \(k W\) with edges of length 4a. The cube rests on a rough horizontal floor. One of the vertical faces of the cube is parallel to a smooth vertical wall and at a distance \(5 a\) from it. A uniform ladder, of length \(10 a\) and weight \(W\), is represented by \(A B\). The ladder rests in equilibrium with \(A\) in contact with the rough top surface of the cube and \(B\) in contact with the wall. The distance \(A C\) is \(a\) and the vertical plane containing \(A B\) is perpendicular to the wall. The coefficients of friction between the ladder and the cube, and between the cube and the floor, are both equal to \(\mu\). A small dog of weight \(\frac { 1 } { 4 } W\) climbs the ladder and reaches the top without the ladder sliding or the cube turning about the edge through \(D\). Show that \(\mu \geqslant \frac { 4 } { 5 }\). Show that the cube does not slide whatever the value of \(k\). Find the smallest possible value of \(k\) for which equilibrium is not broken.
CAIE FP2 2008 November Q11 OR
A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, \(x _ { 1 } \mathrm { ml }\), and from the second machine, \(x _ { 2 } \mathrm { ml }\), are summarised by $$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$ The volumes have distributions with means \(\mu _ { 1 } \mathrm { ml }\) and \(\mu _ { 2 } \mathrm { ml }\) for the first and second machines respectively. Test, at the \(2 \%\) significance level, whether \(\mu _ { 2 }\) is greater than \(\mu _ { 1 }\). Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { 2 } - \mu _ { 1 } > 0.1\).
CAIE FP2 2009 November Q1
1 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 \(120 ^ { \circ }\). Show that the speed of the particle is then \(\sqrt { } \left( \frac { 1 } { 2 } g a \right)\) and find the initial speed of projection.
CAIE FP2 2009 November Q2
2 A circular wheel is modelled as a uniform disc of mass 6 kg and radius 0.25 m . It is rotating with angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) 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.