Questions — CAIE FP2 (515 questions)

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CAIE FP2 2019 June Q5
12 marks Challenging +1.8
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 marks Moderate -0.8
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
8 marks Standard +0.3
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 marks Standard +0.3
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
10 marks Standard +0.3
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
12 marks Moderate -0.3
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
Challenging +1.2
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
Challenging +1.2
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
5 marks Challenging +1.2
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
8 marks Challenging +1.2
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 Challenging +1.2
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
10 marks Standard +0.3
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
11 marks Challenging +1.8
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
3 marks Standard +0.8
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
10 marks Standard +0.3
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
13 marks Standard +0.8
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
Challenging +1.8
\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
Challenging +1.2
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 2011 November Q1
4 marks Standard +0.3
1 A particle is moving in a circle of radius 2 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitude of the resultant acceleration of the particle when \(t = 4\).
CAIE FP2 2011 November Q2
7 marks Standard +0.3
2 A particle \(P\) is moving in simple harmonic motion with centre \(O\). When \(P\) is 5 m from \(O\) its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when it is 9 m from \(O\) its speed is \(\frac { 3 } { 5 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is \(\frac { 15 } { 2 } \sqrt { } 2 \mathrm {~m}\). Given that the greatest speed of \(P\) is \(3 \sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(V\).
CAIE FP2 2011 November Q3
9 marks Challenging +1.2
3 A fixed hollow sphere with centre \(O\) has a smooth inner surface of radius \(a\). A particle \(P\) of mass \(m\) is projected horizontally with speed \(2 \sqrt { } ( a g )\) from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical.
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height that \(P\) reaches above the level of \(O\).
CAIE FP2 2011 November Q4
11 marks Standard +0.8
4 Two smooth spheres \(P\) and \(Q\), of equal radius, have masses \(m\) and \(3 m\) respectively. They are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(P\) has speed \(u\) and collides directly with sphere \(Q\) which has speed \(k u\), where \(0 < k < 1\). Sphere \(P\) is brought to rest by the collision. Show that the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 k + 1 } { 3 ( 1 - k ) }\). One third of the total kinetic energy of the spheres is lost in the collision. Show that $$k = \frac { 1 } { 3 } ( 2 \sqrt { } 3 - 3 )$$
CAIE FP2 2011 November Q5
12 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{96b6c92d-6d13-452f-84ec-37c45651b232-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find
  1. the period of small oscillations,
  2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.