Questions — CAIE (7646 questions)

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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.
CAIE FP2 2011 November Q6
8 marks Standard +0.3
6 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 1 \\ \frac { 1 } { 2 } & 1 \leqslant x \leqslant 3 \\ 0 & x > 3 \end{cases}$$ Find the distribution function of \(X\). The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find
  1. the probability density function of \(Y\),
  2. the expected value and variance of \(Y\).
CAIE FP2 2011 November Q7
11 marks Standard +0.3
7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable \(T\). The probability density function f of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where \(\lambda\) is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .
CAIE FP2 2011 November Q8
11 marks Standard +0.8
8 A sample of 216 observations of the continuous random variable \(X\) was obtained and the results are summarised in the following table.
Interval\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)\(5 \leqslant x < 6\)
Observed frequency13153159107
It is suggested that these results are consistent with a distribution having probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x < 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant. The relevant expected frequencies are given in the following table.
Interval\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)\(5 \leqslant x < 6\)
Expected frequency17\(a\)\(b\)\(c\)91
  1. Show that \(a = 19\) and find the values of \(b\) and \(c\).
  2. Carry out a goodness of fit test at the \(10 \%\) significance level.
CAIE FP2 2011 November Q9
13 marks Standard +0.8
9 A random sample of five metal rods produced by a machine is taken. Each rod is tested for hardness. The results, in suitable units, are as follows. $$\begin{array} { l l l l l } 524 & 526 & 520 & 523 & 530 \end{array}$$ Assuming a normal distribution, calculate a \(95 \%\) confidence interval for the population mean. Some adjustments are made to the machine. Assume that a normal distribution is still appropriate and that the population variance remains unchanged. A second random sample, this time of ten metal rods, is now taken. The results for hardness are as follows. $$\begin{array} { l l l l l l l l l l } 525 & 520 & 522 & 524 & 518 & 520 & 519 & 525 & 527 & 516 \end{array}$$ Stating suitable hypotheses, test at the \(10 \%\) significance level whether there is any difference between the population means before and after the adjustments.
CAIE FP2 2011 November Q10 EITHER
Challenging +1.2
\includegraphics[max width=\textwidth, alt={}]{96b6c92d-6d13-452f-84ec-37c45651b232-5_606_787_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).
CAIE FP2 2011 November Q10 OR
Standard +0.8
The regression line of \(y\) on \(x\) obtained from a random sample of five pairs of values of \(x\) and \(y\) is $$y = 2.5 x - 1.5$$ The data is given in the following table.
\(x\)12426
\(y\)236\(p\)\(q\)
  1. Show that \(p + q = 19\).
  2. Find the values of \(p\) and \(q\).
  3. Determine the value of the product moment correlation coefficient for this sample.
  4. It is later discovered that the values of \(x\) given in the table have each been divided by 10 (that is, the actual values are \(10,20,40,20,60\) ). Without any further calculation, state
    1. the equation of the actual regression line of \(y\) on \(x\),
    2. the value of the actual product moment correlation coefficient.
CAIE FP2 2011 November Q5
12 marks Challenging +1.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-2_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.
CAIE FP2 2011 November Q10 EITHER
Challenging +1.2
\includegraphics[max width=\textwidth, alt={}]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-5_606_787_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).
CAIE FP2 2011 November Q5
12 marks Challenging +1.8
5 \includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_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.
CAIE FP2 2011 November Q10 EITHER
Challenging +1.2
\includegraphics[max width=\textwidth, alt={}]{0d4a352c-4eda-45b4-9284-60c6fc680f02-5_606_789_411_680}
A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find
  1. the value of \(k\),
  2. the horizontal component of the force on \(P\), in terms of \(W\).