Questions — CAIE FP2 (474 questions)

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CAIE FP2 2019 November Q10
10 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 30 } \left( \frac { 8 } { x ^ { 2 } } + 3 x ^ { 2 } - 14 \right) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  2. Find the probability density function of \(Y\).
  3. Find the value of \(y\) such that \(\mathrm { P } ( Y < y ) = 0.8\).
CAIE FP2 2019 November Q11 EITHER
The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac { 2 } { 3 } \mathrm {~kg}\) is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N . The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N , has one end attached to \(P\) and the other end attached to \(B\).
  1. Show that when \(P\) is in equilibrium \(A P = 0.75 \mathrm {~m}\).
    The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
  2. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. Find the speed of \(P\) when it passes through the equilibrium position.
  4. Find the speed of \(P\) when its acceleration is equal to half of its maximum value.
CAIE FP2 2019 November Q11 OR
The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
Number of repairs in a week012345\(\geqslant 6\)
Number of weeks61596310
  1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data.
    Records over a longer period of time indicate that the mean number of repairs in a week is 1.6 . The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
    Number of repairs in a week012345\(\geqslant 6\)
    Expected frequency8.07612.92110.3375.5132.205\(a\)\(b\)
  2. Show that \(a = 0.706\) and find the value of the constant \(b\).
  3. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a \(10 \%\) significance level.
    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 November Q1
5 marks
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5]
\includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
4 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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2019 November Q1
5 marks
1 A particle \(P\) is moving in a circle of radius 2 m . At time \(t\) seconds, its velocity is \(( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the radial component of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the transverse component of the acceleration of \(P\) at this instant.
[0pt] [5]
\includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-04_591_805_262_671} A uniform square lamina \(A B C D\) of side \(4 a\) and weight \(W\) rests in a vertical plane with the edge \(A B\) inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(A B\), where \(B E = 3 a\) (see diagram).
  1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\).
  2. Given that the lamina is about to slip, find the value of \(\mu\).
CAIE FP2 2019 November Q4
4 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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.
    \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
CAIE FP2 2017 Specimen Q2
2 A small uniform sphere \(A\), of mass \(2 m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\).
  1. Find expressions for the speeds of \(A\) and \(B\) immediately after the collision.
    Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is 0.4 . After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal.
  2. Find \(e\).
  3. Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\).
    \(3 A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(A B = 3 a \mathrm {~m}\). One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(m g \mathrm {~N}\), is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). One end of a second light elastic string, of natural length \(k a \mathrm {~m}\) and modulus of elasticity \(2 m g \mathrm {~N}\), is attached to \(B\). The other end of this string is attached to \(P\). It is given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(A B\).
CAIE FP2 2017 Specimen Q4
4 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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\).
  1. Show that the least possible value of \(u\) is \(\sqrt { } ( a g )\).
    It is now given that \(u = \sqrt { } ( a g )\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac { 1 } { 4 } m\).
  2. Find the speed of the combined particle immediately after the collision.
    In the subsequent motion, when \(O P\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\).
  3. Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
  4. Find the value of \(\cos \theta\) when the string becomes slack.
CAIE FP2 2017 Specimen Q5
5 A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar { x }\) is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a \(95 \%\) confidence interval for the population mean.
CAIE FP2 2017 Specimen Q6
6 A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\).
  1. Show that the probability that a head is obtained when the coin is tossed once is \(\frac { 2 } { 3 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-11_69_1571_450_328}
  2. Find \(\mathrm { P } ( X = 4 )\).
  3. Find \(\mathrm { P } ( X > 4 )\).
  4. Find the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.999\).
CAIE FP2 2017 Specimen Q7
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
  1. Show that \(Y\) has probability density function given by $$g ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16
    0 & \text { otherwise } \end{cases}$$
  2. Find the median value of \(Y\).
  3. Find the expected value of \(Y\).
CAIE FP2 2017 Specimen Q8
8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.
CAIE FP2 2017 Specimen Q9
9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \quad \Sigma x ^ { 2 } = 29950 \quad \Sigma y = 400 \quad \Sigma y ^ { 2 } = 21226 \quad \Sigma x y = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination.
  1. Estimate the mark that this student would have obtained in the French examination.
  2. Test, at the \(5 \%\) significance level, whether there is non-zero correlation between marks in Mathematics and marks in French.
CAIE FP2 2017 Specimen Q10 EITHER
\includegraphics[max width=\textwidth, alt={}]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-18_598_601_440_772}
An object is formed by attaching a thin uniform rod \(P Q\) to a uniform rectangular lamina \(A B C D\). The lamina has mass \(m\), and \(A B = D C = 6 a , B C = A D = 3 a\). The rod has mass \(M\) and length \(3 a\). The end \(P\) of the rod is attached to the mid-point of \(A B\). The rod is perpendicular to \(A B\) and in the plane of the lamina (see diagram).
  1. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 1 }\), through \(Q\) and perpendicular to the plane of the lamina, is \(3 ( 8 m + M ) a ^ { 2 }\).
  2. Show that the moment of inertia of the object about a smooth horizontal axis \(l _ { 2 }\), through the mid-point of \(P Q\) and perpendicular to the plane of the lamina, is \(\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }\).
  3. Find expressions for the periods of small oscillations of the object about the axes \(l _ { 1 }\) and \(l _ { 2 }\), and verify that these periods are equal when \(m = M\).
CAIE FP2 2017 Specimen Q10 OR
A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$
  1. Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make.
    A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\) 's Crown plants.
  2. Test, at the \(5 \%\) significance level, whether Farmer \(A\) 's claim is justified.
CAIE FP2 2017 June Q4
  1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of \(\operatorname { disc } A\).
    The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\).
  2. Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\).
CAIE FP2 2017 June Q4
  1. Find the tension in the string in terms of \(W\).
  2. Find the modulus of elasticity of the string in terms of \(W\).
  3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal.
    \includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-10_442_442_260_849} A particle 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 is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(O P\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(Q O P = 90 ^ { \circ }\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac { 4 } { 5 }\).
  4. Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 14 } { 5 } a g\).
    The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).
  5. Obtain another equation relating \(u ^ { 2 } , v ^ { 2 } , a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\).
  6. Find the least tension in the string during the motion.
CAIE FP2 2018 June Q5
  1. Show that the moment of inertia of the object about the axis \(l\) is \(180 M a ^ { 2 }\).
  2. Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.
CAIE FP2 2019 June Q4
  1. Find the moment of inertia of the object, consisting of the rod and two spheres, about \(L\).
    The object is pivoted at \(A\) so that it can rotate freely about \(L\). The object is released from rest with the rod making an angle of \(60 ^ { \circ }\) to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is \(\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)\).
  2. Find the value of \(k\).
CAIE FP2 2017 November Q5
  1. Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac { 169 } { 3 } m a ^ { 2 }\).
  2. Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.
CAIE FP2 2018 November Q3
  1. Show that \(u ^ { 2 } = 2 a g\).
  2. Find the maximum tension in the string as the particle moves from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{a092cd45-dc19-476d-adf7-0198fbb2116e-06_543_807_255_669} A uniform rod \(A B\) of length \(2 a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(A C = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45 ^ { \circ }\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\).
  3. Find an expression for \(x\) in terms of \(a\) and \(\mu\).
  4. Hence show that \(\mu \geqslant \frac { 1 } { 3 }\).
  5. Given that \(x = \frac { 3 } { 2 } a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\).
CAIE FP2 2019 November Q5
  1. Show that the moment of inertia of the object about \(L\) is \(\left( \frac { 408 + 7 \lambda } { 12 } \right) M a ^ { 2 }\).
    The period of small oscillations of the object about \(L\) is \(5 \pi \sqrt { } \left( \frac { 2 a } { g } \right)\).
  2. Find the value of \(\lambda\).
CAIE FP2 2017 Specimen Q3
  1. Find the value of \(k\).
  2. The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).