CAIE
FP2
2015
June
Q3
Challenging +1.2
3 A particle moves on a straight line \(A O B\) in simple harmonic motion, where \(A B = 2 a \mathrm {~m}\). The centre of the motion is \(O\) and the particle is instantaneously at rest at \(A\) and \(B\). The point \(M\) is the mid-point of \(O B\). The particle passes through \(M\) moving towards \(O\) and next achieves its maximum speed one second later. Find the period of the motion.
Find the distance of the particle from \(O\) when its speed is equal to one half of its maximum speed.
At an instant 2.5 seconds after the particle passes through \(M\) moving towards \(O\), the distance of the particle from \(O\) is \(\sqrt { } 2 \mathrm {~m}\). Find, in metres, the amplitude of the motion.
CAIE
FP2
2015
June
Q4
6 marks
Challenging +1.8
4
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422}
The diagram shows a central cross-section CDEF of a uniform solid cube of weight \(W\) and with edges of length \(2 a\). The cube rests on a rough horizontal plane. A thin uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(6 a\), is hinged to the plane at \(A\). The rod rests in smooth contact with the cube at \(C\), with angle \(C A D\) equal to \(30 ^ { \circ }\). The rod is in the same vertical plane as \(C D E F\). The coefficient of friction between the plane and the cube is \(\mu\). Given that the system is in equilibrium, show that \(\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3\). [6]
Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).
CAIE
FP2
2015
June
Q5
Challenging +1.8
5
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598}
The end \(B\) of a uniform rod \(A B\), of mass \(3 M\) and length \(4 a\), is rigidly attached to a point on the circumference of a uniform disc. The disc has centre \(O\), mass \(2 M\) and radius \(a\), and \(A B O\) is a straight line. The disc and the rod are in the same vertical plane. A particle \(P\), of mass \(M\), is attached to the rod at a distance \(k a\) from \(A\), where \(k\) is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis \(l\) through \(A\) perpendicular to the plane of the disc, is \(\left( 67 + k ^ { 2 } \right) M a ^ { 2 }\).
The system is free to rotate about \(l\) and performs small oscillations of period \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\). Find the possible values of \(k\).
CAIE
FP2
2015
June
Q7
Standard +0.8
7 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 3.25 x - 4.27\). The sum of the ten \(x\) values is 15.6 and the product moment correlation coefficient for the sample is 0.56 . Find the equation of the regression line of \(x\) on \(y\).
Test, at the \(5 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
CAIE
FP2
2015
June
Q8
Standard +0.8
8 A large number of long jumpers are competing in a national long jump competition. The distances, in metres, jumped by a random sample of 7 competitors are as follows.
$$\begin{array} { l l l l l l l }
6.25 & 7.01 & 5.74 & 6.89 & 7.24 & 5.64 & 6.52
\end{array}$$
Assuming that distances are normally distributed, test, at the \(5 \%\) significance level, whether the mean distance jumped by long jumpers in this competition is greater than 6.2 metres.
The distances jumped by another random sample of 8 long jumpers in this competition are recorded. Using the data from this sample of 8 long jumpers, a \(95 \%\) confidence interval for the population mean, \(\mu\) metres, is calculated as \(5.89 < \mu < 6.75\). Find the unbiased estimates for the population mean and population variance used in this calculation.
CAIE
FP2
2015
June
Q10 OR
Challenging +1.3
The times taken, in hours, by cyclists from two different clubs, \(A\) and \(B\), to complete a 50 km time trial are being compared. The times taken by a cyclist from club \(A\) and by a cyclist from club \(B\) are denoted by \(t _ { A }\) and \(t _ { B }\) respectively. A random sample of 50 cyclists from \(A\) and a random sample of 60 cyclists from \(B\) give the following summarised data.
$$\Sigma t _ { A } = 102.0 \quad \Sigma t _ { A } ^ { 2 } = 215.18 \quad \Sigma t _ { B } = 129.0 \quad \Sigma t _ { B } ^ { 2 } = 282.3$$
Using a 5\% significance level, test whether, on average, cyclists from club \(A\) take less time to complete the time trial than cyclists from club \(B\).
A test at the \(\alpha \%\) significance level shows that there is evidence that the population mean time for cyclists from club \(B\) exceeds the population mean time for cyclists from club \(A\) by more than 0.05 hours. Find the set of possible values of \(\alpha\).
CAIE
FP2
2016
June
Q4
Challenging +1.2
4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\).
Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)
CAIE
FP2
2016
June
Q5
Challenging +1.8
5
\includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813}
A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\).
The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.
CAIE
FP2
2016
June
Q11 OR
Challenging +1.8
Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve \(A\) and measures the wing span, \(x \mathrm {~cm}\), for each bird. She then calculates a \(95 \%\) confidence interval for the population mean wing span, \(\mu \mathrm { cm }\), for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the \(95 \%\) confidence interval is \(25.17 \leqslant \mu \leqslant 26.83\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.
Petra also measures the wing spans of a random sample of 7 birds from nature reserve \(B\). Their wing spans, \(y \mathrm {~cm}\), are as follows.
$$\begin{array} { l l l l l l l }
23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6
\end{array}$$
She believes that the mean wing span of birds found in nature reserve \(A\) is greater than the mean wing span of birds found in nature reserve \(B\). Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the \(10 \%\) significance level, whether there is evidence to support Petra's belief.