科目：进阶数学 Further Mathematics (9231)
年份：2019 冬季
试卷：Paper 23
总页数：23
章节：9231 Further Mathematics
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1 A particle $P$ is moving in a circle of radius 2 m. At time $t$ seconds, its velocity is $(t - 1)^2 \text{ m s}^{-1}$. At a particular time $T$ seconds, where $T > 0$, the magnitude of the radial component of the acceleration of $P$ is 8 m s$^{-2}$. Find the magnitude of the transverse component of the acceleration of $P$ at this instant.
[5]A uniform square lamina $ABCD$ of side $4a$ and weight $W$ rests in a vertical plane with the edge $AB$ 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 $AB$, where $BE = 3a$ (see diagram).
(i) Find expressions in terms of $W$ for the normal reaction forces at $E$ and $B$. [5](ii) Given that the lamina is about to slip, find the value of $\mu$. [3]
3 Three uniform small spheres $A, B$ and $C$ have equal radii and masses $5m, 5m$ and $3m$ respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, 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 $u$.
(i) Show that the speed of $A$ after its collision with $B$ is $\frac{1}{2}u(1 - e)$ and find the speed of $B$.
Sphere $B$ now collides with sphere $C$. Subsequently there are no further collisions between any of the spheres.
(ii) Find the set of possible values of $e$. [6]7 © UCLES 2019 9231/23/O/N/19 [Turn over]
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 $\sqrt{2ag}$ so that it begins to move along a circular path. The string becomes slack when $OP$ makes an angle $\theta$ with the upward vertical through $O$.
(i) Show that $\cos \theta = \frac{2}{3}$. [5](ii) Find the greatest height, above the horizontal through $O$, reached by $P$ in its subsequent motion. [4]
A thin uniform rod $AB$ has mass $\lambda M$ and length $2a$. The end $A$ of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre $O$, mass $3M$ and radius $a$. The end $B$ of the rod is rigidly attached to the surface of a uniform solid sphere with centre $C$, mass $5M$ and radius $a$. The rod lies along the line joining the centres of the spheres, so that $CBAO$ 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 $L$.
(i) Show that the moment of inertia of the object about $L$ is $\left(\frac{408 + 7\lambda}{12}\right)Ma^2$. [6]The period of small oscillations of the object about $L$ is $5\pi \sqrt{\left(\frac{2a}{g}\right)}$.
(ii) Find the value of $\lambda$. [6]6 A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, $\mu$ metres, is calculated from the data as $1.65 \leq \mu \leq 1.85$. (i) Find an unbiased estimate for the population variance. [3] (ii) Denoting the height of a member of the club by $x$ metres, find $\Sigma x^2$ for this sample of 9 members. [4]
7 The time, $T$ days, before an electrical component develops a fault has distribution function $F$ given by
$$F(t) = \begin{cases}
1 - e^{-at} & t \geqslant 0, \\
0 & \text{otherwise},
\end{cases}$$
where $a$ is a positive constant. The mean value of $T$ is 200.
(i) Write down the value of $a$. [1]
(ii) Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]
A piece of equipment contains $n$ of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the $n$ components has not developed a fault is greater than 0.99.
(iii) Find the smallest possible value of $n$. [4]8 A random sample of 8 elephants from region $A$ is taken and their weights, $x$ tonnes, are recorded. (1 tonne = 1000 kg.) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x^2 = 131.82$$ A random sample of 10 elephants from region $B$ is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of $0.1555$ tonnes$^2$. The distributions of the weights of elephants in regions $A$ and $B$ are both assumed to be normal with the same population variance. Test at the 10% significance level whether the mean weight of elephants in region $A$ is the same as the mean weight of elephants in region $B$. [9]
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9 A random sample of five pairs of values of $x$ and $y$ is taken from a bivariate distribution. The values are shown in the following table, where $p$ and $q$ are constants. | $x$ | 1 | 2 | 3 | 4 | 5 | | :--- | :--- | :--- | :--- | :--- | :--- | | $y$ | 4 | $p$ | $q$ | 2 | 1 | The equation of the regression line of $y$ on $x$ is $y = -0.5x + 3.5$. (i) Find the values of $p$ and $q$. [7]
(ii) Find the value of the product moment correlation coefficient. [3]
10 The random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
\frac{1}{30}\left(\frac{8}{x^2} + 3x^2 - 14\right) & 2 \leqslant x \leqslant 4, \\
0 & \text{otherwise.}
\end{cases}$$
(i) Find the distribution function of $X$. [3]
The random variable $Y$ is defined by $Y = X^2$.
(ii) Find the probability density function of $Y$. [4](iii) Find the value of $y$ such that $P(Y < y) = 0.8$. [3]
11 Answer only one of the following two alternatives.
EITHER
The points $A$ and $B$ are a distance 1.2 m apart on a smooth horizontal surface. A particle $P$ of mass $\frac{2}{3}$ 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 $B$.
(i) Show that when $P$ is in equilibrium $AP = 0.75$ m. [3]
The particle $P$ is displaced by 0.05 m from the equilibrium position towards $A$ and then released from rest.
(ii) Show that $P$ performs simple harmonic motion and state the period of the motion. [6](iii) Find the speed of $P$ when it passes through the equilibrium position. [2] (iv) Find the speed of $P$ when its acceleration is equal to half of its maximum value. [3]
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 week | 0 | 1 | 2 | 3 | 4 | 5 | ≥6 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Number of weeks | 6 | 15 | 9 | 6 | 3 | 1 | 0 | (i) 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. [3] 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 week | 0 |
(iii) Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]
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