科目：进阶数学 Further Mathematics (9231)
年份：2011 冬季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
1. A particle is moving in a circle of radius $2\text{ m}$. At time $t$ its velocity is $(t^2 - 12)\text{ m s}^{-1}$. Find the magnitude of the resultant acceleration of the particle when $t = 4$.
2. A particle $P$ is moving in simple harmonic motion with centre $O$. When $P$ is $5\text{ m}$ from $O$ its speed is $V\text{ m s}^{-1}$, and when it is $9\text{ m}$ from $O$ its speed is $\frac{2}{5}V\text{ m s}^{-1}$. Show that the amplitude of the motion is $\frac{15}{2}\sqrt{2}\text{ m}$.
Given that the greatest speed of $P$ is $3\sqrt{2}\text{ m s}^{-1}$, find $V$.
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 $2\sqrt{(ag)}$ from the lowest point of the inner surface of the sphere. 6 The continuous random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
0 & x < 1, \\
\frac{1}{2} & 1 \leqslant x \leqslant 3, \\
0 & x > 3.
\end{cases}$$
Find the distribution function of $X$. [2]
The random variable $Y$ is defined by $Y = X^3$. Find
(i) the probability density function of $Y$, [3]
(ii) the expected value and variance of $Y$. [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
$$f(t) = \begin{cases}
0 & t < 0, \\
\lambda e^{-\lambda t} & t \geqslant 0,
\end{cases}$$
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 ha9 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. $$524 \quad 526 \quad 520 \quad 523 \quad 530$$ Assuming a normal distribution, calculate a 95% confidence interval for the population mean. [5] 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. $$525 \quad 520 \quad 522 \quad 524 \quad 518 \quad 520 \quad 519 \quad 525 \quad 527 \quad 516$$ Stating suitable hypotheses, test at the 10% significance level whether there is any difference between the population means before and
10 Answer only one of the following two alternatives.
EITHER
A uniform rod $AB$, of weight $W$ and length $2a$, rests with the end $A$ on a rough horizontal plane. A light inextensible string $BC$ 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
$$\mu \geq \frac{2a \cos \theta}{h + 2a \sin \theta}.$$
[7]
Given that the particle attached at $C$ has weight $kW$, angle $ABP = 90^\circ$ and $h = 3a$, find
(i) the value of $BLANK PAGE
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