科目：进阶数学 Further Mathematics (9231)
年份：2015 夏季
试卷：Paper 21
总页数：7
章节：9231 Further Mathematics
1. A particle $P$ is moving in a circle of radius 0.25 m. At time $t$ seconds, its velocity is $(2t^2 - 4t + 3) \text{ m s}^{-1}$. At a particular time $T$ seconds, where $T > 0$, the magnitude of the transverse component of the acceleration of $P$ is $6 \text{ m s}^{-2}$. Find the magnitude of the radial component of the acceleration of $P$ at this instant.
2. A particle $P$ moves on a straight line $AOB$ in simple harmonic motion. The centre of the motion is $O$, and $P$ is instantaneously at rest at $A$ and $B$. The point $C$ is on the line $AOB$, between $A$ and $O$, and $CO = 10 \text{ m}$. When $P$ is at $C$, the magnitude of its acceleration is $0.625 \text{ m s}^{-2}$ and it is moving towards $O$ with speed $6 \text{ m s}^{-1}$. Find
(i) the period of the motion, in terms 6 The independent random variables $X$ and $Y$ have distributions with the same variance $\sigma^2$. Random samples of $N$ observations of $X$ and 10 observations of $Y$ are taken, and the results are summarised by $$\Sigma x = 5, \quad \Sigma x^2 = 11, \quad \Sigma y = 10, \quad \Sigma y^2 = 160.$$ These data give a pooled estimate of 12 for $\sigma^2$. Find $N$. [4] 7 A random sample of 8 sunflower plants is taken from the large number grown by a gardener, and the heights of the plants are measured. A 95% confidence interval for the population mean, $\mu$ metres, is calculated from the sample data as $1.17 < \mu < 2.03$. Given that the height of a sunflower plant is denoted by $x$ metres, find the values of $\Sigma x$ and $\Sigma x^2$ for this sample of 8 plants. [7] 8 (a) Fo
10 Young children at a primary school are learning to throw a ball as far as they can. The distance thrown at the beginning of the school year and the distance thrown at the end of the same school year are recorded for each child. The distance thrown, in metres, at the beginning of the year is denoted by $x$; the distance thrown, in metres, at the end of the year is denoted by $y$. For a random sample of 10 children, the results are shown in the following table. | Child | A | B | C | D | E | F | G | H | I | J | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $x$ | 5.2 | 4.1 | 3.7 | 5.4 | 7.6 | 6.1 | 3.2 | 4.0 | 3.5 | 8.0 | | $y$ | 6.2 | 4.8 | 5.0 | 5.6 | 7.7 | 7.0 | 4.0 | 4.5 | 3.6 | 8.5 | $$\Sigma x = 50.8, \quad \Sigma x^2 = 284.16, \quad \Sigma
11 Answer only one of the following two alternatives.
EITHER
A uniform disc, with centre $O$ and radius $a$, is surrounded by a uniform concentric ring with radius $3a$. The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass $\frac{3}{2}m$ and length $2a$. The disc and the ring each have mass $2m$. The rods meet the ring at the points $A, B, C$ and $D$. The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through $O$ perpendicular to the plane of the object is $45ma^2$.
[4]
Find the moment of inertia of the object about an axis $l$ through $A$ in the plane of the object and tangential to the ring.
[3]
A particle of mass $3m$ is now attaBLANK PAGE
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