科目：进阶数学 Further Mathematics (9231)
年份：2009 夏季
试卷：Paper 2
总页数：7
章节：9231 Further Mathematics
1 A line $OP$ of fixed length $l$ rotates in a plane about the fixed point $O$. At time $t = 0$, the line is at the position $OA$. At time $t$, angle $AOP = \theta$ radians and $\frac{d\theta}{dt} = \sin \theta$. Show that, for all $t$, the magnitude of the acceleration of $P$ is equal to the magnitude of its velocity. [5]
2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm. It takes 2.25 s to perform 100 complete oscillations. Find, in m s$^{-1}$, the maximum speed of the tip of the needle. [5]
Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is 2.50 m s$^{-1}$, correct to 3 significant figures. [2]
3
A uniform lamina of mass $m$ is bounded by concentric circleA uniform sphere rests on a horizontal plane. The sphere has centre $O$, radius 0.6 m and weight 36 N. A uniform rod $AB$, of weight 14 N and length 1 m, rests with $A$ in contact with the plane and $B$ in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is $C$, and $A, B, C$ and $O$ lie in the same vertical plane (see diagram). The contacts at $A, B$ and $C$ are rough and the system is in equilibrium. By taking moments about $C$ for the system, show that the magnitude of the normal contact force at $A$ is 10 N. Show that the magnitudes of the frictional forces at $A, B$ and $C$ are equal. [2] The coefficients of friction at $A, B$ and $C$ are all equal to $\mu$. Find the smallest possible value of $\mu$. [7] 5 Two sphe
7 An experiment was carried out to determine how much weedkiller to apply per 100 m² in a large field. Ten 100 m² areas of the field were randomly chosen and sprayed with predetermined volumes of the weedkiller. The volume of the weedkiller is denoted by $x$ litres and the number of weeds that survived is denoted by $y$. The results are given in the table.
$$\begin{tabular}{cccccccccc}
x & 0.10 & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 & 0.40 & 0.45 & 0.50 & 0.55 \\
y & 48 & 40 & 44 & 35 & 39 & 24 & 10 & 13 & 9 & 6 \\
\end{tabular}$$
$$[\Sigma x = 3.25, \Sigma x^2 = 1.2625, \Sigma y = 268, \Sigma y^2 = 9548, \Sigma xy = 66.10.$$
It is given that the product moment correlation coefficient for the data is $-0.951$, correct to 3 decimal places.
(i) Calculate the equation of a suitable10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by $T$ minutes. Show that $P(T > t) = e^{-0.8t}$ and hence show that $T$ has a negative exponential distribution.
Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.
11 Answer only one of the following two alternatives.
EITHER
Two particles $A$ and $B$, of equal mass $m$, are connected by a light elastic string of natural length $a$ and modulus of elasticity $4mg$. Particle $A$ rests on a rough horizontal table at a distance $a$ from the edge of the table. The string passes over a small smooth pulley $P$ fixed at the edge of the tablBLANK PAGE
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