科目：进阶数学 Further Mathematics (9231)
年份：2014 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1. The equation $x^3 + px + q = 0$, where $p$ and $q$ are constants, with $q \neq 0$, has one root which is the reciprocal of another root. Prove that $p + q^2 = 1$. [5]
2. Expand and simplify $(r + 1)^4 - r^4$. [1]
Use the method of differences together with the standard results for $\sum_{r=1}^{n} r$ and $\sum_{r=1}^{n} r^2$ to show that
$$\sum_{r=1}^{n} r^3 = \frac{1}{4} n^2 (n + 1)^2.$$ [4]
3. Prove by mathematical induction that, for all non-negative integers $n$,
$$11^{2n} + 25^n + 22$$
is divisible by 24. [6]
4. Obtain the general solution of the differential equation
$$\frac{d^2 x}{dt^2} - 6 \frac{dx}{dt} + 25x = 195 \sin 2t.$$ [6]
5. The curve $C$ has polar equation $r = a(1 + \sin \theta)$, where $a$ is a positive constant and $0 \leq \theta < 2\pi$. Draw a sketc8 The curve $C$ has parametric equations
$$x = t^2, \quad y = t - \frac{1}{3}t^3, \quad \text{for } 0 \leqslant t \leqslant 1.$$
Find
(i) the arc length of $C$, [5]
(ii) the surface area generated when $C$ is rotated through $2\pi$ radians about the $x$-axis. [5]
9 The matrix $M$, where
$$M = \begin{pmatrix}
-2 & 2 & 2 \\
2 & 1 & 2 \\
-3 & -6 & -7
\end{pmatrix},$$
has an eigenvector $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$. Find the corresponding eigenvalue. [2]
It is given that if the eigenvalues of a general $3 \times 3$ matrix $A$, where
$$A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix},$$
are $\lambda_1, \lambda_2$ and $\lambda_3$ then
$$\lambda_1 + \lambda_2 + \lambda_3 = a + e + i$$
and the determinant of $A$ has t12 Answer only one of the following two alternatives.
EITHER
The curve $C$ has parametric equations
$$x = t^2, \quad y = (2 - t)^{\frac{1}{2}}, \quad \text{for } 0 \leq t \leq 2.$$
Find
(i) $\frac{d^2y}{dx^2}$ in terms of $t$, [5]
(ii) the mean value of $y$ with respect to $x$ over the interval $0 \leq x \leq 4$, [6]
(iii) the $y$-coordinate of the centroid of the region enclosed by $C$, the $x$-axis and the $y$-axis. [3]
OR
The curve $C$ has equation
$$y = \frac{ax^2 + bx + c}{x + d},$$
where $a, b, c$ and $d$ are constants. The curve cuts the $y$-axis at $(0, -2)$ and has asymptotes $x = 2$ and $y = x + 1.$
(i) Write down the value of $d$. [1]
(ii) Determine the values of $a, b$ and $c$. [6]
(iii) Show that, at all points on $C$, either $y \leq 3