科目：进阶数学 Further Mathematics (9231)
年份：2015 夏季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. The quartic equation $x^4 - px^2 + qx - r = 0$, where $p, q$ and $r$ are real constants, has two pairs of equal roots. Show that $p^2 + 4r = 0$ and state the value of $q$.
2. The curve $C$ has polar equation $r = e^{4\theta}$ for $0 \leq \theta \leq \alpha$, where $\alpha$ is measured in radians. The length of $C$ is 2015. Find the value of $\alpha$.
3. Prove by mathematical induction that, for all positive integers $n$, $\sum_{r=1}^{n} \frac{1}{(2r)^2 - 1} = \frac{n}{2n+1}$.
State the value of $\sum_{r=1}^{\infty} \frac{1}{(2r)^2 - 1}$.
4. Use the formula for $\tan(A - B)$ in the List of Formulae (MF10) to show that
$$\tan^{-1}(x+1) - \tan^{-1}(x-1) = \tan^{-1}\left(\frac{2}{x^2}\right).$$
Deduce the sum to $n$ terms of the series
$$\tan^{-1}\left(\frac{2}{1^2}\rig8 A line, passing through the point $A(3, 0, 2)$, has vector equation $\mathbf{r} = 3\mathbf{i} + 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$. It meets the plane $\Pi$, which has equation $\mathbf{r}.(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = 3$, at the point $P$. Find the coordinates of $P$.
Write down a vector $\mathbf{n}$ which is perpendicular to $\Pi$, and calculate the vector $\mathbf{w}$, where
$$\mathbf{w} = \mathbf{n} \times (2\mathbf{i} + \mathbf{j} - 2\mathbf{k}).$$
The point $Q$ lies in $\Pi$ and is the foot of the perpendicular from $A$ to $\Pi$. Use the vector $\mathbf{w}$ to determine an equation of the line $PQ$ in the form $\mathbf{r} = \mathbf{u} + \mu\mathbf{v}$.
9 Find the particular solution of the differential equation
$$\frac{d^2x}{dt11 Answer only one of the following two alternatives.
EITHER
The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
1 & 2 & 3 & 4 \\
1 & -1 & 2 & 3 \\
1 & -3 & 3 & 5 \\
1 & 4 & 2 & 2
\end{pmatrix}.$$
The range space of $T$ is denoted by $V$.
(i) Determine the dimension of $V$. [3]
(ii) Show that the vectors $\begin{pmatrix}
1 \\
1 \\
1 \\
1
\end{pmatrix}, \begin{pmatrix}
2 \\
-1 \\
-3 \\
4
\end{pmatrix}, \begin{pmatrix}
3 \\
2 \\
3 \\
2
\end{pmatrix}$ are a basis of $V$. [5]
The set of elements of $\mathbb{R}^4$ which do not belong to $V$ is denoted by $W$.
(iii) State, with a reason, whether $W$ is a vector space. [1]
(iv) Show that if the vector $\begin{pmatrix}
x \