科目：进阶数学 Further Mathematics (9231)
年份：2015 冬季
试卷：Paper 13
总页数：3
章节：9231 Further Mathematics
1. The curve $C$ is defined parametrically by
$$x = 2\cos^3 t \quad \text{and} \quad y = 2\sin^3 t, \quad \text{for} \quad 0 < t < \frac{1}{2}\pi.$$
Show that, at the point with parameter $t$,
$$\frac{d^2y}{dx^2} = \frac{1}{6}\sec^4 t \cosec t.$$
2. Find the general solution of the differential equation
$$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = 7 - 2t^2.$$
3. Given that $a$ is a constant, prove by mathematical induction that, for every positive integer $n$,
$$\frac{d^n}{dx^n}(x e^{\alpha x}) = n a^{n-1} e^{\alpha x} + a^n x e^{\alpha x}.$$
4. The sequence $a_1, a_2, a_3, \ldots$ is such that, for all positive integers $n$,
$$a_n = \frac{n+5}{\sqrt{(n^2-n+1)}} - \frac{n+6}{\sqrt{(n^2+n+1)}}.$$
The sum $\sum_{n=1}^{N} a_n$ is denoted by $S_N$. Find
7 The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
1 & -2 & -3 & 1 \\
3 & -5 & -7 & 7 \\
5 & -9 & -13 & 9 \\
7 & -13 & -19 & 11
\end{pmatrix}.$$
Find the rank of $M$ and a basis for the null space of $T$. [6]
The vector $\begin{pmatrix}
1 \\
2 \\
3 \\
4
\end{pmatrix}$ is denoted by $e$. Show that there is a solution of the equation $Mx = Me$ of the form
$$x = \begin{pmatrix}
a \\
b \\
-1 \\
-1
\end{pmatrix},$$
where the constants $a$ and $b$ are to be found. [4]
8 The curve $C$ has equation $y = \frac{2x^2 + kx}{x + 1}$, where $k$ is a constant. Find the set of values of $k$ for which $C$ has no stationary points. [5]
For the case $k = 4$, find the equations of the asymptot11 Answer only one of the following two alternatives.
EITHER
The points $A, B$ and $C$ have position vectors $\mathbf{i}, \mathbf{2j}$ and $\mathbf{4k}$ respectively, relative to an origin $O$. The point $N$ is the foot of the perpendicular from $O$ to the plane $ABC$. The point $P$ on the line-segment $ON$ is such that $OP = \frac{3}{4}ON$. The line $AP$ meets the plane $OBC$ at $Q$. Find a vector perpendicular to the plane $ABC$ and show that the length of $ON$ is $\frac{4}{\sqrt{(21)}}$.
Find the position vector of the point $Q$. [5]
Show that the acute angle between the planes $ABC$ and $ABQ$ is $\cos^{-1}\left(\frac{2}{5}\right)$. [5]
OR
The curve $C$ has polar equation $r = a(1 - \cos \theta)$ for $0 \leqslant \theta < 2\pi$. Sketch $C$. [2]
Find the area of the