科目：进阶数学 Further Mathematics (9231)
年份：2016 夏季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1. The roots of the cubic equation $2x^3 + x^2 - 7 = 0$ are $\alpha, \beta$ and $\gamma$. Using the substitution $y = 1 + \frac{1}{x}$, or otherwise, find the cubic equation whose roots are $1 + \frac{1}{\alpha}, 1 + \frac{1}{\beta}$ and $1 + \frac{1}{\gamma}$, giving your answer in the form $ay^3 + by^2 + cy + d = 0$, where $a, b, c$ and $d$ are constants to be found.
2. Express $\frac{4}{r(r+1)(r+2)}$ in partial fractions and hence find $\sum_{r=1}^{n} \frac{4}{r(r+1)(r+2)}$.
3. Prove by mathematical induction that, for all positive integers $n$, $10^n + 3 \times 4^{n+2} + 5$ is divisible by 9.
4. A curve $C$ has polar equation $r^2 = 8 \cos 2\theta$ for $0 < \theta < \frac{1}{2}\pi$. Find a cartesian equation of $C$.
5. Let $I_n = \int_0^{4\pi} \cos^n x \sin^2 x \, dx$, for 7 A curve $C$ has equation $y = \frac{x^2}{x - 2}$. Find the equations of the asymptotes of $C$. [3]
Show that there are no points on $C$ for which $0 < y < 8$. [4]
Sketch $C$, giving the coordinates of the turning points. [3]
8 Find a cartesian equation of the plane $\Pi_1$ passing through the points with coordinates $(2, -1, 3), (4, 2, -5)$ and $(-1, 3, -2)$. [4]
The plane $\Pi_2$ has cartesian equation $3x - y + 2z = 5$. Find the acute angle between $\Pi_1$ and $\Pi_2$. [3]
Find a vector equation of the line of intersection of the planes $\Pi_1$ and $\Pi_2$. [4]
9 Find the value of the constant $k$ such that $y = kx^2e^{2x}$ is a particular integral of the differential equation
$$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 4e^{2x}.$$ [4]
Hence find the general solu11 Answer only one of the following two alternatives.
EITHER
A curve $C$ has parametric equations
$$x = e^{2t} \cos 2t, \quad y = e^{2t} \sin 2t, \quad \text{for } -\frac{1}{2}\pi \leq t \leq \frac{1}{2}\pi.$$
Find the arc length of $C$. [6]
Find the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis. [8]
OR
The linear transformation $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ is represented by the matrix $M$, where
$$M = \begin{pmatrix}
1 & -2 & 3 & -4 \\
2 & -4 & 7 & -9 \\
4 & -8 & 14 & -18 \\
5 & -10 & 17 & -22
\end{pmatrix}.$$
Find the rank of $M$. [3]
Obtain a basis for the null space $K$ of $T$. [3]
Evaluate
$$M \begin{pmatrix}
1 \\
-2 \\
2 \\
-1
\end{pmatrix},$$
and hence show that any solution