科目：进阶数学 Further Mathematics (9231)
年份：2011 冬季
试卷：Paper 12
总页数：3
章节：9231 Further Mathematics
1 The equation $x^3 + px + q = 0$ has a repeated root. Prove that $4p^3 + 27q^2 = 0$. [5]
2 The position vectors of points $A, B, C$, relative to the origin $O$, are $a, b, c$, where
$$\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k}, \quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}, \quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.$$
Find $\mathbf{a} \times \mathbf{b}$ and deduce the area of the triangle $OAB$. [3]
Hence find the volume of the tetrahedron $OABC$, given that the volume of a tetrahedron is $\frac{1}{3} \times$ area of base $\times$ perpendicular height. [2]
3 Prove by mathematical induction that, for all positive integers $n,$
$$\frac{d^n}{dx^n} (e^x \sin x) = 2^{\frac{1}{2}n} e^x \sin(x + \frac{1}{4}n\pi).$$ [7]
4 The linear transformati7 The curve $C$ has equation $y = \frac{x^2 + px + 1}{x - 2}$, where $p$ is a constant. Given that $C$ has two asymptotes, find the equation of each asymptote.
Find the set of values of $p$ for which $C$ has two distinct turning points.
Sketch $C$ in the case $p = -1$. Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.
8 The vector $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{A}$, with corresponding eigenvalue $\lambda$, and is also an eigenvector of the matrix $\mathbf{B}$, with corresponding eigenvalue $\mu$. Show that $\mathbf{e}$ is an eigenvector of the matrix $\mathbf{AB}$ with corresponding eigenvalue $\lambda \mu$.
State the eigenvalues of the matrix $\mathbf{C}$, where
$$\11 Answer only one of the following two alternatives.
EITHER
Let $\omega = \cos \frac{1}{5}\pi + i \sin \frac{1}{5}\pi$. Show that $\omega^5 + 1 = 0$ and deduce that
$$\omega^4 - \omega^3 + \omega^2 - \omega = -1. \tag{2}$$
Show further that
$$\omega - \omega^4 = 2 \cos \frac{1}{5}\pi \text{ and } \omega^3 - \omega^2 = 2 \cos \frac{3}{5}\pi. \tag{4}$$
Hence find the values of
$$\cos \frac{1}{5}\pi + \cos \frac{3}{5}\pi \text{ and } \cos \frac{1}{5}\pi \cos \frac{3}{5}\pi. \tag{4}$$
Find a quadratic equation having roots $\cos \frac{1}{5}\pi$ and $\cos \frac{3}{5}\pi$ and deduce the exact value of $\cos \frac{1}{5}\pi$. \tag{4}
OR
Given that
$$x^2 \frac{d^2y}{dx^2} + 4x(1+x) \frac{dy}{dx} + 2(1+4x+2x^2)y = 8x^2$$
and that $x^2y = z$, show that
$$\frac{