科目：进阶数学 Further Mathematics (9231)
年份：2024 冬季
试卷：Paper 23
总页数：18
章节：9231 Further Mathematics
2 BLANK PAGE © UCLES 2024 9231/23/O/N/24
1. Find the set of values of $k$ for which the system of equations $$x + 5y + 6z = 1,$$ $$kx + 2y + 2z = 2,$$ $$-3x + 4y + 8z = 3,$$ has a unique solution and interpret this situation geometrically. [4]
2 It is given that
$$x = 1 + \frac{1}{t} \quad \text{and} \quad y = \cos^{-1} t \quad \text{for } 0 < t < 1.$$
(a) Show that $$\frac{dy}{dx} = \frac{r^2}{\sqrt{1 - t^2}}.$$ [2](b) Show that $\frac{d^2y}{dx^2} = -t^a(1-t^2)^b(2-t^2)$, where $a$ and $b$ are constants to be determined. [4]3 A curve has equation $y = e^x$ for $\ln \frac{4}{5} \leq x \leq \ln \frac{12}{5}$. The area of the surface generated when the curve is rotated through $2\pi$ radians about the $x$-axis is denoted by $A$.
(a) Use the substitution $u = e^x$ to show that
$$A = 2\pi \int_{\frac{4}{5}}^{\frac{12}{5}} \sqrt{1 + u^2} du.$$
[2]
(b) Use the substitution $u = \sinh v$ to show that
$$A = \pi \left( \frac{904}{225} + \ln \frac{5}{3} \right).$$
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4 The matrix $A$ is given by
$$A = \begin{pmatrix}
-11 & 1 & 8 \\
0 & -2 & 0 \\
-16 & 1 & 13
\end{pmatrix}.$$
(a) Show that $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ is an eigenvector of $A$ and state the corresponding eigenvalue. [2]
(b) Show that the characteristic equation of $A$ is $\lambda^3 - 19\lambda - 30 = 0$ and hence find the other eigenvalues of $A$. [3](c) Use the characteristic equation of $A$ to find $A^{-1}$. [4]5 Find the particular solution of the differential equation
$$6 \frac{d^2 x}{d t^2} - 5 \frac{dx}{dt} + x = t^2 + t + 1,$$
given that, when $t = 0$, $x = 12$ and $\frac{dx}{dt} = -6$. [10]
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The diagram shows the curve with equation $y = \left(\frac{1}{2}\right)^x$ for $0 \leq x \leq 1$, together with a set of $N$ rectangles each of width $\frac{1}{N}$.
(a) By considering the sum of the areas of these rectangles, show that $\int_0^1 \left(\frac{1}{2}\right)^x dx > L_N$, where
$$L_N = \frac{1}{2N\left(2^{\frac{1}{N}} - 1\right)}.$$
[4](b) Use a similar method to find, in terms of $N$, an upper bound $U_N$ for $\int_0^1 \left(\frac{1}{2}\right)^x dx$.
(c) Find the least value of $N$ such that $U_N - L_N \leq 10^{-3}$.(d) Given that $\int_{0}^{\frac{1}{2}} \frac{1}{2} x \, dx = \frac{1}{2} \ln 2$, use the value of $N$ found in part (c) to find upper and lower bounds for $\ln 2$.
7 (a) Show that an appropriate integrating factor for
$$\sqrt{x^2 + 16} \frac{dy}{dx} + y = x \sqrt{x^2 + 16}$$
is $\frac{1}{4}x + \frac{1}{4}\sqrt{x^2 + 16}$. [4](b) Hence find the solution of the differential equation
$$\sqrt{x^2 + 16} \frac{dy}{dx} + y = x\sqrt{x^2 + 16}$$
for which $y = 6$ when $x = 3$. [6]16 BLANK PAGE © UCLES 2024 9231/23/O/N/24
8 (a) By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^7$, where $z = \cos\theta + i\sin\theta$, use de Moivre’s theorem to show that
$$\cos^7\theta = a\cos 7\theta + b\cos 5\theta + c\cos 3\theta + d\cos\theta,$$
where $a, b, c$ and $d$ are constants to be determined. [5]Let $I_n = \int_0^{\frac{1}{2}\pi} \cos^n \theta \mathrm{d}\theta$.
(b) Show that
$$nI_n = 2^{-\frac{1}{2}n} + (n-1)I_{n-2}.$$
[4](c) Using the results given in parts (a) and (b), find the exact value of $I_g$. [5]