科目：进阶数学 Further Mathematics (9231)
年份：2021 冬季
试卷：Paper 22
总页数：14
章节：9231 Further Mathematics
1 It is given that $y = \sinh(x^2) + \cosh(x^2)$.
(a) Use standard results from the list of formulae (MF19) to find the Maclaurin’s series for $y$ in terms of $x$ up to and including the term in $x^4$.
[2]
(b) Deduce the value of $\frac{d^4y}{dx^4}$ when $x = 0$.
[1]
(c) Use your answer to part (a) to find an approximation to $\int_0^{\frac{1}{2}} y \, dx$, giving your answer as a rational fraction in its lowest terms.
[2]2 Find the solution of the differential equation
$$\frac{dy}{dx} + \frac{4x^3y}{x^4 + 5} = 6x$$
for which $y = 1$ when $x = 1$. Give your answer in the form $y = f(x)$. [7]The diagram shows the curve with equation $y = 1 - x^2$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.
(a) By considering the sum of the areas of the rectangles, show that
$$\int_0^1 (1-x^2)dx < \frac{4n^2 + 3n - 1}{6n^2}.$$
[4](b) Use a similar method to find, in terms of $n$, a lower bound for $\int_{0}^{1}(1-x^2)dx$. [4]4 (a) Write down all the roots of the equation $x^5 - 1 = 0$. [2] (b) Use de Moivre’s theorem to show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$. [4]
(c) Use the results of parts (a) and (b) to express each real root of the equation $$8x^9 - 8x^7 + x^5 - 8x^4 + 8x^2 - 1 = 0$$ in the form $\cos k\pi$, where $k$ is a rational number. [4]
5 The curve $C$ has parametric equations
$$x = 3t + 2t^{-1} + at^3, \quad y = 4t - \frac{3}{2}t^{-1} + bt^3, \quad \text{for } 1 \leq t \leq 2,$$
where $a$ and $b$ are constants.
(a) It is given that $a = \frac{2}{3}$ and $b = -\frac{1}{2}$.
Show that $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \frac{25}{4}(t^2 + t^{-2})^2$ and find the exact length of $C$. [6](b) It is given instead that $a = b = 0$.
Find the value of $\frac{d^2y}{dx^2}$ when $t = 1$. [4]6 The matrix $P$ is given by
$$P = \begin{pmatrix}
1 & 6 & 6 \\
0 & 2 & 6 \\
0 & 0 & -3
\end{pmatrix}.$$
(a) Use the characteristic equation of $P$ to find $P^{-1}$. [5](b) Find the matrix $A$ such that
$$P^{-1}AP = \begin{pmatrix}
4 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 6
\end{pmatrix}.$$
[4]
(c) State the eigenvalues and corresponding eigenvectors of $A^3$.
[2]7 It is given that $y = x^2 w$ and
$$x^2 \frac{d^2 w}{dx^2} + 4x(x+1) \frac{dw}{dx} + (5x^2 + 8x+2)w = 5x^2 + 4x+2.$$
(a) Show that
$$\frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + 5y = 5x^2 + 4x+2.$$
[4](b) Find the general solution for $w$ in terms of $x$. [7]
8 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
$$1 - \tanh^2 x = \text{sech}^2 x.$$
[3]
(b) Using the substitution $u = \tanh x$, or otherwise, find $\int \text{sech}^2 x \tanh^2 x \text{d}x.$
[2]
It is given that, for $n \geqslant 0$, $I_n = \int_0^{\ln 3} \text{sech}^n x \tanh^2 x \text{d}x$.
(c) Show that, for $n \geqslant 2$,
$$(n+1) I_n = \left(\frac{4}{5}\right)^3 \left(\frac{3}{5}\right)^{n-2} + (n-2) I_{n-2}.$$
[5]
[You may use the result that $\frac{\mathrm{d}}{\mathrm{d}x}(\text{sech} x) = -\tanh x \text{sech} x$.](d) Find the value of $I_4$. [3]