Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further June 2025 Question Paper 21
9231 Mathematics - Further June 2025 Question Paper 21
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Find the roots of the equation $z^3 = 27 - 27\mathrm{i}$, giving your answers in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r > 0$ and $-\pi \leqslant \theta < \pi$.
Let $I_n = \displaystyle\int_0^1 (1-x)^n \sinh x\,dx$, where $n$ is a non-negative integer.
Show that, for $n \geqslant 2$, $I_n = -1 + n(n-1)I_{n-2}$.
Find the exact value of $I_2$.
By considering the binomial expansion of $\left(z - \dfrac1z\right)^5$, where $z = \cos\theta + \mathrm{i}\sin\theta$, use de Moivre's theorem to show that $\mathrm{cosec}^5\theta = \dfrac{a}{\sin 5\theta + b\sin 3\theta + c\sin\theta}$, where $a$, $b$ and $c$ are integers to be determined.
The diagram shows the curve with equation $y = \dfrac{1}{\sqrt x}\mathrm{e}^{\sqrt x}$ for $x \geqslant 1$, together with a set of $n-1$ rectangles of unit width.
By considering the sum of the areas of these rectangles, show that $\displaystyle\sum_{r=1}^n \dfrac{1}{\sqrt r}\mathrm{e}^{\sqrt r} < \left(2 + \dfrac{1}{\sqrt n}\right)\mathrm{e}^{\sqrt n} - 2\mathrm{e}$.
Use a similar method to find, in terms of $n$, a lower bound for $\displaystyle\sum_{r=1}^n \dfrac{1}{\sqrt r}\mathrm{e}^{\sqrt r}$.
Find the particular solution of the differential equation $6\dfrac{d^2x}{dt^2} + 3\dfrac{dx}{dt} + 6x = \mathrm{e}^{-t}$, given that, when $t = 0$, $x = \dfrac{dx}{dt} = 0$.
Starting from the definitions of $\tanh$ and $\mathrm{sech}$ in terms of exponentials, prove that $1 - \tanh^2 u = \mathrm{sech}^2 u$.
Show that $\dfrac{d}{dt}(\mathrm{sech}^{-1}t) = -\dfrac{1}{t\sqrt{1-t^2}}$.
It is given that $x = \tanh^{-1}t$ and $y = t\,\mathrm{sech}^{-1}t$, for $0 < t < 1$. Show that $\dfrac{dy}{dx} = -\sqrt{1-t^2} + (1-t^2)\mathrm{sech}^{-1}t$.
Find $\dfrac{d^2y}{dx^2}$ in terms of $t$.
Find the solution of the differential equation $\dfrac{dy}{dx} - \dfrac{x+5}{x^2 + 10x + 61}y = 1$, given that $y = 0$ when $x = 3$. Give your answer in an exact form.
It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $\mathbf{A}$, with corresponding eigenvector $\mathbf{e}$. Show that $\mathbf{e}$ is an eigenvector of $\mathbf{A}^3$ with corresponding eigenvalue $\lambda^3$.
The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} -1 & 3 & 4 \\ 0 & 1 & 0 \\ 0 & -2 & 5 \end{pmatrix}$. Show that the eigenvalues of $\mathbf{A}$ are $-1$, 1 and 5.
Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A} - 2\mathbf{I} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$.
Use the characteristic equation of $\mathbf{A}$ to show that $(\mathbf{A} - 2\mathbf{I})^3 = a\mathbf{A}^2 + b\mathbf{A} + c\mathbf{I}$, where $a$, $b$ and $c$ are constants to be determined.