Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further June 2025 Question Paper 23
9231 Mathematics - Further June 2025 Question Paper 23
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Find the Maclaurin's series for $\mathrm{e}^{\left(\frac{1}{x+2}\right)}$ up to and including the term in $x^2$.
Starting from the definitions of $\tanh$ and $\mathrm{sech}$ in terms of exponentials, prove that $\tanh^2 t + \mathrm{sech}^2 t = 1$.
The curve $C$ has parametric equations $x = \ln(\cosh t)$, $y = \tan^{-1}(\sinh t)$, for $0 \leqslant t \leqslant 1$. Find the length of $C$.
The curve $C$ has equation $9y^2 - 3\sinh^{-1}(xy) = 1 - 3\ln 3$.
Show that, at the point $\left(4, \tfrac13\right)$ on $C$, $\dfrac{dy}{dx} = -\tfrac12$.
Find the value of $\dfrac{d^2y}{dx^2}$ at the point $\left(4, \tfrac13\right)$.
Find the particular solution of the differential equation $\dfrac{d^2x}{dt^2} + \dfrac{dx}{dt} - 2x = 2t^2 + t - 1$, given that, when $t = 0$, $x = \dfrac{dx}{dt} = 0$.
Use de Moivre's theorem to show that $\sec 5\theta = \dfrac{\sec^5\theta}{5\sec^4\theta - 20\sec^2\theta + 16}$.
Hence, obtain the roots of the equation $\sqrt3\,x^5 - 10x^4 + 40x^2 - 32 = 0$ in the form $\sec(q\pi)$, where $q$ is rational.
The diagram shows the curve with equation $y = \dfrac{1}{x^2 + 1}$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\tfrac1n$.
By considering the sum of the areas of these rectangles, show that $\displaystyle\sum_{r=1}^n \dfrac{n}{n^2 + r^2} < \tfrac14\pi$.
Use a similar method to find a lower bound for $\displaystyle\sum_{r=1}^n \dfrac{n}{n^2 + r^2}$. Give your answer in terms of $n$ and $\pi$.
Deduce the exact value of $\displaystyle\lim_{n\to\infty}\sum_{r=1}^n \dfrac{n}{n^2 + r^2}$.
Find the solution of the differential equation $\dfrac{dy}{dx} - \dfrac{2x+6}{x^2 + 6x + 5}y = 4$, given that $y = 0$ when $x = 0$. Give your answer in an exact form.
Find the values of $a$ for which the system of equations $\tfrac32 x + 3y + 8z = 1$, $ax + 3y + 4z = 2$, $ay - z = 3$ does not have a unique solution.
The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} \tfrac32 & 3 & 8 \\ 0 & 3 & 4 \\ 0 & 0 & -1 \end{pmatrix}$. Given that $\mathbf{B} = \mathbf{A}^{-1}$, use the characteristic equation of $\mathbf{A}$ to show that $\mathbf{B}^2 = p\mathbf{I} + q\mathbf{A}$, where $p$ and $q$ are constants to be determined.
Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{A}^{-1} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$.