Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 21
9231 Mathematics - Further November 2025 Question Paper 21
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Find the values of $k$ for which the system of equations $x - y + z = 0$, $x + ky + 3z = 0$, $x + 2y + kz = 0$ does not have a unique solution.
The curve $C$ has parametric equations $x = \sinh t$ and $y = t + \cosh t$.
Find $\dfrac{dy}{dx}$ in terms of $t$.
Show that $\dfrac{d^2y}{dx^2} = \dfrac{1 - \sinh t}{\cosh^3 t}$.
Find the Maclaurin's series for $y$ in terms of $x$ up to and including the term in $x^2$.
The integral $I_n$ is defined by $I_n = \displaystyle\int_0^1 (1+x^2)^n\,dx$.
By considering $\dfrac{d}{dx}\left(x(1+x^2)^n\right)$, or otherwise, show that $(2n+1)I_n = 2^n + 2nI_{n-1}$.
Find the exact value of $I_{-2}$.
Find the particular solution of the differential equation $5\dfrac{d^2y}{dx^2} + 2\dfrac{dy}{dx} + y = x^2 + 5x + 3$, given that, when $x = 0$, $y = \dfrac{dy}{dx} = 0$.
The diagram shows the curve with equation $y = \tfrac13 x^3 + x$ 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 $\displaystyle\int_0^1\left(\tfrac13 x^3 + x\right)dx < U_n$, where $U_n = \tfrac{1}{12}\left(1 + \tfrac1n\right)\left(7 + \tfrac1n\right)$.
Use a similar method to find, in terms of $n$, a lower bound $L_n$ for $\displaystyle\int_0^1\left(\tfrac13 x^3 + x\right)dx$.
Show that $\displaystyle\lim_{n\to\infty}(U_n - L_n) = 0$.
The matrix $\mathbf{P}$ has non-zero eigenvalues and is given by $\mathbf{P} = \begin{pmatrix} a & 1 & 1 \\ 0 & 2a & -1 \\ 0 & 0 & -3a \end{pmatrix}$.
State, in terms of $a$, the eigenvalues of $\mathbf{P}$.
Find $\mathbf{P}^2$ in terms of $a$.
Use the characteristic equation of $\mathbf{P}$ to find $\mathbf{P}^{-1}$ in terms of $a$.
The $3\times3$ matrix $\mathbf{A}$ has eigenvalues 1, 2, 3 with corresponding eigenvectors $\begin{pmatrix}a\\0\\0\end{pmatrix}$, $\begin{pmatrix}1\\2a\\0\end{pmatrix}$, $\begin{pmatrix}1\\-1\\-3a\end{pmatrix}$ respectively. Find $\mathbf{A}$ in terms of $a$.
Show that $\dfrac{d}{dx}\left(\tanh^{-1}x\right) = \dfrac{1}{1 - x^2}$.
Find the solution of the differential equation $x\dfrac{dy}{dx} - y = x^2\tanh^{-1}x$, for $0 < x < 1$, given that $y = 0$ when $x = \tfrac12$. Give your answer in an exact form.
State the sum of the series $z + z^2 + \ldots + z^n$, for $z \neq 1$.
By letting $z = \tfrac12(\cos\theta + \mathrm{i}\sin\theta)$ use de Moivre's theorem to deduce that $\displaystyle\sum_{m=1}^n \left(\tfrac12\right)^m \sin m\theta = \dfrac{\left(\tfrac12\right)^{n+2}\sin n\theta - \left(\tfrac12\right)^{n+1}\sin(n+1)\theta + \tfrac12\sin\theta}{\tfrac54 - \cos\theta}$.
Use the result in (b) to find $\displaystyle\sum_{m=1}^n \left(\tfrac12\right)^m m\cos m\theta$ in terms of $n$ and $\theta$.
Hence find $\displaystyle\sum_{m=1}^\infty \left(\tfrac12\right)^m m\cos m\theta$ in terms of $\cos\theta$.