Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 12
9231 Mathematics - Further November 2025 Question Paper 12
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Use standard results from the list of formulae (MF19) to find $\displaystyle\sum_{r=1}^{n}(r^3 - r)$ in terms of $n$, fully factorising your answer.
Express $\dfrac{r+3}{r^3 - r}$ in the form $\dfrac{A}{r-1} + \dfrac{B}{r} + \dfrac{C}{r+1}$, where $A$, $B$ and $C$ are constants to be determined, and hence use the method of differences to find $\displaystyle\sum_{r=2}^{n}\dfrac{r+3}{r^3 - r}$.
Deduce the value of $\displaystyle\sum_{r=2}^{\infty}\dfrac{r+3}{r^3 - r}$.
The cubic equation $x^3 + bx^2 + cx + d = 0$, where $b$, $c$ and $d$ are constants, has roots $\alpha$, $\beta$ and $\gamma$. It is given that $\alpha + \beta + \gamma = 2$, $\alpha^2 + \beta^2 + \gamma^2 = 3$ and $\alpha^4 + \beta^4 + \gamma^4 = 5$.
Find the values of $b$ and $c$.
Find the value of $d$.
The sequence of positive numbers $u_1, u_2, u_3, \ldots$ is such that $u_1 < 5$ and, for $n \geqslant 1$, $u_{n+1} = \dfrac{6u_n + 5}{u_n + 2}$.
By considering $5 - u_{n+1}$, prove by mathematical induction that $u_n < 5$ for all positive integers $n$.
Show that $u_{n+1} > u_n$ for $n \geqslant 1$.
Let $k$ and $m$ be non-zero constants. The matrices are $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 1 \\ 1 & 1 \end{pmatrix}$, $\mathbf{B} = \begin{pmatrix} k & 0 \\ 0 & m \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 1 & 2 \end{pmatrix}$.
Give full details of the geometrical transformation in the $x$-$y$ plane represented by the matrix $\mathbf{B}$ in each of the following cases.
$m = 1$
$m = k$
Show that the matrix $\mathbf{ABC}$ is singular.
The curve $C$ has polar equation $r^2 = \tan 2\theta$, where $0 \leqslant \theta \leqslant \tfrac{1}{8}\pi$.
Sketch $C$ and state the greatest distance of a point on $C$ from the pole.
Find the exact value of the area of the region bounded by $C$ and the half-line $\theta = \tfrac{1}{8}\pi$.
Show that $C$ has Cartesian equation $x^4 - 2xy - y^4 = 0$, given that $0 \leqslant x \leqslant \cos\!\left(\tfrac{1}{8}\pi\right)$ and $0 \leqslant y \leqslant \sin\!\left(\tfrac{1}{8}\pi\right)$.
Using your answer to (b), deduce the exact value of the area bounded by $C$, the $x$-axis and the line $x = \cos\!\left(\tfrac{1}{8}\pi\right)$.
The plane $\Pi$ has equation $x + 3y + 2z = 1$.
Find the perpendicular distance from the origin $O$ to the plane $\Pi$.
Relative to $O$, the points $A$, $B$, $C$ have position vectors $-\mathbf{j} + 2\mathbf{k}$, $2\mathbf{i} - \mathbf{k}$ and $2\mathbf{i} - \mathbf{j} - \mathbf{k}$ respectively. Find the acute angle between the planes $OAB$ and $\Pi$.
Find an equation for the common perpendicular to the lines $OC$ and $AB$.
The curve $C$ has equation $y = \dfrac{x^2 + x + 1}{x + 1}$.
Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
Sketch $C$.
Sketch the curve with equation $y = \dfrac{|x|^2 + |x| + 1}{|x| + 1}$.
Find, in exact form, the set of values of $x$ for which $\dfrac{|x|^2 + |x| + 1}{|x| + 1} < 3$.