Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 14
9231 Mathematics - Further November 2025 Question Paper 14
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Prove by mathematical induction that $\left(\tfrac65\right)^n \geqslant 1 + \tfrac15 n$ for every positive integer $n$.
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 $\gamma = \dfrac1\beta$.
Show that $\beta + \dfrac1\beta = d - b$.
Show that $\beta + \dfrac1\beta = \dfrac{1-c}{d}$.
It is now given that $b = 3$ and $c = -3$, and that $d > 0$.
Find the value of $d$.
Find the value of $\alpha^2 + \beta^2 + \gamma^2$.
Show that $\displaystyle\sum_{r=1}^n (r+1)(r+2)(r+3) = \tfrac14 n(n^3 + bn^2 + cn + d)$, where the constants $b$, $c$ and $d$ are to be found.
Use the method of differences to find $\displaystyle\sum_{r=1}^n \dfrac{2}{(r+1)(r+2)(r+3)}$, expressing your answer in terms of $n$.
Deduce the value of $\displaystyle\sum_{r=1}^\infty \dfrac{2}{(r+1)(r+2)(r+3)}$.
Relative to the origin $O$, the points $A$, $B$, $C$ and $D$ have position vectors $\overrightarrow{OA} = 3\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}$, $\overrightarrow{OB} = 2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$, $\overrightarrow{OC} = 2\mathbf{i} + \mathbf{j} - \mathbf{k}$ and $\overrightarrow{OD} = \mathbf{i} - \mathbf{j} + 3\mathbf{k}$.
Find an equation of the plane through $A$, $B$ and $C$, giving your answer in the form $ax + by + cz = e$.
Find the shortest distance between the lines $AB$ and $CD$.
Show that $\cos\tfrac{1}{12}\pi = \tfrac14(\sqrt6 + \sqrt2)$.
Show that $\sin\tfrac{1}{12}\pi = \tfrac14(\sqrt6 - \sqrt2)$.
The matrix $\mathbf{M} = \begin{pmatrix} \tfrac14(\sqrt6+\sqrt2) & \tfrac14(\sqrt6-\sqrt2) \\ \tfrac14(\sqrt6-\sqrt2) & -\tfrac14(\sqrt6+\sqrt2) \end{pmatrix}$ represents a single transformation that is equivalent to the combination of two transformations. Give full details of these two transformations and state the order in which they are applied.
Express $\mathbf{M}^{-1}$ as the product of two matrices, neither of which is the identity matrix.
The line $y = mx$ is invariant under the transformation represented by $\mathbf{M}$. Show that $\sin\tfrac{1}{12}\pi\, m^2 + 2m\cos\tfrac{1}{12}\pi - \sin\tfrac{1}{12}\pi = 0$, and hence find the possible values of $m$ in the form $a\cot\tfrac{1}{12}\pi + b\,\mathrm{cosec}\,\tfrac{1}{12}\pi$.
The curve $C$ has polar equation $r = \cos\tfrac12\theta$, for $0 \leqslant \theta \leqslant \pi$.
Sketch $C$.
Find the exact value of the area of the region enclosed by $C$ and the initial line.
Find the maximum distance of a point on $C$ from the initial line, giving your answer in exact form.
The curve $C$ has equation $y = \dfrac{10x^2 - 11x - 18}{10x - 18}$.
Write down the equations of the asymptotes of $C$.
Show that $C$ has no stationary points.
Sketch $C$, stating the coordinates of any points of intersection with the axes.
On a separate diagram, sketch the curve with equation $y = \left|\dfrac{10x^2 - 11x - 18}{10x - 18}\right|$.
It is given that $\left|\dfrac{10x^2-11x-18}{10x-18}\right| < 4$ for $\dfrac{-29-\sqrt{4441}}{20} < x < p$ and $\dfrac{-29+\sqrt{4441}}{20} < x < q$. Find the exact values of $p$ and $q$.