Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further June 2025 Question Paper 11
9231 Mathematics - Further June 2025 Question Paper 11
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Use standard results from the list of formulae (MF19) to show that $\displaystyle\sum_{r=1}^n (2-3r)(5-3r) = an^3 + bn^2 + cn$, where $a$, $b$ and $c$ are integers to be determined.
Use the method of differences to find $\displaystyle\sum_{r=1}^n \dfrac{1}{(2-3r)(5-3r)}$ in terms of $n$.
Deduce the value of $\displaystyle\sum_{r=1}^\infty \dfrac{1}{(2-3r)(5-3r)}$.
The cubic equation $x^3 + 2x + 1 = 0$ has roots $\alpha$, $\beta$, $\gamma$.
Find a cubic equation whose roots are $\alpha^3 - 1$, $\beta^3 - 1$, $\gamma^3 - 1$.
Find the value of $(\alpha^3-1)^2 + (\beta^3-1)^2 + (\gamma^3-1)^2$.
Find the value of $(\alpha^3-1)^3 + (\beta^3-1)^3 + (\gamma^3-1)^3$.
The sequence $u_1, u_2, u_3, \ldots$ is such that $u_1 = 5$ and $u_{n+1} = 6u_n + 5$ for $n \geqslant 1$.
Prove by induction that $u_n = 6^n - 1$ for all positive integers $n$.
Deduce that $u_{2n}$ is divisible by $u_n$ for $n \geqslant 1$.
The matrix $\mathbf{M}$ is given by $\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, where $0 < \theta < 2\pi$.
The matrix $\mathbf{M}$ represents a sequence of two geometrical transformations in the $x$-$y$ plane. State the type of each transformation, and make clear the order in which they are applied.
Find the value of $\theta$ for which the transformation represented by $\mathbf{M}$ has a line of invariant points.
The curve $C$ has polar equation $r = \theta\,\mathrm{e}^{\frac18\theta}$, for $0 \leqslant \theta \leqslant 2\pi$.
Sketch $C$.
Find the area of the region bounded by $C$ and the initial line, giving your answer in the form $(p\pi^2 + q\pi + r)\mathrm{e}^{\frac12\pi} + s$, where $p$, $q$, $r$ and $s$ are integers to be determined.
Show that, at the point of $C$ furthest from the initial line, $\theta\cos\theta + \left(\tfrac18\theta + 1\right)\sin\theta = 0$, and verify that this equation has a root between 5 and 5.05.
The points $A$, $B$, $C$ have position vectors $\mathbf{i} - 2\mathbf{k}$, $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$, $2\mathbf{i} - \mathbf{j} - \mathbf{k}$ respectively.
Find the equation of the plane $ABC$, giving your answer in the form $ax + by + cz = d$.
A point $D$ has position vector $\mathbf{i} + t\mathbf{k}$, where $t \neq -2$. Find the acute angle between the planes $ABC$ and $ABD$.
Find the values of $t$ such that the shortest distance between the lines $AB$ and $CD$ is $\sqrt2$.
The curve $C$ has equation $y = \dfrac{2x^2 - 5x}{2x^2 - 7x - 4}$.
Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
Sketch $C$, stating the coordinates of the intersections with the axes.
Sketch the curve with equation $y = \left|\dfrac{2x^2 - 5x}{2x^2 - 7x - 4}\right|$.
Find in exact form the set of values of $x$ for which $\left|\dfrac{2x^2 - 5x}{2x^2 - 7x - 4}\right| < \dfrac19$.