Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further June 2025 Question Paper 13
9231 Mathematics - Further June 2025 Question Paper 13
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The matrix $\mathbf{M}$ represents the sequence of two transformations in the $x$-$y$ plane given by a stretch parallel to the $x$-axis, scale factor 14, followed by a rotation anticlockwise about the origin through angle $\tfrac13\pi$.
Show that $2\mathbf{M} = \begin{pmatrix} 14 & -\sqrt3 \\ 14\sqrt3 & 1 \end{pmatrix}$.
Find the equations of the invariant lines, through the origin, of the transformation in the $x$-$y$ plane represented by $\mathbf{M}$.
The unit square $S$ in the $x$-$y$ plane is transformed by $\mathbf{M}$ onto the rectangle $P$. Find the matrix which transforms $P$ onto $S$.
Prove by mathematical induction that $2025^n + 47^n - 2$ is divisible by 46 for all positive integers $n$.
The quartic equation $x^4 + 7x^2 + 3x + 22 = 0$ has roots $\alpha$, $\beta$, $\gamma$, $\delta$.
Find the value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$.
Find the value of $\alpha^4 + \beta^4 + \gamma^4 + \delta^4$.
Use standard results from the list of formulae (MF19) to find the value of $\displaystyle\sum_{r=1}^{10}\left((\alpha^2+r)^2 + (\beta^2+r)^2 + (\gamma^2+r)^2 + (\delta^2+r)^2\right)$.
Let $w_r = r(r+1)(r+2)\ldots(r+9)$.
Show that $w_{r+1} - w_r = 10(r+1)(r+2)\ldots(r+9)$.
Given that $u_r = (r+1)(r+2)\ldots(r+9)$, find $\displaystyle\sum_{r=1}^n u_r$ in terms of $n$.
Given that $v_r = x^{w_{r+1}} - x^{w_r}$, find the set of values of $x$ for which the infinite series $v_1 + v_2 + v_3 + \ldots$ is convergent and give the sum to infinity when this exists.
The plane $\Pi$ has equation $\mathbf{r} = 2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} - \mathbf{k}) + \mu(3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$.
Find a Cartesian equation of $\Pi$, giving your answer in the form $ax + by + cz = d$.
The point $P$ has position vector $4\mathbf{i} + 2\mathbf{j} + 9\mathbf{k}$. Find the position vector of the foot of the perpendicular from $P$ to $\Pi$.
The line $l$ is parallel to the vector $3\mathbf{i} + 5\mathbf{j} - \mathbf{k}$. Find the acute angle between $l$ and $\Pi$.
The curve $C$ has equation $y = \dfrac{x^2 + a}{x + a}$, where $a$ is a positive constant.
Find the equations of the asymptotes of $C$.
Find, in terms of $a$, the $x$-coordinates of the stationary points on $C$.
Sketch $C$, stating the coordinates of any intersections with the axes.
Sketch the curve with equation $y = \left|\dfrac{x^2 + a}{x + a}\right|$.
Find the set of values of $a$ for which $\left|\dfrac{x^2 + a}{x + a}\right| = a$ has two real solutions.
The curve $C$ has polar equation $r^2 = \mathrm{e}^{\sin\theta}\cos\theta$, for $-\tfrac12\pi \leqslant \theta \leqslant \tfrac12\pi$.
Find the polar coordinates of the point on $C$ that is furthest from the pole, giving your answers correct to 3 decimal places.
Find the polar coordinates of the point on $C$ that is furthest from the half-line $\theta = \tfrac12\pi$, giving your answers correct to 3 decimal places.
Sketch $C$.
Find the area of the region bounded by $C$, giving your answer in exact form.