Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 11
9231 Mathematics - Further November 2025 Question Paper 11
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Use standard results from the list of formulae (MF19) to find $\displaystyle\sum_{r=1}^n (8r^3 + 12r^2 + 4r + 3)$ in terms of $n$, simplifying your answer.
Show that $\dfrac{1}{r^4} - \dfrac{1}{(r+1)^4} = \dfrac{4r^3 + 6r^2 + 4r + 1}{r^4(r+1)^4}$ and hence use the method of differences to find $\displaystyle\sum_{r=1}^n \dfrac{4r^3 + 6r^2 + 4r + 1}{r^4(r+1)^4}$.
Deduce the value of $\displaystyle\sum_{r=1}^\infty \dfrac{4r^3 + 6r^2 + 4r + 1}{r^4(r+1)^4}$.
The matrices $\mathbf{A}$ and $\mathbf{B}$ are given by $\mathbf{A} = \begin{pmatrix} 1 & \frac32 \\ 0 & 1 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 0 \\ \frac32 & 1 \end{pmatrix}$.
Give full details of the geometrical transformation in the $x$-$y$ plane represented by $\mathbf{A}$.
Give full details of the geometrical transformation in the $x$-$y$ plane represented by $\mathbf{B}$.
The triangle $DEF$ in the $x$-$y$ plane is transformed by $\mathbf{AB}$ onto triangle $PQR$. Show that the triangles $DEF$ and $PQR$ have the same area.
Find the equations of the invariant lines, through the origin, of the transformation in the $x$-$y$ plane represented by $\mathbf{AB}$.
Prove by mathematical induction that, for every positive integer $n$, $\dfrac{d^{2n-1}}{dx^{2n-1}}(x\cos x) = (-1)^n(x\sin x - (2n-1)\cos x)$.
The quartic equation $x^4 + x^3 + x^2 + x + 1 = 0$ has roots $\alpha$, $\beta$, $\gamma$, $\delta$.
Show that a quartic equation with roots $2\alpha+1$, $2\beta+1$, $2\gamma+1$, $2\delta+1$ is $y^4 - 2y^3 + 4y^2 + 2y + 11 = 0$.
The sum $(2\alpha+1)^n + (2\beta+1)^n + (2\gamma+1)^n + (2\delta+1)^n$ is denoted by $S_n$. Find the value of $S_2$.
Given that $S_3 = -22$, find the value of $S_4$.
The plane $\Pi_1$ has equation $\mathbf{r} = -3\mathbf{i} - \mathbf{j} - \mathbf{k} + \lambda(\mathbf{j} + 2\mathbf{k}) + \mu(\mathbf{i} + 3\mathbf{j} + \mathbf{k})$.
Find an equation for $\Pi_1$ in the form $ax + by + cz = d$.
Find the perpendicular distance from the point with position vector $-\mathbf{i} - 2\mathbf{k}$ to $\Pi_1$.
The plane $\Pi_2$ has equation $3x + 2y - z = 14$. Find a vector equation of the line of intersection of $\Pi_1$ and $\Pi_2$.
The curve $C$ has polar equation $r = \sin 3\theta$, for $0 \leqslant \theta \leqslant \tfrac13\pi$.
Sketch $C$ and state the equation of the line of symmetry.
Find the exact value of the area of the region enclosed by $C$.
(You may use the identity $\sin 3\theta \equiv 3\sin\theta - 4\sin^3\theta$.) Find the maximum distance of a point on $C$ from the initial line.
Find a Cartesian equation for $C$.
The curve $C$ has equation $y = \dfrac{x+2}{x^2 + 3x + 1}$.
Find the equations of the asymptotes of $C$.
Show that $C$ has no stationary points.
Sketch $C$, stating the coordinates of the intersections with the axes.
Sketch the curve with equation $y = \left|\dfrac{x+2}{x^2+3x+1}\right|$.
Find in exact form the set of values of $x$ for which $\left|\dfrac{x+2}{x^2+3x+1}\right| > 2$.