A Level Further Mathematics 9231 Paper 11, Oct/Nov 2025

Substitutes correct formulae from MF19.

Substitutes correct formulae from MF19.

Simplifies.

Puts over a common denominator and expands, AG.

Puts over a common denominator and expands, AG.

Shows three complete terms, including first and last.

Shows three complete terms, including first and last.

$=1-\frac{1}{(n+1{)}^4}$

1

Shear, [x-axis fixed]

With $(0,1)$ mapped to $(\frac{3}{2},1)$.

Shear, [y-axis fixed],

with $(1,0)$ mapped to $(1,\frac{3}{2})$.

Finds AB or uses product of determinants. Or suitable reasoning that area does not change.

$\det \mathbf{A}\mathbf{B}=1$

Transforms $\left(\begin{array}{c}x\\ y\end{array}\right)$ to $\left(\begin{array}{c}X\\ Y\end{array}\right)$.

Uses $Y=mX$.

Uses $Y=mX$.

$\frac{3}{2}+m=\frac{13}{4}m+\frac{3}{2}{m}^2\Rightarrow m^2+\frac{3}{2}m-1=0$

$y=-2x$ and $y=\frac{1}{2}x$.

Checks base case using product rule.

States inductive hypothesis.

Differentiates once.

Differentiates once.

Differentiates again.

Differentiates again.

States conclusion.

Correct substitution used in equation.

Expands.

Expands.

Simplifies, AG.

Uses $(\alpha +\beta +\gamma +\delta {)}^2={\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2+2(\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta )$

$S_2=-4$

Sums over roots.

$S_4=-76$

Finds perpendicular to ${\mathrm{\Pi }}_1$.

Finds perpendicular to ${\mathrm{\Pi }}_1$.

Uses point on ${\mathrm{\Pi }}_1$.

$5x-2y+z=-14$

Uses correct formula for perpendicular distance from point to ${\mathrm{\Pi }}_1$.

Uses correct formula for perpendicular distance from point to ${\mathrm{\Pi }}_1$.

$\frac{7}{\sqrt{30}}$

States point common to both planes e.g. $\left(\begin{array}{c}0\\ 7\\ 0\end{array}\right)$.

Finds direction of line.

Finds direction of line.

$\mathbf{r}=\left(\begin{array}{c}0\\ 7\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)$

Initial line drawn, pole clear and one loop in the first quadrant.

Symmetrical with correct shape at extremities.

States the equation of the line of symmetry.

Uses $\frac{1}{2}\int r^2\mathrm{d}\theta$ with correct limits.

Applies double angle formula correctly and integrates.

Applies double angle formula correctly and integrates.

$=\frac{1}{12}\pi$

Uses $y=r\sin \theta$.

Differentiates correct function and sets equal to 0. Might see completing the square.

$\sin \theta \cos \theta \mathrm{\ne }0\Rightarrow 6-16{\sin }^2\theta =0$

Solves quadratic in $\sin \theta$.

$y=\frac{9}{16}\mathrm{.}$

Uses $y=r\sin \theta$

Uses $r^2=x^2+y^2$

Rearranges to a form with no fractions.

Vertical asymptotes.

Horizontal asymptote.

Finds $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Sets equal to 0 and forms quadratic equation.

Discriminant is $4^2-20=-4<0$ so no real roots.

Axes and asymptotes labelled.

Correct shape and position.

States coordinates of intersections with axes, $(0,2)$ and $(-2,0)$, may be seen on diagram.

FT from sketch in part 7(c)

From a correct graph and with correct shape at $(-2,0)$.

Finds critical points, award M1 for each case.

$x=-\frac{5}{2},x=0$ or $x=-\frac{7}{4}-\frac{1}{4}\sqrt{17},x=-\frac{7}{4}+\frac{1}{4}\sqrt{17}$

$-\frac{7}{4}-\frac{1}{4}\sqrt{17}<x<-\frac{5}{2}$, $-\frac{7}{4}+\frac{1}{4}\sqrt{17}<x<0$, $x\mathrm{\ne }-\frac{3}{2}\pm \frac{1}{2}\sqrt{5}$