A Level Further Mathematics 9231 Paper 11, May/June 2023

States base case.

States inductive hypothesis.

Multiplies $2{\mathbf{A}}^k$ with $\mathbf{A}$.

Multiplies $2{\mathbf{A}}^k$ with $\mathbf{A}$ correctly.

States conclusion.

$\det {\mathbf{A}}^n=\det \left(\begin{array}{cc}{3}^n& 0\\ \frac{1}{2}(3^n-1)& 1\end{array}\right)=3^n$ Or a multiple of ${\mathbf{A}}^{-n}=2^{-1}{3}^{-n}\left(\begin{array}{cc}2& 0\\ 1-3^n& 2\times 3^n\end{array}\right)$ seen.

${\mathbf{A}}^{-n}=3^{-n}\left(\begin{array}{cc}1& 0\\ \frac{1}{2}(1-3^n)& 3^n\end{array}\right)$

Uses formula for sum of squares.

4

Expands.

Collects like terms and uses $\alpha +\beta +\gamma =-4$ and their result from part (a).

Correct expression ${\sum }_{r=1}^n(4+2(-4)r+3r^2)$.

Applies formulae from MF19.

Simplifies.

$n(n^2-\frac{5}{2}n+\frac{1}{2})$

Finds partial fractions.

Correct partial fractions.

Writes at least three complete terms, including last.

$\frac{1}{k}(1-\frac{1}{kn+1})$

$\frac{1}{k}$

Or applies the method of differences again.

$\frac{1}{k}(\frac{1}{k(n-1)+1}-\frac{1}{k{n}^2+1})$

Uses correct matrix for rotation.

$b^2=-c^2$ which is impossible since $b$ and $c$ are real and $b\mathrm{\ne }0$.

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

Uses $y=mx$ and $Y=mX$.

Correct substitution into transformed coordinates.

$c^2+am=ma+b^2{m}^2⟹c^2=b^2{m}^2$

$y=\frac{c}{b}x$ and $y=-\frac{c}{b}x$

Product of shear and enlargement matrices.

Correct product in correct order $\left(\begin{array}{cc}1& 5\\ 0& 1\end{array}\right)\left(\begin{array}{cc}5& 0\\ 0& 5\end{array}\right)$.

$\left(\begin{array}{cc}5& 5^2\\ 0& 5\end{array}\right)$

Using their $\mathbf{M}$.

$300{\text{ cm}}^2$

Correct shape and domain, polar graph with $r$ strictly decreasing but condone if not strictly decreasing close to $\theta =\pi$.

Fully correct including shape at $\theta =0$ correct shape at $\theta =\pi$ correct

Identified as point furthest from the pole and given as coordinates $(1,0)$.

Uses correct formula with correct limits.

Integrates $\frac{1}{{\theta }^2+1}$.

Correct integration $\frac{1}{2}[{\tan }^{-1}\theta {]}_0^{\pi }$.

$\frac{1}{2}{\tan }^{-1}\pi =0.631$

Uses $y=r\sin \theta$

Sets derivative equal to zero.

Correct derivative expression set to zero.

$\theta \mathrm{\ne }0⟹(\theta +\frac{1}{\theta })\cot \theta -1=0$

Shows sign change (1sf or better).

States vertical asymptote.

Finds oblique asymptote.

$y=x+4$

Differentiates.

Forms quadratic equation or simplifies $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Correct conclusion.

Axes and asymptotes.

Branches correct.

States coordinates of intersections with axes.

FT from sketch in (c).

Correct shape at infinity and on $x$ axis.

Finds critical points for positive case.

Finds critical points for negative case.

$x=0,\frac{11}{2}$ or $x=-12,\frac{5}{2}$

$-12<x<0$ or $\frac{5}{2}<x<\frac{11}{2}$

Finds common perpendicular.

Correct direction vector $\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right)$.

Substitutes point.

$y+z=-7$

States point common to both planes e.g. $\left(\begin{array}{c}-7\\ -2\\ -5\end{array}\right)$.

Finds direction of line.

Correct direction vector $\left(\begin{array}{c}2\\ -5\\ 5\end{array}\right)$.

$\mathbf{r}=\left(\begin{array}{c}-7\\ -2\\ -5\end{array}\right)+\lambda \left(\begin{array}{c}2\\ -5\\ 5\end{array}\right)$

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

$a=1$

Uses correct formula.

Correct equation set up.

Solves for $b$.

$b=-1+\sqrt{2}$