A Level Further Mathematics 9231 Paper 23, May/June 2023

Correct first derivative.

Correct second derivative.

Differentiates their $f^{\mathrm{\prime }\mathrm{\prime }}(x)$ using product rule.

Evaluates their derivatives at $x=0$. Must have attempted all three derivatives.

CWO.

$\frac{151}{750}$

Auxiliary equation.

Complementary function. Allow with 'x =' missing.

Particular integral and its derivatives.

Substitutes and equates coefficients.

$r=1,q=0,p=1$

Must have $x=$.

Must have 'x ='. Accept 'x is approximately' but do not accept '$x\to$'. FT on their PI.

Use of $z-z^{-1}=2\text{i}\sin \theta$ and $z+z^{-1}=2\cos \theta$

Expands and groups $(z+z^{-1}{)}^4$.

$(z+z^{-1}{)}^4=(z^4+z^{-4})+4(z^2+z^{-2})+6$

Expands and groups $(z-z^{-1}{)}^4$.

$(z-z^{-1}{)}^4=(z^4+z^{-4})-4(z^2+z^{-2})+6$

Substitutes $z^n+z^{-n}=2\cos n\theta$ once in LHS.

${\cot }^4\theta =\frac{\cos 4\theta +4\cos 2\theta +3}{\cos 4\theta -4\cos 2\theta +3}$

Differentiates $y^3$ correctly.

Differentiates $(x+y{)}^6$ correctly.

Substitutes $(-4,3)$, AG.

Differentiates $y^2{y}^{\mathrm{\prime }}$.

Differentiates first part of $(x+y{)}^5(1+y^{\mathrm{\prime }})$.

Differentiates second part of $(x+y{)}^5(1+y^{\mathrm{\prime }})$.

Substitutes $(-4,3)$.

$y^{\mathrm{\prime }\mathrm{\prime }}=-\frac{96}{289}$

$\cosh x=\frac{1}{2}({\text{e}}^x+{\text{e}}^{-x})$

Expands.

$\cosh 2x+1$, AG.

Finds integrating factor.

${\text{e}}^{2\ln \cosh x}={\cosh }^{2}x$

Correct form on LHS, $\frac{\text{d}}{\text{d}x}(yI)$ for their integrating factor $I$, and attempt to integrate their RHS.

Uses ${\cosh }^{2}x=\frac{1}{2}(\cosh 2x+1)$.

$\frac{1}{2}(\frac{1}{2}\sinh 2x+x)+C$ (M1 A1 is for RHS).

Finds $C$.

Divides through by their coefficient of $y$.

$y={\text{sech}}^{2}x(\frac{1}{4}\sinh 2x+\frac{1}{2}x+1)$

Forms the sum of the areas of the rectangles.

Must have first two terms and last term for A1. Accept $\left(\frac{1}{n}\right){(1-\frac{n}{n})}^2$ for the last term. Accept written in summation form with correct limits.

Setting up correct series so that formulae from MF19 can be applied. Can also recognise sum as $\frac{1}{n^3}{\sum }_{r=1}^n{r}^2$

Correct intermediate expanded summation expression.

AG. Must have gained previous accuracy mark.

Forms the sum of the areas of appropriate rectangles.

Must have first two and last terms for A1. Accept $\left(\frac{1}{n}\right){(1-\frac{n}{n})}^2$ for the last term. Accept written in summation form with correct limits.

Setting up correct series so that formulae from MF19 can be applied. Recognising sum as $\frac{1}{n^3}{\sum }_{r=1}^{n-1}{r}^2=\frac{1}{n^3}{\sum }_{r=1}^n{r}^2-\frac{1}{n}$ without wrong working scores M1 A1 M1.

$\frac{2n^2-3n+1}{6n^2}$

Simplifies $U_n-L_n$ to $\frac{c}{n}$.

$\frac{1}{n}\to 0\text{ as }n\to \mathrm{\infty }$, AG.

Recognises integral.

$\ln 3$

Uses the product rule to differentiate.

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

Uses $x^2=1+x^2-1$.

Integrates both sides using the limits given. Requires previous method mark.

Substitutes limits and rearranges. AG.

Forms correct integral with correct limits.

$\frac{1}{2}{\int }_0^{\frac{4}{3}}\sqrt{1+u^2}\text{ d}u=\frac{1}{2}{I}_1$, AG.

Applies reduction formula with $n=1$.

$s=\frac{1}{4}\ln 3+\frac{5}{9}$

Shows that determinant is non-zero or solves equations.

$a(a-1)\mathrm{\ne }0$ or correct solution for $x,y,z$.

Must be clear that there is just one point where all 3 planes intersect.

Equates determinant to zero or working to find characteristic equation.

AG. Factorisation must be clear.

Uses vector product (or equations) to find corresponding eigenvectors.

Eigenvector for $\lambda =a$.

Eigenvector for $\lambda =-1$.

Eigenvector for $\lambda =1-a$.

Correctly matched permutations of columns. Their eigenvectors must be non-zero for M1.

$\mathbf{P}$ and $\mathbf{D}$ matrices correct.

Finds characteristic equation multiplied out.

Multiplies through by $\mathbf{A}$.

${\mathbf{A}}^4=(a^2-a+1){\mathbf{A}}^2+(a^2-a)\mathbf{A}$