A Level Further Mathematics 9231 Paper 21, May/June 2026

Finds first derivative. Accept $\frac{1}{x+1+\sqrt{(x+1{)}^2+1}}(1+\frac{2x+2}{2\sqrt{(x+1{)}^2+1}})$ or implicit $y^{\mathrm{\prime }}\cosh y=1$.

Finds second derivative. Allow not simplified or implicit $y^{\mathrm{\prime }\mathrm{\prime }}\cosh y+(y^{\mathrm{\prime }}{)}^2\sinh y=0$.

${\sinh }^{-1}1=\ln (1+\sqrt{2})$

Evaluates their derivatives at zero only. Must have attempted both derivatives.

CWO.

Applies integration by parts. Allow limits missing. M1 for correct trigonometric functions and powers of $(1+t)$.

Correct first integration by parts result.

Applies integration by parts again. M1 for correct trigonometric functions and powers of $(1+t)$.

AG. Must have seen limits and correct use of brackets. CWO.

Finds $I_0$.

Uses reduction formula from (a).

$I_2=\frac{5}{\pi }-\frac{4}{{\pi }^3}$

$u=\cosh x⟹\mathrm{d}u=\sinh x\mathrm{d}x$

Applies correct identity ${\sinh }^{2}u={\cosh }^{2}u-1$ to find integral in terms of $u$.

Correct integral in terms of $u$.

$=\frac{1}{3}{u}^3-u(+C)=\frac{1}{3}{\cosh }^{3}x-\cosh x(+C)$

Finds integrating factor.

Integrating factor correctly simplified to $\sinh x$.

Correct form on LHS and attempt to integrate RHS.

Integrates RHS using their part (a). For M1, their answer to (a) must be of the correct form $a{\cosh }^{3}x+b\cosh x$ with $a,b$ non-zero.

Correct integrated form $y\sinh x=\frac{1}{3}{\cosh }^{3}x-\cosh x+C$.

Substitutes initial conditions. M0 if expression incorrectly manipulated before substitution. E.g. not all terms divided by $\sinh x$.

OE.

Differentiates $(x-y{)}^5$ correctly.

Differentiates $\ln (xy)$ correctly.

Substitutes $(-1,-2)$ into $5(x-y{)}^4(1-y^{\mathrm{\prime }})+y^{\mathrm{\prime }}{y}^{-1}+x^{-1}=0$, AG. CWO. Might see $y^{\mathrm{\prime }}(y^{-1}-5(x-y{)}^4)=-5(x-y{)}^4-x^{-1}$.

Differentiates first part of $5(x-y{)}^4(1-y^{\mathrm{\prime }})$.

Differentiates second part of $5(x-y{)}^4(1-y^{\mathrm{\prime }})$.

Differentiates $y^{\mathrm{\prime }}{y}^{-1}+x^{-1}$. Might see $\frac{x{y}^{\mathrm{\prime }}+y}{xy}$ differentiated using product or quotient rules. E.g. $\frac{xy(x{y}^{\mathrm{\prime }\mathrm{\prime }}+2y^{\mathrm{\prime }})-(x{y}^{\mathrm{\prime }}+y{)}^2}{(xy{)}^2}$ or $-(xy{)}^{-2}(x{y}^{\mathrm{\prime }}+y{)}^2+(xy{)}^{-1}(x{y}^{\mathrm{\prime }\mathrm{\prime }}+2y^{\mathrm{\prime }})$.

Substitutes $(-1,-2)$ and $y^{\mathrm{\prime }}=\frac{8}{11}$.

$y^{\mathrm{\prime }\mathrm{\prime }}=\frac{86}{1331}$

Forms the sum of the areas of the rectangles. M1 for sum of $n$ complete terms.

A1 for at least 3 correct terms including first and last. May be written in summation form.

Applies formula for $\sum r^3$ from MF19.

AG. CWO.

Forms the sum of the areas of appropriate rectangles. M1 for sum of $n$ complete terms.

A1 for at least 3 correct terms including first and last.

Applies formulae from MF19. Allow $U_n+\frac{1}{10n}-\frac{12}{10n}$.

OE. e.g. $\frac{11}{40}{(1+\frac{1}{n})}^2+\frac{1}{10}-\frac{11}{10n}$.

Their $U_n-L_n$ simplifies to $\frac{c}{n}$. Accept $\frac{11}{40}(1{)}^2+\frac{1}{10}-\frac{11}{40}(1{)}^2-\frac{1}{10}$.

AG. Their $L_n$ must be correct.

Auxiliary equation. M1 for correct type of roots $a\pm b\mathrm{i}$.

Complementary function. Allow ‘$y=$’ missing.

Particular integral and its derivatives.

Substitutes and equates coefficients.

$q=-1r=-5$

General solution. Must have ‘$y=$’. FT on CF.

Differentiates. General solution must be of the correct form. Must use product rule.

Uses initial conditions.

A1 is both $A$ and $B$ correct.

Must have ‘$y=$’. CWO.

$\frac{z-z^{n+1}}{1-z}$ or $\frac{z^{n+1}-z}{z-1}$ or $\frac{z(z^n-1)}{z-1}$

States one root of $z^7=1,z\mathrm{\ne }1$.

Five other roots. No repeats or incorrect roots, $z=1$ must be omitted.

Applies sum to infinity and substitutes for $z$. Expect $\frac{{\mathrm{e}}^{\mathrm{i}\theta }}{\mathrm{e}-{\mathrm{e}}^{\mathrm{i}\theta }}$.

Correct expression for sum to infinity with $z$ substituted.

Rationalises denominator. Expect $\frac{{\mathrm{e}}^{\mathrm{i}\theta }}{\mathrm{e}-{\mathrm{e}}^{\mathrm{i}\theta }}\left(\frac{\mathrm{e}-{\mathrm{e}}^{-\mathrm{i}\theta }}{\mathrm{e}-{\mathrm{e}}^{-\mathrm{i}\theta }}\right)=\frac{{\mathrm{e}}^{1+\mathrm{i}\theta }-1}{{\mathrm{e}}^2-\mathrm{e}({\mathrm{e}}^{\mathrm{i}\theta }+{\mathrm{e}}^{-\mathrm{i}\theta })+1}$.

Applies ${\mathrm{e}}^{\mathrm{i}\theta }+{\mathrm{e}}^{-\mathrm{i}\theta }=2\cos \theta$ (or ${\sin }^2\theta +{\cos }^2\theta =1$).

Correct simplified complex form with real and imaginary parts separated.

Takes the real part.

AG.

Finds determinant in terms of $a$ and $b$.

Determinant correctly evaluated as $4a+ab-b-2$.

$a=\frac{b+2}{b+4}[\text{since }b\mathrm{\ne }-4]$

Finds characteristic equation.

Replaces $\lambda$ with $\mathbf{A}$ and multiplies through by $\mathbf{A}$.

Substitutes ${\mathbf{A}}^3=2{\mathbf{A}}^2+\mathbf{A}-2\mathbf{I}$.

CWO

Upper diagonal matrix or characteristic equation.

Uses vector product (or equations) to find corresponding eigenvectors. M0 if all three vectors are zero vectors.

First eigenvector correct.

Second eigenvector correct.

Third eigenvector correct.

Constructs $\mathbf{P}$ and $\mathbf{D}$. A column of zeros or repeated column in $\mathbf{P}$ is M0.

Or correctly matched permutations of columns.