A Level Further Mathematics 9231 Paper 23, Oct/Nov 2024

Evaluates determinant.

$8-5(8k+6)+6(4k+6)$

$k\mathrm{\ne }\frac{7}{8}$

Three planes intersect at a unique point.

Uses chain rule

$\frac{t^2}{\sqrt{1-t^2}}$, AG.

Applies Quotient rule.

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)=\frac{2t\sqrt{1-t^2}+t^3(1-t^2{)}^{-\frac{1}{2}}}{1-t^2}$

Uses chain rule.

$=-t^3(1-t^2{)}^{-\frac{3}{2}}(2-t^2)$

Correct formula and applies substitution

$2\pi {\int }_{\frac{4}{3}}^{\frac{12}{5}}u\sqrt{1+u^2}\left(\frac{1}{u}\right)\mathrm{d}u$, AG.

$u=\sinh v\Rightarrow \mathrm{d}u=\cosh v\mathrm{d}v$

Applies substitution and ${\cosh }^{2}x=1+{\sinh }^{2}x$.

$A=2\pi {\int }_{\ln 3}^{\ln 5}{\cosh }^{2}v\mathrm{d}v$. Need limits for A1.

Applies $2{\cosh }^{2}x=\cosh 2x+1$.

Correct integration.

$\pi (\frac{904}{225}+\ln \frac{5}{3})$, AG.

Multiplies matrix with eigenvector.

$\lambda =-2$

Sets determinant equal to zero.

Expands determinant, AG.

$\lambda =5,-3$

States that $\mathbf{A}$ satisfies its characteristic equation.

Multiplies through by ${\mathbf{A}}^{-1}$.

M1 for substituting ${\mathbf{A}}^2$.

${\mathbf{A}}^{-1}=\frac{1}{30}\left(\begin{array}{ccc}-26& -5& 16\\ 0& -15& 0\\ -32& -5& 22\end{array}\right)$

Auxiliary equation, two distinct real roots.

Complementary function.

Particular integral and its derivatives.

Substitutes and equates coefficients.

$p=1q=11r=44$

General solution. Need to see ' $x = $ '. FT on CF.

Differentiates.

Uses initial conditions.

$A=-38,B=6$

Need to see ' $x = $ '.

Forms the sum of the areas of the rectangles. M1 for correct number of rectangles.

$\left(\frac{1}{N}\right){\left(\frac{1}{2}\right)}^{\frac{1}{N}}+\cdots +\left(\frac{1}{N}\right){\left(\frac{1}{2}\right)}^{\frac{N}{N}}$

Applies ${\sum }_{n=1}^N{r}^n=\frac{r(r^N-1)}{r-1}$

$\frac{1}{2N(2^{\frac{1}{N}}-1)}$, AG.

Forms the sum of the areas of appropriate rectangles.

${\int }_0^1{\left(\frac{1}{2}\right)}^x\mathrm{d}x<\left(\frac{1}{N}\right)+\cdots +\left(\frac{1}{N}\right){\left(\frac{1}{2}\right)}^{\frac{N-1}{N}}$

Applies ${\sum }_{n=0}^{N-1}{r}^n=\frac{r^N-1}{r-1}$.

$\frac{2^{\frac{1}{N}}}{2N(2^{\frac{1}{N}}-1)}$

Simplifies $U_N-L_N$ to $\frac{c}{N}$.

Least value of $N$ is 500

Forms inequality.

$\frac{1}{2U_N}<\ln 2<\frac{1}{2L_N}$ FT on their $U_N$.

$0.693$

$0.694$

Divides through by $\sqrt{x^2+16}$.

Finds integrating factor

${\mathrm{e}}^{{\sinh }^{-1}\left(\frac{x}{4}\right)}$

$=\frac{1}{4}x+\frac{1}{4}\sqrt{x^2+16}$, AG.

Correct form on LHS and RHS.

$\frac{\mathrm{d}}{\mathrm{d}x}(y(x+\sqrt{x^2+16}))=x^2+x\sqrt{x^2+16}$

Integrates RHS. RHS of the correct form.

$y(x+\sqrt{x^2+16})=\frac{1}{3}{x}^3+\frac{1}{3}(x^2+16{)}^{\frac{3}{2}}+C$

Substitutes initial conditions into their expression.

$y(x+\sqrt{x^2+16})=\frac{1}{3}{x}^3+\frac{1}{3}(x^2+16{)}^{\frac{3}{2}}-\frac{8}{3}$, OE.

For LHS.

Expands and groups.

$(z^7+z^{-7})+7(z^5+z^{-5})+21(z^3+z^{-3})+35(z+z^{-1})$. A0 if not grouped clearly.

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

${\cos }^7\theta =\frac{1}{64}\cos 7\theta +\frac{7}{64}\cos 5\theta +\frac{21}{64}\cos 3\theta +\frac{35}{64}\cos \theta$

Applies integration by parts.

$I_n={[{\cos }^{n-1}\theta \sin \theta ]}_0^{\frac{1}{4}\pi }+(n-1){\int }_0^{\frac{1}{4}\pi }{\cos }^{n-2}\theta {\sin }^2\theta \mathrm{d}\theta$

Applies ${\sin }^2\theta =1-{\cos }^2\theta$

$n{I}_n=2^{-\frac{n}{2}}+(n-1)I_{n-2}$, AG.

Applies reduction formula from (b).

$9I_9=2^{-\frac{9}{2}}+8I_7=\frac{1}{16}\left(2^{-\frac{1}{2}}\right)+8I_7$

Applies identity from (a). Allow missing/incorrect limits.

$I_7=\frac{1}{64}{[\frac{1}{7}\sin 7\theta +\frac{7}{5}\sin 5\theta +\frac{21}{3}\sin 3\theta +35\sin \theta ]}_0^{\frac{1}{4}\pi }$

Check their exact answer when $I_9$ is the subject, like terms collected. ISW. 0.402237