A Level Mathematics 9709 Paper 33, May/June 2023

Use exponentials or law for the logarithm of a product, quotient or power

Correctly remove logarithms

Obtain a correct equation in $x$

Obtain 0.034

Divide to obtain quotient $2x^2\pm 2x+k$ ($k\mathrm{\ne }0$)

Obtain [quotient] $2x^2-2x-4$

Obtain [remainder] $10x-15$

Show a circle with centre $3+\mathrm{i}$

Show a circle with radius 3 and centre not at the origin

Show the line $y=2$

Shade the correct region

State $\frac{\mathrm{d}y}{\mathrm{d}\theta }=1-2\sin \theta$

Use correct quotient rule, or product rule if rewrite $x$ as $\cos \theta (2-\sin \theta {)}^{-1}$

Obtain $\frac{\mathrm{d}x}{\mathrm{d}\theta }=\frac{-(2-\sin \theta )\sin \theta +{\cos }^2\theta }{(2-\sin \theta {)}^2}$ o.e.

Use $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}\theta }÷\frac{\mathrm{d}x}{\mathrm{d}\theta }$

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}=(2-\sin \theta {)}^2$.

Use correct product rule

Obtain correct derivative in any form

Equate derivative to zero and obtain $a=\frac{1}{3}{\tan }^{-1}\left(\frac{2}{3a}\right)$.

Use the iterative process $a_{n+1}=\frac{1}{3}{\tan }^{-1}\left(\frac{2}{3a_n}\right)$ correctly at least twice during successive iterations in the numerous iterations

Obtain final answer 0.36

Show sufficient iterations to 4 or more d.p. to justify 0.36 to 2 d.p. or show there is a sign change in the interval $(0.355,0.365)$

Expand $\cos (x-{60}^{\circ })$ correctly and evaluate $3\cos x+2\cos (x-{60}^{\circ })$ to obtain $4\cos x+\sqrt{3}\sin x$ or unsimplified coefficients

State $R=\sqrt{19}$ [$R\cos \alpha =4,R\sin \alpha =\sqrt{3}$]

Use correct trig formulae for their expansion to find $\alpha$ e.g. $\alpha ={\tan }^{-1}\frac{\sqrt{3}}{4}$ or ${\cos }^{-1}\frac{4}{\sqrt{19}}$ or ${\sin }^{-1}\frac{\sqrt{3}}{\sqrt{19}}$

Obtain $\alpha ={23.41}^{\circ }$

${\cos }^{-1}\left(\frac{2.5}{R}\right)$

Use a correct method to find a value of $2\theta$ (not $x$) in the interval. Allow sign error in moving $\alpha$ to right side

Obtain one correct answer e.g. ${39.2}^{\circ }$

Obtain second correct answer e.g. ${164.2}^{\circ }$ and no others in the interval

$\frac{\mathrm{d}u}{\mathrm{d}x}=-\sin x$

Use double angle formula and substitute for $x$ and $\mathrm{d}x$ throughout the integral

Obtain $\pm \int 2u{\mathrm{e}}^{2u}\mathrm{d}u$

Justify new limits and obtain ${\int }_{-1}^{1}2u{\mathrm{e}}^{2u}\mathrm{d}u$ from correct working

Commence integration and reach $au{\mathrm{e}}^{2u}+b\int {\mathrm{e}}^{2u}\mathrm{d}u$, where $ab\mathrm{\ne }0,b<0$

Complete integration and obtain $u{\mathrm{e}}^{2u}-\frac{1}{2}{\mathrm{e}}^{2u}$

Use correct limits correctly in $cu{\mathrm{e}}^{2u}+d{\mathrm{e}}^{2u}$ having integrated twice or in $c\cos x{\mathrm{e}}^{2\cos x}+d{\mathrm{e}}^{2\cos x}$

Obtain $\frac{1}{2}{\mathrm{e}}^2+\frac{3}{2}{\mathrm{e}}^{-2}$

Separate the variables correctly

Obtain $\ln x$

Split the fraction and integrate to obtain $p\ln (y^2+4)$ or $q{\tan }^{-1}\frac{y}{2}$ correctly

Obtain $\frac{1}{2}\ln (y^2+4)$

Obtain $2{\tan }^{-1}\frac{y}{2}$

Use $(4,2\sqrt{3})$ in an expression containing at least 2 of $a\ln x$, $b\ln (y^2+4)$ and $c{\tan }^{-1}\frac{y}{2}$ to obtain constant of integration

Correct solution (any form) e.g. $\frac{1}{2}\ln (y^2+4)+2{\tan }^{-1}\frac{y}{2}=\ln x+\frac{2\pi }{3}$ or $\frac{1}{2}\ln (y^2+4)+2{\tan }^{-1}\frac{y}{2}=\ln x+2{\tan }^{-1}\sqrt{3}+\frac{1}{2}\ln 16-\ln 4$

Obtain $\sqrt{8}{\mathrm{e}}^{-\frac{1}{6}\pi }$ or 1.68 or more accurate or $2\sqrt{2}{\mathrm{e}}^{-\frac{1}{6}\pi }$ or $\frac{\sqrt{8}}{{\mathrm{e}}^{\frac{1}{6}\pi }}$ or ${\mathrm{e}}^{0.516}$

Perform scalar product of direction vectors and set result equal to zero

Use $P$ to find the value of $\lambda$

Obtain $c=-5$ or $b=6$

$a=-6,b=6\text{ and }c=-5\text{ all correct}$

Find $\overrightarrow{PQ}$ (or $\overrightarrow{QP}$) for a general point $Q$ on $m$ $=\pm ((1+2\mu ,2-3\mu ,3+\mu )-(a+\lambda c,3-2\lambda ,b+4\lambda ))$

Equate the scalar product of $\overrightarrow{PQ}$ (or $\overrightarrow{QP}$) and a direction vector for $m$ to zero and obtain an equation in $\mu$

Solve and obtain $\mu =-1$

Obtain $\overrightarrow{OQ}=-\mathbf{i}+5\mathbf{j}+2\mathbf{k}$ or $\overrightarrow{PQ}=-5\mathbf{i}-2\mathbf{j}+4\mathbf{k}$ Must be labelled correctly

Carry out a method to find the position vector of $R$

Obtain $-\frac{17}{2}\mathbf{i}+2\mathbf{j}+8\mathbf{k}$ from correct working

State or imply the form $\frac{A}{1+2x}+\frac{B}{3-x}+\frac{C}{(3-x{)}^2}$

Use a correct method to find a constant

Obtain one of $A=2,B=2\text{ and }C=-3$

Obtain a second value

Obtain the third value

Use a correct method to obtain the first two terms of one of the unsimplified expansions $(1+2x{)}^{-1},{(1-\frac{1}{3}x)}^{-1},{(1-\frac{1}{3}x)}^{-2},(3-x{)}^{-1},(3-x{)}^{-2}$

Obtain the correct unsimplified expansion up to the term in $x^2$ for the first partial fraction

Obtain the correct unsimplified expansion up to the term in $x^2$ for the second partial fraction

Obtain the correct unsimplified expansion up to the term in $x^2$ for the third partial fraction

Obtain final answer $\frac{7}{3}-4x+\frac{215}{27}{x}^2$

Multiply numerator and denominator by $(3-a\mathrm{i})$

Use ${\mathrm{i}}^2=-1$ at least once and separate real and imaginary parts

Obtain $\frac{13a-\mathrm{i}(5a^2+6)}{9+a^2}$ or $\frac{13a-5a^2\mathrm{i}-6\mathrm{i}}{9+a^2}$

Use $\arg z$ to form equation in $a$ $\frac{5a^2+6}{13a}=\pm \tan (\pm \frac{\pi }{4})$ or $-\frac{13a}{5a^2+6}=\pm \tan (\pm \frac{\pi }{4})$ or ${\tan }^{-1}(-\frac{5a^2+6}{13a})=\pm \frac{\pi }{4}$ or ${\tan }^{-1}(-\frac{13a}{5a^2+6})=\pm \frac{\pi }{4}$

Obtain $a=2$

Obtain $z=2-2\mathrm{i}\text{ only}$

State $\arg (z^3)=-\frac{3}{4}\pi$ or evaluate from $z=b-b\mathrm{i}$ or from $-2b^3(1+\mathrm{i})$

Complete method to obtain $r$ from their $z$

$r=16\sqrt{2}$