A Level Mathematics 9709 Paper 31, Oct/Nov 2023

Use correct quotient or product rule

Obtain correct derivative in any form

Equate derivative to 8 and solve for $x$

Obtain answers $x=\frac{2}{5}$ and $\frac{4}{15}$

Obtain answers $y=-\frac{4}{5}$ and $-\frac{16}{45}$

Show points representing $2\mathrm{i}$ and $-2+\mathrm{i}$

Show perpendicular bisector of _their_ ($2\mathrm{i}$ and $-2+\mathrm{i}$)

Show correct half–line of gradient 1 from point (–1, 0)

Correct loci and shade correct region

State or imply that $\ln y=\ln a+x\ln b$

Carry out a completely correct method for finding $\ln a$ or $\ln b$

Obtain value $a=4.06$

Obtain value $b=9.97$

Multiply numerator and denominator by $a+5\mathrm{i}$

Use ${\mathrm{i}}^2=-1$

Obtain answer $\frac{3a-10}{a^2+25}+\frac{2a+15}{a^2+25}\mathrm{i}$

State or imply $\mathrm{Im}(a)÷\mathrm{Re}(a)=1$

Obtain answer $a=25$

Use correct trig formulae and obtain an equation in $\sin x$ and $\cos x$

Obtain a correct equation in any form

Substitute exact trig ratios and obtain an expression for $\tan x$

Obtain answer $\tan x=-\frac{1}{\sqrt{3}}$

Obtain answer, e.g. $x=\frac{5\pi }{6}$

Obtain second answer, e.g. $x=\frac{11\pi }{6}$ and no others in the interval

State correct derivative of $x$ or $y$ with respect to $t$

Use $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}t}\times \frac{\mathrm{d}t}{\mathrm{d}x}$

Obtain answer $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{\sqrt{t}}$

State or imply _their_ $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2}$

Obtain $\sqrt{t}=4$

Obtain answer $(7,\ln 16)$

Separate variables correctly

Obtain term $-\ln (\cos \theta )$

Obtain term of the form $a\ln (x^2+3)$

Obtain term $\frac{1}{2}\ln (x^2+3)$

Use $x=1,\theta =0$ to evaluate a constant or as limits in a solution containing terms of the form $a\ln (x^2+3)$ and $b\ln (\cos \theta )$

Obtain correct answer in any form

Obtain final answer $x^2=\frac{4}{{\cos }^2\theta }-3$

Sketch a relevant graph, e.g. $y={\mathrm{e}}^x-3$ $y={\mathrm{e}}^x-3$ Should cut vertical axis at (0, –2) and have increasing gradient.

Sketch a second relevant graph, e.g. $y=\sqrt{x}$ and justify the given statement $y=\sqrt{x}$ should start at (0, 0) and have reducing grading

Calculate the values of a relevant expression or pair of expressions at $x=1$ and $x=2$

Complete the argument correctly with correct calculated values

State $x=\ln (3+\sqrt{x})$ and rearrange to the given equation $\sqrt{x}={\mathrm{e}}^x-3$

Use the iterative process correctly at least once

Obtain final answer 1.43

Show sufficient iterations to at least 4 d.p. to justify 1.43 to 2 d.p. or show there is a sign change in the interval (1.425, 1.435) Condone recovery and small differences in the final figure in the iteration

Use the correct product rule

Obtain correct derivative in any form

Equate derivative to zero and solve for $x$

Obtain answer $(\sqrt{2},\sqrt{2}{\mathrm{e}}^{-\frac{1}{2}})$

State or imply $\mathrm{d}x=\frac{1}{2}{u}^{-\frac{1}{2}}\mathrm{d}u$

Substitute for $x$ and $\mathrm{d}x$

Obtain correct integral $\frac{1}{2}\int {\mathrm{e}}^{-\frac{1}{4}u}\mathrm{d}u$

Use correct limits in an integral of the form $a{\mathrm{e}}^{-\frac{1}{4}u}$ or $a{\mathrm{e}}^{-\frac{1}{4}{x}^2}$

Obtain answer $2-2{\mathrm{e}}^{-\frac{9}{4}}$

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

Use a correct method for finding a coefficient

Obtain one of $A=4$, $B=2$ and $C=-7$

Obtain a second value

Obtain the third value

Use a correct method to find the first two terms of the expansion of $(1-2x{)}^{-1}$, $(2+x{)}^{-1}$, $(2+x{)}^{-2}$, ${(1+\frac{x}{2})}^{-1}$ or ${(1+\frac{x}{2})}^{-2}$

Obtain correct un-simplified expansions up to the term in $x^2$ of each partial fraction: $A(1+(-1)(-2x)+\frac{(-1)(-2)}{2}(-2x{)}^2+\mathrm{.}\mathrm{.})$ $A=4$

Obtain correct un-simplified expansions up to the term in $x^2$ of each partial fraction: $\frac{B}{2}(1+(-1)\left(\frac{x}{2}\right)+\frac{(-1)(-2)}{2}{\left(\frac{x}{2}\right)}^2+\mathrm{.}\mathrm{.})$ $B=2$

Obtain correct un-simplified expansions up to the term in $x^2$ of each partial fraction: $\frac{C}{4}(1+(-2)\left(\frac{x}{2}\right)+\frac{(-2)(-3)}{2}{\left(\frac{x}{2}\right)}^2+\mathrm{.}\mathrm{.})$ $C=-7$

Obtain final answer $\frac{13}{4}+\frac{37}{4}x+\frac{239}{16}{x}^2$

$\mathrm{\mid }x\mathrm{\mid }<\frac{1}{2}$

Obtain $3\mathbf{i}+2\mathbf{j}+\mathbf{k}$

$\overrightarrow{AM}$ or $\overrightarrow{AP}$ correct soi

Carry out correct process for evaluating the scalar product of $\overrightarrow{AM}$ and $\overrightarrow{AP}$

Using the correct process for the moduli, divide the scalar product by the product of the moduli and obtain the inverse cosine of the result

Obtain answer ${53.4}^{\circ }$ or ${0.932}^{\mathrm{c}}$

Find $\overrightarrow{PQ}$ (or $\overrightarrow{QP}$) for a general point $Q$ on the line passing through $O$ and $M$,

Calculate the scalar product of $\overrightarrow{PQ}$ and a direction vector for the line passing through $O$ and $M$ and equate to zero

Solve and obtain correct solution e.g. $\mu =-\frac{1}{2}$

Carry out method to calculate $PQ$

Obtain answer $\frac{\sqrt{10}}{2}$