A Level Mathematics 9709 Paper 32, May/June 2023

State or imply non-modular inequality $(5x-3{)}^2<2^2(3x-7{)}^2$, or corresponding quadratic equation, or pair of linear equations $(5x-3)=\pm 2(3x-7)$

Solve a 3-term quadratic, or solve two linear equations for $x$

Obtain critical values $x=\frac{17}{11}$ and $x=11$

State final answer $x<\frac{17}{11}$, $x>11$

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

Remove logarithms and obtain a correct equation in $x$

Obtain final answer $x=\sqrt{2}$ only

Show a circle with centre –3 + 2i.

Show a circle with radius 2

Carry out a correct method for finding the least value of $\mathrm{\mid }z\mathrm{\mid }$

Obtain answer $\sqrt{13}-2$ or $\sqrt{17-4\sqrt{13}}$

Use correct double angle formula to obtain an equation in $\cos (\frac{x}{2})$ only

Obtain a 3 term quadratic in $\cos (\frac{x}{2})$,

Obtain $\cos (\frac{x}{2})=-\frac{3}{4}$ and $\cos (\frac{x}{2})=1$

Solve for the original $x$

Obtain $x=0$ and $4.84$ and no others in the interval

Substitute $2+y\text{i}$ in $a^3-a^2-2a$ and attempt expansions of $a^2$ and $a^3$

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

Obtain final answer $-5y^2+(6y-y^3)\text{i}$

Equate *their* $-5y^2$ to $-20$ and solve for $y$

Obtain $y=-2$

Obtain final answer $\arg a=-\frac{\pi }{4}$

Calculate the values of a relevant expression or pair of expressions at $x=0.5$ and $x=1$

Complete the argument correctly with conclusion about change of sign or change of inequalities and with correct calculated values. Can all be in symbols – an explanation in words is not required.

State $x=\frac{1}{3}(x+4{\tan }^{-1}\frac{1}{3x})$

Rearrange to the given equation $\cot (\frac{x}{2})=3x$ Need intermediate step between $\frac{x}{2}={\tan }^{-1}\frac{1}{3x}$ and $\cot (\frac{x}{2})=3x$

Use the iterative process correctly at least once

Obtain final answer 0.79

Show sufficient iterations to at least 4 d.p. to justify 0.79 to 2 d.p. or show there is a sign change in the interval (0.785, 0.795)

State or imply $6y\frac{\text{d}y}{\text{d}x}$ as the derivative of $3y^2$

State or imply $4x\frac{\text{d}y}{\text{d}x}+4y$ as the derivative of $4xy$

Equate derivative of LHS to zero and solve for $\frac{\text{d}y}{\text{d}x}$

Obtain $\frac{\text{d}y}{\text{d}x}=-\frac{3x+2y}{2x+3y}$

Equate $\frac{\text{d}y}{\text{d}x}$ to –2 and solve for $x$ in terms of $y$ or for $y$ in terms of $x$

Obtain $x=-4y$ or $y=-\frac{x}{4}$

Substitute *their* $x=-4y$ or *their* $y=-\frac{x}{4}$ in curve equation

Obtain $y=\pm \frac{1}{\sqrt{7}}$ or $x=\pm \frac{4}{\sqrt{7}}$

Obtain both pairs of values

Separate variables correctly

Obtain term $-\frac{1}{2}{\text{e}}^{-2x-1}$

Obtain term of the form $a{\tan }^{-1}(\frac{3y}{2})$

Obtain term $\frac{1}{6}{\tan }^{-1}(\frac{3y}{2})$

Use $x=1$, $y=0$ to evaluate a constant or as limits in a solution containing or derived from terms of the form $a{\tan }^{-1}(by)$ and $c{\text{e}}^{\pm (2x+1)}$

Obtain correct answer in any form

Obtain final answer $y=\frac{2}{3}\tan (3{\text{e}}^{-3}-3{\text{e}}^{-2x-1})$

State that $y$ approaches $\frac{2}{3}\tan (3{\text{e}}^{-3})$

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 $A=-4$, $B=-3$ and $C=5$

Obtain a second value

Obtain the third value

Integrate and obtain term $-2\ln (1+2x)$

Integrate and obtain term $3\ln (2-x)$

Integrate and obtain term $\frac{5}{2-x}$

Substitute limits correctly in an integral with two terms (obtained correctly) of the form $a\ln (1+2x)+b\ln (2-x)+\frac{c}{2-x}$, where $abc\mathrm{\ne }0$

Obtain answer $\frac{5}{2}-\ln 72$ after full and correct working

Use the product rule correctly to obtain $p(x+5)(3-2x{)}^n+q(3-2x{)}^{\frac{1}{2}}$

Obtain correct derivative in any form

Equate derivative to zero and obtain a linear equation

Obtain a correct linear equation.

Obtain answer $(-\frac{2}{3},\frac{13\sqrt{39}}{9})$

Use the given substitution and reach $a\int (\frac{13}{2}-\frac{u}{2})u^{\frac{1}{2}}\text{d}u$

Obtain correct integral $-\frac{1}{2}\int (\frac{13}{2}-\frac{u}{2})u^{\frac{1}{2}}\text{d}u$

$x=-5$ and $\frac{3}{2}$

Use correct limits the right way round in an integral of the form $a(\frac{26}{3}{u}^{\frac{3}{2}}-\frac{2}{5}{u}^{\frac{5}{2}})$

Obtain answer $\frac{169}{15}\sqrt{13}$ or $a=\frac{169}{15}$

Carry out correct method for finding a vector equation for $AB$

Obtain $[\mathbf{r}=]\mathbf{i}+2\mathbf{j}-2\mathbf{k}+\lambda (\mathbf{i}-3\mathbf{j}+3\mathbf{k})$

Equate two pairs of components of general points on their $AB$ and $l$ and evaluate $\lambda$ or $\mu$

Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda =-1$, $\mu =-2$

Verify that all three equations are not satisfied and the lines fail to intersect ($\mathrm{\ne }$ is sufficient justification e.g. $0\mathrm{\ne }-3$).

Find $\overrightarrow{AP}$ for a general point $P$ on $l$, e.g. $-3\mathbf{j}+5\mathbf{k}+\mu (2\mathbf{i}-3\mathbf{j}+4\mathbf{k})$

Calculate scalar product of *their* $\overrightarrow{AP}$ and a direction vector for $l$ and equate the result to zero

Obtain $\mu =-1$

Obtain answer $-\mathbf{i}+2\mathbf{j}-\mathbf{k}$