A Level Mathematics 9709 Paper 31, May/June 2026

State or imply non-modular inequality $3^2(x+4{)}^2>(2x-1{)}^2$, or corresponding quadratic equation, or pair of linear equations

Make a reasonable attempt at solving a 3-term quadratic, or solve two linear equations for $x$

Obtain critical values $x=-13$ and $x=-\frac{11}{5}$

State final answer $x<-13$, $x>-\frac{11}{5}$

State or imply $3\ln y=\ln A+kx$

Substitute values of $\ln y$ and $x$ in equation of correct form and solve for $k$

Obtain $k=-7.51$

Solve for $A$

Obtain $A=212$

Use the correct product or quotient rule

Obtain the correct derivative in any form, e.g $\frac{(3x^2-1)(4x+7)-(2x^2+7x)(6x)}{(3x^2-1{)}^2}$

Substitute $x=1$ into their $\frac{\mathrm{d}y}{\mathrm{d}x}$

Find gradient of normal $=-1÷\text{their }\frac{\mathrm{d}y}{\mathrm{d}x}$

Substitute their gradient of normal, $x=1$ and their $y$ into $y=mx+c$

$x-8y+35=0$

Square $x+\mathrm{i}y$ and equate real and imaginary parts to $-71$ and $-20\sqrt{3}$ respectively

Obtain equations $x^2-y^2=-71$ and $2xy=-20\sqrt{3}$

Eliminate one variable and find an equation in the other

Obtain $x^4+71x^2-300=0$ or $y^4-71y^2-300=0$, or three-term equivalents

Obtain answers $\pm (2-5\sqrt{3}\mathrm{i})$

Expand the brackets to obtain $\cos x\mathrm{cosec}x-1+1-\sin x\sec x$ or equivalent

Simplify to $\cot x-\tan x$ or $\frac{1}{\tan x}-\tan x$

Use double angle formula

Obtain $2\cot 2x$ following full and correct working

Find a value of $x$ in the given interval

Obtain one answer, e.g. 0.294

Obtain all answers 0.294, 1.86 and no others in the interval

Use the correct product rule

Obtain the correct derivative in any form

Equate $\frac{\mathrm{d}y}{\mathrm{d}x}$ to 0 and obtain $\tan x=x+2$ with no omissions and without wrong working

Sketch $y=\tan x$ for $-\pi <x<\pi$

Sketch $y=x+2$ for $-\pi <x<\pi$ and justify the given statement

Calculate the values of a relevant expression or pair of expressions at $x=1.2$ and $x=1.3$

Complete the argument correctly with correct calculated values

Use the iterative process at least once

Obtain final answer 1.274

Show sufficient iterations to 5 d.p. to justify 1.274 to 3 d.p. or show there is a sign change in the interval (1.2735, 1.2745)

Obtain $\mathrm{Re}z⩾-2$ Or $\mathrm{\mid }z+3\mathrm{\mid }⩾\mathrm{\mid }z+1\mathrm{\mid }$ o.e.

Obtain answer of the form $\mathrm{\mid }z-a\mathrm{\mid }\le b$

Obtain answer $\mathrm{\mid }z+3+\mathrm{i}\mathrm{\mid }\le 2$

Attempt to solve $(x+3{)}^2+(y+1{)}^2=2^2$ with $x=-2$

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

Obtain answer 3.39 or $\sqrt{8+2\sqrt{3}}$

Commence integration by parts and reach $A{x}^2{\mathrm{e}}^{3x}-\int Bx{\mathrm{e}}^{3x}\mathrm{d}x$

Obtain $x^2{\mathrm{e}}^{3x}-\int 2x{\mathrm{e}}^{3x}\mathrm{d}x$

Complete integration by parts and reach $A{x}^2{\mathrm{e}}^{3x}-Bx{\mathrm{e}}^{3x}+C{\mathrm{e}}^{3x}$

Obtain $x^2{\mathrm{e}}^{3x}-\frac{2}{3}x{\mathrm{e}}^{3x}+\frac{2}{9}{\mathrm{e}}^{3x}$

Substitute limits correctly in an expression of the form $A{x}^2{\mathrm{e}}^{3x}-Bx{\mathrm{e}}^{3x}+C{\mathrm{e}}^{3x}$

Obtain answer $\frac{5}{9}{\mathrm{e}}^3-\frac{2}{9}$

Use a correct method to form an equation for line $t$

Obtain $\mathbf{r}=\left(\begin{array}{c}2\\ -5\\ 3\end{array}\right)+\lambda \left(\begin{array}{c}-8\\ 9\\ -2\end{array}\right)$

Justify lines are not parallel, e.g. $\left(\begin{array}{c}-1\\ 2\\ -1\end{array}\right)\mathrm{\ne }d\left(\begin{array}{c}-8\\ 9\\ -2\end{array}\right)$

Express $s$ or $t$ in component form e.g. $(1-\mu ,3+2\mu ,4-\mu )$ or $(2-8{\lambda }_1,-5+9{\lambda }_1,3-2{\lambda }_1)$ or $(-6-8{\lambda }_2,4+9{\lambda }_2,1-2{\lambda }_2)$ or $(-2-8{\lambda }_3,-\frac{1}{2}+9{\lambda }_3,2-2{\lambda }_3)$

Equate two pairs of components of general points on $s$ and *their* $t$ and solve simultaneously for $\lambda$ or for $\mu$

Obtain correct answer for $\lambda$ or $\mu$, e.g. $\mu =-\frac{55}{7}$, ${\lambda }_1=-\frac{6}{7}$

Determine that all three equations are not satisfied and the lines fail to intersect, and conclude the lines are skew Conclusion needs to follow correct working

Substitute $x=2$ or $-2$ into $\mathrm{p}(x)$ and equate to 0

Obtain $8a-44+2b+4=0$

Substitute $x=2$ into ${\mathrm{p}}^{\mathrm{\prime }}(x)$ and equate to 24

Obtain $12a-44+b=24$

Obtain $a=6$ and $b=-4$

Equate $(x-2)(6x^2+Ax+B)$ to $6x^3-11x^2-4x+4$ and obtain equations to solve for $A$ and $B$ or divide $6x^3-11x^2-4x+4$ by $x-2$ and reach $a{x}^2\pm kx$

$(x-2)(6x^2+x-2)$

Obtain $(x-2)(3x+2)(2x-1)$

Obtain one correct region $-\frac{2}{3}\le x\le \frac{1}{2}$ or $x\ge 2$

Obtain both regions $-\frac{2}{3}\le x\le \frac{1}{2}$, $x\ge 2$

Use correct quotient rule to obtain expression of the form $\frac{\sin 4x\frac{\mathrm{d}}{\mathrm{d}x}(\cos 4x)-\cos 4x\frac{\mathrm{d}}{\mathrm{d}x}(\sin 4x)}{(\sin 4x{)}^2}$

Obtain $\frac{\sin 4x(-4\sin 4x)-\cos 4x(4\cos 4x)}{{\sin }^{2}4x}$

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}=-4{\mathrm{cosec}}^{2}4x$ correctly

Separate variables correctly and reach $a{\mathrm{cosec}}^{2}4y$ or $b{\mathrm{e}}^{-3x-2}$

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

Obtain term $-\frac{1}{4}\cot 4y$

Use $x=-\frac{2}{3}$, $y=\frac{1}{16}\pi$ to evaluate a constant or as limits in a solution containing terms of the form $a\cot by$ and $c{\mathrm{e}}^{\pm (3x+2)}$

Obtain correct answer in any form

Obtain $y=-0.314$