A Level Mathematics 9709 Paper 32, Oct/Nov 2025

Roughly symmetrical. Condone some inaccuracy, but needs to look as if they intended symmetry and straight lines. If not, then B0. Needs to be in the correct position. Needs to have two solid line segments. Needs to exist in both quadrants above the axis. Ignore dotted lines below the axis. Solid line below the axis is B0. Condone if no scale shown, but need to see 3a and -3a marked. Ignore $y=a-2x$ if seen.

Obtain critical value $-\frac{2}{3}a$ from $x+3a=a-2x$ (or from $(x+3a{)}^2=(a-2x{)}^2$)

State final answer $x>-\frac{2}{3}a$

Use a correct law for the logarithm of a product or the logarithm of a power (or use a correct law of indices for a product or a power)

Obtain $\ln 3+(x+1)\ln 2=\ln 4+(2x-3)\ln 3$ (or $(\frac{2}{9}{)}^x=\frac{2}{81}$)

Solve for $x$

Obtain 2.46 only

Complete method to obtain equation in $y$ only or Obtain the $y$ coordinate of $P$

Obtain $2+\text{i}(1-\sqrt{3})$

State one relevant angle, e.g. ${\tan }^{-1}\frac{1}{3}$ or ${\sin }^{-1}\frac{2}{\sqrt{10}}$ (or find $m=\frac{3+2\sqrt{6}}{5}$ choosing positive root)

Complete method to obtain the required angle

Obtain 1.01 radians or ${57.7}^{\circ }$

Begin integration by parts and obtain $p{x}^2{\tan }^{-1}x+q\int \frac{x^2}{1+x^2}\text{d}x$

Obtain $\frac{x^2}{2}{\tan }^{-1}x-\frac{1}{2}\int \frac{x^2}{1+x^2}\text{d}x$

Use $\int \frac{x^2}{1+x^2}\text{d}x=\int 1-\frac{1}{1+x^2}\text{d}x$ and integrate to obtain $\beta x+\gamma {\tan }^{-1}x$

Obtain $\frac{x^2}{2}{\tan }^{-1}x-\frac{1}{2}(x-{\tan }^{-1}x)$

Substitute correct limits correctly in an expression of the form $\alpha x^2{\tan }^{-1}x+\beta x+\gamma {\tan }^{-1}x$

Obtain $\frac{\pi }{4}-\frac{1}{2}$

Use correct product rule

Obtain correct derivative and state a clear conclusion

Use $\text{f}(3)=0$ (or attempt division of $\text{f}(x)$ by $(x-3{)}^2$ as far as $(2x+Q)$)

Obtain $3p+q=-18$ (or obtain quotient $(2x+8)$)

Use ${\text{f}}^{\mathrm{\prime }}(3)=0$ (or equate linear remainder to zero and compare coefficients)

Obtain $p=-30$ (or obtain one of $p=-30,q=72$)

Obtain $p=-30,q=72$

Sketch a relevant graph, e.g. $y=\cot 2x$

Sketch a second relevant graph, e.g. $y=2\sin 2x-1$ and justify the given statement

Calculate the values of a relevant expression or pair of expressions at $x=0.4$ and $x=0.6$

Justify the given statement with correct calculated values

Use the iterative process $x_{n+1}=\frac{1}{2}{\tan }^{-1}(\frac{1}{2\sin 2x_n-1})$ correctly at least once

Obtain final answer 0.49

Show sufficient iterations to 4 d.p. to justify 0.49 to 2 d.p. or show that there is a sign change in $(0.485,0.495)$

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

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

Complete the differentiation and equate the derivative of the LHS to zero and solve for $\frac{\text{d}y}{\text{d}x}$ (the $=0$ can be implied)

Obtain $\frac{\text{d}y}{\text{d}x}=\frac{x^2+2xy}{2y^2-x^2}$ from correct working

Equate derivative to 0 and solve for $x$ or for $x$ in terms of $y$.

Obtain $(0,2)$

Use their $x=-2y$ to form an equation in one unknown Or equivalent e.g. substitute $y=-\frac{x^2}{2x}$

Obtain $(4,-2)$

Use correct double angle formula to expand $\sin 4x$

Use correct double angle formulae to obtain an expression in $\sin x$ and $\cos x$

Obtain $\sin 4x\equiv 4\sin x(2{\cos }^{3}x-\cos x)$ from fully correct working

Use the identity to obtain $p\int \sin x{\cos }^{6}x\text{d}x+q\int \sin x{\cos }^{4}x\text{d}x$

Obtain $r{\cos }^{7}x+s{\cos }^{5}x$

Obtain $-\frac{8}{7}{\cos }^{7}x+\frac{4}{5}{\cos }^{5}x$

Use the correct limits correctly in an expression of the form $r{\cos }^{7}x+s{\cos }^{5}x$

Obtain $\frac{1}{35}(\sqrt{2}+12)$

State or imply the form $A+\frac{B}{x+2a}+\frac{C}{x+3a}$

Use a correct method for finding a coefficient

Obtain one of $A=1,B=2a$ and $C=-3a$

Obtain a second value

Obtain the third value

Integrate and obtain terms $Ax+B\ln (x+2a)+C\ln (x+3a)$

Substitute limits correctly in an integral containing at least 2 terms from the form $rx+s\ln (x+2a)+t\ln (x+3a)$

Obtain $a(2+\ln (\frac{9}{8}))$ from correct working

State $\frac{\text{d}V}{\text{d}t}=5000-k{h}^2$

Use $V=2500h$ and chain rule to obtain $\frac{\text{their }\frac{\text{d}V}{\text{d}t}}{2500}$

Use $h=20,\frac{\text{d}h}{\text{d}t}=0.4$ to obtain $k$

Obtain $\frac{\text{d}h}{\text{d}t}=\frac{500-h^2}{250}$ from correct working

Separate variables correctly

Obtain $\frac{1}{250}t$

Obtain $\frac{1}{2\sqrt{500}}\ln \mathrm{\mid }\frac{\sqrt{500}+h}{\sqrt{500}-h}\mathrm{\mid }$

Use $t=0,h=0$ in an expression of the form $pt+q\ln (a+h)+r\ln (b-h)$ to obtain the constant of integration or as limits on a definite integral

Obtain $t=16.1(\text{s})$

$\overrightarrow{MB}=-\mathbf{i}+5\mathbf{j}+2\mathbf{k}$

$\overrightarrow{MC}=-5\mathbf{i}-3\mathbf{j}+4\mathbf{k}$

Carry out the correct process for evaluating the scalar product of $\overrightarrow{MB}$ and $\overrightarrow{MC}$ (or process for finding lengths of sides of triangle $BMC$)

Using the correct process for the moduli, divide the scalar product by the product of the moduli for their two vectors (or correct use of cosine rule as far as $\cos \theta =\dots$)

Obtain $-\frac{\sqrt{15}}{75}$ (or $\frac{-1}{\sqrt{375}}$)

Working with triangle $MBC$: Correct method to calculate the exact value of $\sin (\mathrm{\angle }CMB)$

$\sin (\mathrm{\angle }CMB)=\sqrt{\frac{374}{375}}$

Correct method to calculate the area of triangle $ABC$ using $MB$ and $MC$

$2\times \frac{1}{2}\sqrt{30}\sqrt{50}\times \sqrt{\frac{374}{375}}=2\sqrt{374}$