A Level Mathematics 9709 Paper 32, Oct/Nov 2024

Obtain a correct unsimplified version of the $x$ or $x^2$ term of the expansion of $(9-3x{)}^{-\frac{1}{2}}$ or ${(1-\frac{1}{3}x)}^{-\frac{1}{2}}$

State correct first term 3

Obtain the next term $-\frac{1}{2}x$

Obtain the next term $-\frac{1}{24}{x}^2$

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

Sketch a second relevant graph on the same axes, e.g. $y=\sec x$ and justify the given statement

State $x=\frac{1}{2}{\tan }^{-1}(\cos x)$ and rearrange to the given equation $\cot 2x=\sec x$ Note: If using the alternative approach in (a), can stop at $\tan 2x=\cos x$

Square $x+\mathrm{i}y$ and equate real and imaginary parts to 6 and $-8$ respectively

Obtain equations $x^2-y^2=6$ and $2xy=-8$

Eliminate one variable and find an equation in the other (from 2 equations each in 2 unknowns)

Obtain $x^4-6x^2-16=0$ or $y^4+6y^2-16=0$

Obtain answers $\pm (2\sqrt{2}-\sqrt{2}\mathrm{i})$ or exact equivalents

Use laws of indices correctly and solve for $5^x$

Use a correct method for solving an equation of the form $5^x=a$, where $a>0$

Obtain answer $-0.544$

$\frac{4\pi }{7}$ and/or $-\frac{\pi }{7}$

Obtain $\arg u=\frac{5}{7}\pi$

Reflection (in the) real axis

$\arg u^{\ast }=-\frac{5}{7}\pi$

Form a pair of equations in $a$ and $b$

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

Obtain value $a=0.3$

Obtain value $b=8$

Use correct double angle formula to obtain an equation in $\tan x$

Obtain a correct equation in $\tan x$ in any form without fractions

Reduce to the given answer of ${\tan }^{4}x-2{\tan }^{2}x-3=0$ correctly

A complete correct method to solve the equation to obtain a value for $\theta$

Obtain two of $(\theta =)\frac{1}{6}\pi ,\frac{1}{3}\pi ,\frac{2}{3}\pi$ and $\frac{5}{6}\pi$

Obtain the other two of $(\theta =)\frac{1}{6}\pi ,\frac{1}{3}\pi ,\frac{2}{3}\pi$ and $\frac{5}{6}\pi$ and no others in the interval

Use correct product rule or chain rule to find derivative of $x$ with respect to $t$

Obtain $\frac{\mathrm{d}x}{\mathrm{d}t}=4\tan 2t{\sec }^{2}2t\text{ oe}$

$\frac{\mathrm{d}y}{\mathrm{d}t}=-2\sin 2t$

Use $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}t}\times \frac{\mathrm{d}t}{\mathrm{d}x}$ to obtain given answer $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{2}{\cos }^{3}2t$

Obtain $x=1$ and $y=\frac{\sqrt{2}}{2}$

State or imply gradient of tangent is $-\frac{\sqrt{2}}{8}$ or gradient of normal is $4\sqrt{2}$

Use correct method to find equation of normal using *their* values

Obtain equation of normal is $y=4\sqrt{2}x-\frac{7\sqrt{2}}{2}$ or equivalent 3 term equation

Use a correct method to find $\overrightarrow{OD}$

Obtain position vector of $D$ is $3\mathbf{i}-11\mathbf{j}-10\mathbf{k}$

Carry out correct method for finding a vector equation for $\overrightarrow{AC}$ or $\overrightarrow{BD}$

Both diagonal equations correct.

Equate at least two pairs of corresponding components and solve for $\lambda$ or for $\mu$

Obtain $\lambda =-\frac{1}{4}$ or $\mu =\frac{1}{4}$

Obtain position vector of $P$ is $\frac{3}{4}\mathbf{i}+\frac{1}{4}\mathbf{j}-\frac{7}{4}\mathbf{k}$

Find direction vector $\overrightarrow{BA}=2\mathbf{i}-3\mathbf{j}-4\mathbf{k}$ and $\overrightarrow{BC}=-3\mathbf{i}-6\mathbf{j}+\mathbf{k}$ or equivalent

Carry out correct process for evaluating the scalar product of two relevant vectors

Using the correct process for the moduli, divide *their* scalar product by the product of their moduli and evaluate the inverse cosine of the result to obtain an angle

Obtain answer ${77.3}^{\circ }$ (or 1.35 radians)

Obtain $\frac{\mathrm{d}V}{\mathrm{d}t}=40\pi -0.8\pi r$ or equivalent

Obtain $\frac{\mathrm{d}V}{\mathrm{d}r}=4\pi r^2$ or equivalent e.g. $\frac{\mathrm{d}V}{\mathrm{d}t}=4\pi r^2\frac{\mathrm{d}r}{\mathrm{d}t}$

Use the chain rule to obtain given answer (including the derivative)

Commence division and reach quotient of the form $-5r\pm 250$ or $5r^2=(50-r)(Ar+B)+C$ and reach $A=-5$ and $B=\pm 250$

Obtain quotient $-5r-250$

Obtain remainder $12500$

Prepare to integrate e.g. separate variables correctly Or express in the form $\frac{\mathrm{d}t}{\mathrm{d}r}=\frac{5r^2}{50-r}(=-(5r+250)+\frac{12500}{50-r})$

Obtain term $t$

Obtain terms $\frac{A}{2}{r}^2+Br-C\ln (50-r)$

Obtain terms $-\frac{5}{2}{r}^2-250r-12500\ln (50-r)$

Use $t=0,r=0$ to evaluate a constant or as limits in a solution containing terms of the form $r^2,r,\ln (50-r)$ and $t$

Obtain final answer $t=-\frac{5}{2}{r}^2-250r-12500\ln (50-r)+12500\ln 50$

Obtain $t=70.5$

Use correct quotient rule NB the question asks for ${\mathrm{f}}^{\mathrm{\prime }}(x)$ so need complete form

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

Equate *their* derivative to zero

Solve for $x$ to obtain $x=\ln a$

Obtain $x=\ln \frac{4}{3}$ and $y=-16$

State or imply $\frac{\mathrm{d}u}{\mathrm{d}x}={\mathrm{e}}^x$

Obtain $\int \frac{2u}{u^2-3u+2}\mathrm{d}u$ or equivalent ${\mathrm{O}\mathrm{r}}_1\int (\frac{2}{u}+\frac{8}{u(u-2)}-\frac{2}{u(u-1)})\mathrm{d}u$

State or imply partial fractions of the form $\frac{A}{u-1}+\frac{B}{u-2}$ ${\mathrm{O}\mathrm{r}}_1\frac{C}{u}+\frac{D}{u-2}+\frac{E}{u-1}{\mathrm{O}\mathrm{r}}_2\frac{2u-3}{u^2-3u+2}+\frac{3}{u^2-3u+2}=\frac{2u-3}{u^2-3u+2}+\frac{F}{u-2}+\frac{G}{u-1}$

Use a correct method for finding a constant

Obtain correct $\frac{-2}{u-1}+\frac{4}{u-2}$ ${\mathrm{O}\mathrm{r}}_2\frac{2u-3}{u^2-3u+2}+\frac{3}{u-2}-\frac{3}{u-1}$

Integrate to obtain $a\ln (u-1)+b\ln (u-2)$ or equivalent

Obtain correct $-2\ln (u-1)+4\ln (u-2)$ or equivalent

Correctly use limits $u=5$ and 3 in an expression of the form $a\ln (u-1)+b\ln (u-2)$ or $x=\ln 5$ and $\ln 3$ in an expression of the form $a\ln ({\mathrm{e}}^x-1)+b\ln ({\mathrm{e}}^x-2)$

Obtain $\ln \frac{81}{4}$