A Level Mathematics 9709 Paper 23, Oct/Nov 2023

State or imply that $\cos \theta =\frac{1}{3}\sqrt{5}$

Substitute appropriate values into $\sin \theta \cos {60}^{\circ }+\cos \theta \sin {60}^{\circ }$

Obtain $\frac{1}{3}+\frac{1}{6}\sqrt{15}$

State or imply derivative of $\tan \frac{1}{2}x$ is $\frac{1}{2}{\sec }^2\frac{1}{2}x$ or derivative of $\cos 2x$ is $-2\sin 2x$

Attempt use of product rule to find first derivative

Obtain correct $\frac{3}{2}{\sec }^2\frac{1}{2}x\cos 2x-6\tan \frac{1}{2}x\sin 2x$

Substitute $\frac{1}{3}\pi$ into attempt at first derivative and attempt evaluation to find the gradient

Obtain $-4$

Obtain $2\ln (2x-5)$

Apply limits correctly

Use one relevant logarithm property correctly

Apply second logarithm property correctly and obtain $\ln 25$

Integrate to obtain $\frac{1}{2}{\text{e}}^{2x-5}$

Obtain final answer $\frac{1}{2}{\text{e}}^{15}-\frac{1}{2}{\text{e}}^3$

Draw V-shaped graph with vertex on positive $x$-axis

Draw (more or less) correct graph of $y=2x+7$ with smaller gradient

Solve $3x-5=2x+7$ to obtain $x=12$

Attempt solution of linear equation where signs of $3x$ and $2x$ are different

Obtain $x=-\frac{2}{5}$

Apply logarithms and use power law for $3^y=k$ where $k>0$ or correct equivalent

Obtain 2.26

Substitute $x=-2$ and equate to zero

Substitute $x=-1$ and equate to $-11$

Obtain $4a-2b-68=0$ and $a-b-26=-11$ or equivalents

Solve a pair of relevant simultaneous linear equations to find $a$ or $b$

Obtain $a=19$ and $b=4$

Divide by $x+2$ at least as far as the $x$ term

Obtain $(x+2{)}^2(6x-5)$

Replace (or imply replacement of) $x$ by $3x$ in factorised form

Obtain $-\frac{2}{3}$ and $\frac{5}{18}$

Use $\text{cosec}\theta =\frac{1}{\sin \theta }$

Express in terms of $\sin \theta$ and $\cos \theta$ only

Obtain given result $4+6\cos \theta -4{\cos }^2\theta$ with sufficient detail

Attempt use of formula to solve 3-term quadratic equation as far as $\cos \theta =k_1$

Solve $4{\cos }^2\theta -6\cos \theta -7=0$ to obtain at least $\cos \theta =-0.770\dots$

Obtain $-2.45$

Express ${\cos }^2\theta$ term in the form $k_2+k_3\cos 2\theta$

Obtain integrand $6\cos \theta +2-2\cos 2\theta$

Integrate to obtain correct $6\sin \theta +2\theta -\sin 2\theta$

Differentiate $y^3$ to obtain $3y^2\frac{\text{d}y}{\text{d}x}$

Differentiate complete equation to produce at least one term involving $\frac{\text{d}y}{\text{d}x}$ using implicit differentiation.

Obtain $2{\text{e}}^{2x}-18+3y^2\frac{\text{d}y}{\text{d}x}+\frac{\text{d}y}{\text{d}x}=0$

Substitute $\frac{\text{d}y}{\text{d}x}=0$ to obtain either $p=\frac{1}{2}\ln 9$ or $p=\ln 3$

Substitute value of $p$ in original equation and rearrange as far as $y^3=\dots$ or $q^3=\dots$

Obtain given result $q=\sqrt[3]{2+18\ln 3-q}$ or $y=\sqrt[3]{2+18\ln 3-y}$ with sufficient detail

Consider sign of $q-\sqrt[3]{2+18\ln 3-q}$ or equivalent for 2.5 and 3.0

Obtain $-0.18\dots$ and $0.34\dots$ with sufficient detail and justify conclusion

Use iteration process correctly at least once

Obtain final answer $q=2.673$

Show sufficient iterations to 6 sf to justify answer or show sign change in the interval $[2.6725,2.6735]$