A Level Mathematics 9709 Paper 23, May/June 2024

Solve $5x+7=2x-3$ to obtain $-\frac{10}{3}$

Attempt solution of linear equation where $5x$ and $2x$ have different signs

Obtain $-\frac{4}{7}$

State $x<-\frac{10}{3}$, $x>-\frac{4}{7}$

$(2x-1)\ln 6$

$\ln 5+3x+2$

Attempt solution of linear equation

Obtain 9.256

Differentiate to obtain form $k_1{\text{e}}^{-x}+k_2{\text{e}}^{2x}$

Obtain $-8{\text{e}}^{-x}-2{\text{e}}^{2x}$

Substitute $x=0$ to obtain $-10$

Attempt to find $x$-coordinate of $B$

Obtain ${\text{e}}^{3x}=8$ and hence $x=\ln 2$

Integrate to obtain $-8{\text{e}}^{-x}-\frac{1}{2}{\text{e}}^{2x}$

Use limits 0 and $\ln 2$ correctly to find area

Obtain $\frac{5}{2}$

Obtain forms $\frac{\text{d}x}{\text{d}t}=k_1\cos t\sin t$ or $\frac{\text{d}y}{\text{d}t}=k_2\cos 2t$

Obtain correct $-8\cos t\sin t$ or $-4\sin 2t$ and $2\sqrt{3}\cos 2t$

Attempt value of $\frac{\text{d}y}{\text{d}x}$ when $t=\frac{1}{6}\pi$

Obtain $\frac{\text{d}y}{\text{d}x}=-\frac{1}{2}$

State or imply gradient of normal is 2

Attempt equation of normal

Obtain $4x-2y-9=0$

Carry out division at least as far as $3x^2+k_{1}x$

Obtain quotient $3x^2+4x-1$

Confirm remainder is 6

Identify integrand as $3x^2+4x-1+\frac{6}{3x+2}$

Integrate to obtain at least $x^3$ and $k_2\ln (3x+2)$ terms

Obtain $x^3+2x^2-x+2\ln (3x+2)$

Apply limits and appropriate logarithm properties

Obtain $14+\ln 16$

Attempt use of quotient rule

Obtain $\frac{(x+3)\frac{2}{2x+1}-\ln (2x+1)}{(x+3{)}^2}$

Equate first derivative to zero and arrange as far as $2x+1=\dots$

Confirm $x=\frac{x+3}{\ln (2x+1)}-0.5$

Consider sign of $x-\frac{x+3}{\ln (2x+1)}+0.5$ or equivalent for 2.5 and 3.0

Obtain $-0.07$ (0.0696...) and 0.4 (0.4166...) and justify conclusion

Use iterative process correctly at least once

Obtain final answer 2.569

Show sufficient iterations to 6 sf to justify answer or show sign change in interval $[2.5685,2.5695]$

Express left-hand side in terms of $\sin \theta$ and $\cos \theta$ using $\text{cosec}2\theta =\frac{1}{\sin 2\theta }$

Obtain $\frac{1}{\cos \theta }$ and confirm $\sec \theta$

Attempt to obtain quadratic equation in $\sec \theta$ or $\cos \theta$ only

Obtain ${\sec }^2\theta -1+\frac{7}{2}\sec \theta =8$ involving one trigonometric ratio

Attempt to solve 3-term quadratic equation for $\sec \theta$, using a correct method, to find at least one value of $\theta$

Obtain any two of the four correct solutions $\pm 0.952$, $\pm 1.76$

Obtain remaining two correct solutions

Identify integrand as $2{\sec }^2\frac{1}{2}x$

Integrate $k{\sec }^2\frac{1}{2}x$ to obtain $2k\tan \frac{1}{2}x$

Obtain correct $4\tan \frac{1}{2}x$