A Level Mathematics 9709 Paper 23, May/June 2025

Integrate to obtain $2\ln (4x+1)$

Apply limits correctly to $k\ln (4x+1)$ and use at least one relevant logarithm property

Obtain $2\ln 45-2\ln 9=2\ln 5$ or equivalent, and conclude $\ln 25$

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

Draw correct graph of $y=4x-5$

Solve linear equation or inequality with signs of $2x$ and $4x$ different

Obtain $-2x+9=4x-5$ and hence $x=\frac{7}{3}$

Conclude $x>\frac{7}{3}$

State or imply non-modulus equation (or inequality) $(2x-9{)}^2=(4x-5{)}^2$

Attempt solution of three-term quadratic equation (or inequality)

Obtain at least $\frac{7}{3}$ and conclude $x>\frac{7}{3}$

Attempt use of quotient rule (or equivalent) to differentiate $\frac{8x}{2x+3}$ or $\frac{-12x^2+15}{(2x+3)}$

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{24}{(2x+3{)}^2}-6$

Equate first derivative to zero, and attempt solution of a quadratic equation to find two values of $x$

Obtain at least one of the stationary points $(-\frac{5}{2},30)$ and $(-\frac{1}{2},6)$, or both $x$-values, $-\frac{5}{2}$ and $-\frac{1}{2}$

Obtain both stationary points

State $4{\mathrm{e}}^{-2x}=1+0.5\sin 3x$ and confirm $x=-0.5\ln (0.25+0.125\sin 3x)$

Use iterative process correctly at least once

Obtain final answer 0.4912

Show sufficient iterations to 6 significant figures to justify answer or show a sign change in the interval [0.49115, 0.49125]

Integrate to obtain the form $k_1{\mathrm{e}}^{-2x}+k_{2}x+k_3\cos 3x$

Obtain correct $-2{\mathrm{e}}^{-2x}-x+\frac{1}{6}\cos 3x$

Apply limits 0 and *their* part (b) answer correctly and attempt evaluation

Obtain 0.61

Substitute $x=\frac{1}{2}$ and $x=3$, and equate each to zero to produce two equations

Obtain $\frac{1}{16}a+\frac{1}{8}b=-\frac{3}{4}$

Obtain $81a+27b=-27$

Solve simultaneous equations to obtain $a=2$ and $b=-7$

Divide by $2x^2-7x+3$ or successively by $2x-1$ and $x-3$

Obtain quotient $x^2+5$

State fully factorised form $(2x-1)(x-3)(x^2+5)$

Attempt solution of at least $\cot 2\theta =3$

Obtain $\tan 2\theta =\frac{1}{3}$ and hence $\theta =0.161$

Substitute $y={\mathrm{e}}^{-1}$ in equation and simplify to quadratic equation in $x$

Evaluate discriminant

Obtain $-x^2+6x-11=0$, giving discriminant $-8$ and confirm result

Attempt use of product rule for differentiation of $(x^2-3)\ln y$

Obtain $2x\ln y+\frac{x^2-3}{y}\frac{\mathrm{d}y}{\mathrm{d}x}$

Obtain complete $2x\ln y+\frac{x^2-3}{y}\frac{\mathrm{d}y}{\mathrm{d}x}+6=0$

Substitute $x=2$ and $y={\mathrm{e}}^2$ to find value of gradient of tangent

Obtain gradient $-14{\mathrm{e}}^2$

Obtain equation $y=-14{\mathrm{e}}^{2}x+29{\mathrm{e}}^2$ or equivalent of required form

Expand to obtain the form $k_1\cos \theta \sin \theta +k_2{\cos }^2\theta$

Obtain $2\sqrt{3}\cos \theta \sin \theta +2{\cos }^2\theta$ with exact coefficients

Obtain $\sqrt{3}\sin 2\theta +\cos 2\theta +1$

State $R=2$

Use appropriate trigonometry to find value of $\alpha$

Conclude $2\cos (2\theta -60)+1$

Attempt to express equation in form $\cos (4\varphi -\alpha )=k$

Use correct process to find one value of $\varphi$

Obtain $\cos (4\varphi -60)=\frac{1}{3}$ and hence 32.6

Use correct process to find second value of $\varphi$

Obtain 87.4 and no others between 0 and 90