A Level Mathematics 9709 Paper 22, Oct/Nov 2025

Apply relevant logarithm property

Obtain correct equation without logarithms $\frac{3x+5}{x-2}={\text{e}}^4$ or equivalent

Solve equation to find exact value of $x$

Obtain $\frac{2{\text{e}}^4+5}{{\text{e}}^4-3}$

Attempt identity involving ${\tan }^2\theta$ and ${\sec }^2\theta$ to obtain quadratic equation in $\sec \theta$ or $\cos \theta$

Obtain $2{\sec }^2\theta +3\sec \theta -20=0$ or $20{\cos }^2\theta -3\cos \theta -2=0$

Solve three-term quadratic equation to obtain at least one value of $\theta$

Obtain at least two values of $\pm 66.4$ and $\pm 104.5$

Obtain all four values and no others between –180 and 180

Solve $3x-4=2x+5$ to obtain $x=9$

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

Obtain $x=-\frac{1}{5}$

Conclude $-\frac{1}{5}⩽x⩽9$

State $7^{0.01N}=9$

Use logarithms to find value of $N$

Conclude with single integer 112

Carry out division as far as $x^2\pm 10x$

Obtain quotient $x^2-10x+17$

Confirm remainder is –11

State or imply that equation is $(x^2+3)(x^2-10x+17)=0$

Attempt solution of *their* three-term quadratic quotient from (a) to give two exact roots

Obtain $5\pm \sqrt{8}$ or $5\pm 2\sqrt{2}$ and no other solutions

Differentiate to obtain $-8\sin 2x+8\cos x$

Attempt to solve equation of form $k_1\sin 2x+k_2\cos x=0$ using correct identity

Obtain at least $\sin x=\frac{1}{2}$

Attempt to find both coordinates of at least one maximum point

Obtain $(\frac{1}{6}\pi ,6)$ and $(\frac{5}{6}\pi ,6)$ and no others between 0 and $\pi$

Integrate to obtain expression of form $k_1\sin 2x+k_2\cos x$

Obtain correct $2\sin 2x-8\cos x$

Apply *their* x-limits correctly to evaluate area under curve between $A$ and $B$

Obtain $6\sqrt{3}$ or exact unsimplified equivalent

Carry out correct process to find area of shaded region

Obtain $6(\frac{5}{6}\pi -\frac{1}{6}\pi )-6\sqrt{3}$, and hence $4\pi -6\sqrt{3}$

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

Obtain correct $\frac{1}{4}{\text{e}}^{2x}-\frac{1}{4}{\text{e}}^{-x}$

Apply limits and arrange at least as far as $\frac{1}{2}{\text{e}}^{2a}=\dots$ or ${\text{e}}^{2a}=\dots$

Confirm $a=\frac{1}{2}\ln (10+\frac{1}{2}{\text{e}}^{-a}+\frac{1}{2}{\text{e}}^{-4a})$

Consider sign of $a-\frac{1}{2}\ln (10+\frac{1}{2}{\text{e}}^{-a}+\frac{1}{2}{\text{e}}^{-4a})$ or equivalent for values 1.0 and 1.2

Obtain –0.16… and 0.041… or equivalents and justify conclusion

Use iteration process correctly at least once

Obtain final answer 1.159

Show sufficient iterations to justify answer or show a sign change in the interval [1.1585, 1.1595]

Attempt use of product rule for differentiating $5x^{2}y$

Obtain $10xy+5x^2\frac{\text{d}y}{\text{d}x}$

Differentiate $4{\text{e}}^{2y}$ and obtain $8{\text{e}}^{2y}\frac{\text{d}y}{\text{d}x}$

Obtain $10xy+5x^2\frac{\text{d}y}{\text{d}x}+8{\text{e}}^{2y}\frac{\text{d}y}{\text{d}x}-7=0$ and hence $\frac{\text{d}y}{\text{d}x}=\frac{7-10xy}{5x^2+8{\text{e}}^{2y}}$

Attempt to find relevant value of $x$ and substitute that and $y=0$ to find gradient

Obtain $x=2$ and gradient $\frac{1}{4}$

Attempt to show that denominator of derivative cannot be zero

State that both terms are non-negative or equivalent and conclude appropriately