A Level Mathematics 9709 Paper 11, Oct/Nov 2025

Use of $b^2-4ac$

Obtain $k^2-20k+64$

Attempt solution of quadratic equation or inequality

Obtain $k<4$, $k>16$ or clear equivalent

State or imply that second and fifth terms are $a\cos \theta$ and $a{\cos }^4\theta$ or $r^3=\frac{1}{64}$

Attempt solution of equation of form ${\cos }^3\theta =k$ as far as $\cos \theta =\dots$, where $0<k<1$

Obtain $\cos \theta =0.25$, and hence $\theta =1.32$ only

Attempt to find $a$ using their $\theta$ and substitute in correct formula for sum to infinity

Obtain $a=32$ and use exact value of $r$ to obtain $\frac{128}{3}$

$x^4$ term from $(px+3{)}^5$ is $5p^4{x}^4\times 3$

Identify term involving $x^6\times \frac{p^2}{x^2}$ as relevant term from $(x^3+\frac{p}{x}{)}^4$

Obtain $6p^2{x}^4$ or $-6p^2{x}^4$

Attempt solution for $p^2$ of equation that is quadratic in $p^2$

Obtain $15p^4-6p^2-216=0$ or equivalent and hence $p=2$

10

+ (or correct middle sign)

3

State reflection in $x$-axis

Obtain $m=-3$

Obtain $n=10$

Use ${\tan }^4\theta =\frac{{\sin }^4\theta }{{\cos }^4\theta }$ or ${\tan }^4\theta -1=(\frac{{\sin }^2\theta }{{\cos }^2\theta }+1)(\frac{{\sin }^2\theta }{{\cos }^2\theta }-1)$

Attempt to express ${\tan }^4\theta -1$ as a single fraction in terms of $\cos \theta$ only

Confirm result $\frac{1-2{\cos }^2\theta }{{\cos }^4\theta }$

Use identity to form equation using $\cos \theta$ only

Obtain $9{\cos }^2\theta =1$ or similarly simplified equivalent

Solve equation of form ${\cos }^2\theta =k$ to obtain at least one solution

Obtain 70.5 and 109.5

State $\mathrm{f}(x)⩾-3$ or $y⩾-3$ or clear equivalent

Attempt to arrange to $x=\dots$ in terms of $y$ or equivalent

Obtain $-3+\sqrt{x+12}$

Attempt expression for $\mathrm{g}\mathrm{f}(x)$

Obtain $2(x+3{)}^2-29=69$

Attempt solution of quadratic equation to find at least one value of $x$

Obtain $x=4$ only

Obtain correct $\frac{1}{2}{r}^2\times \frac{2}{3}\pi -\frac{1}{2}{r}^2\sin \frac{2}{3}\pi$

$0.614r^2$

Obtain $\left[\frac{\mathrm{d}A}{\mathrm{d}r}\right]=1.228r$

Use $\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{d}A}{\mathrm{d}r}\times \frac{\mathrm{d}r}{\mathrm{d}t}$, or equivalent, with $r=20$

Obtain 9.8

State or imply $\frac{2}{3}\pi r$ for length of arc $AB$

Differentiate and apply correct use of chain rule

Obtain 0.84

Integrate to obtain expression $\frac{1}{2}{x}^{\frac{3}{2}}÷\frac{3}{2}$

Evaluate integral of the form $k{x}^{\frac{3}{2}}$ between limits 0 and 9 and subtract result from $\frac{27}{2}$

Obtain $\frac{1}{3}{x}^{\frac{3}{2}}$ giving area under curve is 9 and shaded area is $\frac{9}{2}$

Attempt to express $x^2$ in terms of $y$

Obtain or imply volume is $[\pi ]\int 16y^4[\mathrm{d}y]$

Integrate to obtain $k{y}^5$ and evaluate between limits 0 and $\frac{3}{2}$

Obtain $\frac{16}{5}{y}^5$ and hence $\frac{243}{10}\pi$ or exact equivalent

Attempt to express each of $S_2-1$, $S_4$, $S_9$ in terms of $d$ (with at least one correct)

Obtain $S_2-1=3+d$, $S_4=8+6d$, $S_9=18+36d$

Attempt equation in $d$ by linking the three values

Obtain correct $(3+d)+(18+36d)=2(8+6d)$ or equivalent, and hence $d=-\frac{1}{5}$

$2+14\times \text{their }d$

Second progression: obtain either first term $=\frac{14}{5}$ or common difference $=4$

Second progression: attempt to find 15th term

Obtain 58.8, and hence difference $=59.6$

Attempt to substitute for $x$ or $y$ to produce quadratic equation in one variable

Obtain $5x^2-40x+75=0$ or $5y^2-20=0$

Find two values of one variable

Obtain coordinates $(3,2)$ and $(5,-2)$

Obtain centre $(0,-2)$ [and radius 5]

Attempt area of $ABC$ using mid-point of $AB$ or $\frac{1}{2}\times \text{base}\times \text{height}$

Obtain $\frac{1}{2}\sqrt{20}\times \sqrt{20}$ or $\frac{1}{2}\times 5\times 4$ (or equivalent) and hence 10

Substitute 4 to obtain gradient of curve is $\frac{1}{10}$

Attempt equation of normal using $(4,3)$ and $-\frac{1}{m}$ for their gradient

$y=-10x+43$

$-16x^{-3}$

$+40(2x-3{)}^{-3}$

Substitute 4 to obtain $\frac{7}{100}$

$-8x^{-1}$

$+5(2x-3{)}^{-1}$

Substitute $x=4,y=3$ in an integrated expression to find value of $c$

Obtain $3=-2+1+c$ and hence $y=-8x^{-1}+5(2x-3{)}^{-1}+4$

Substitute $x=-1$ to obtain $q=11$