A Level Mathematics 9709 Paper 31, May/June 2025

Symmetrical. In correct position.

Obtain critical value $\frac{4}{5}$ from $3x-1=3-2x$

State final answer $x<\frac{4}{5}$

Use logarithm of a root or a power

Obtain $p^2(p-1)={\text{e}}^3\sqrt{q+1}$

Obtain $q={\left(\frac{p^2(p-1)}{{\text{e}}^3}\right)}^2-1$

$\frac{x+\text{i}(y+5)}{(x-5)+\text{i}y}\times \frac{(x-5)-\text{i}y}{(x-5)-\text{i}y}$

Equate imaginary part of numerator to zero

Obtain $x-y=5$

Correct use of modulus and solve for $x$ and $y$

$\Rightarrow z=4-\text{i}$

$\Rightarrow z=1-4\text{i}$

At $(\text{e},3)$, $\tan t=1$ or $t=\frac{\pi }{4}$

$\frac{\text{d}x}{\text{d}t}={\sec }^{2}t\times {\text{e}}^{\tan t}$

$\frac{\text{d}y}{\text{d}t}=6\tan t\times {\sec }^{2}t$

$\frac{\text{d}y}{\text{d}x}=\frac{6\tan t\times {\sec }^{2}t}{{\sec }^{2}t\times {\text{e}}^{\tan t}}(=6\tan t\times {\text{e}}^{-\tan t})$

$y-3=\frac{6}{\text{e}}(x-\text{e})$

$y=\frac{6}{\text{e}}x-3$

Use $\text{f}(-2a)=-22a^3$

Obtain $-24a^3+4p{a}^3-14a^3+q{a}^3=-22a^3$

Use $\text{f}\left(\frac{a}{3}\right)=-a^3$

Obtain $\frac{1}{9}{a}^3+\frac{1}{9}p{a}^3+\frac{7}{3}{a}^3+q{a}^3=-a^3$

Obtain $p=5,q=-4$

Obtain $\omega z_1=6{\text{e}}^{\text{i}\frac{3}{4}\pi }$

Obtain $\omega z_2=3{\text{e}}^{\text{i}\frac{2}{3}\pi }$

A and B plotted correctly

C and D plotted with angles relatively correct and the same stretch implied.

Rotation $\frac{\pi }{2}$ radians (anticlockwise), or rotation $\frac{3\pi }{2}$ radians clockwise

Enlargement (scale) factor 2

Expand $5\sin (x+\frac{\pi }{6})-4\cos x$ to obtain $\frac{5}{2}\sqrt{3}\sin x-\frac{3}{2}\cos x$

State $R=\sqrt{21}$

Use correct trig formulae to obtain $\tan \alpha$

Obtain $\alpha =0.333$

${\sin }^{-1}\left(\frac{\sqrt{7}}{\sqrt{21}}\right)$

Use a correct method to obtain a value of $\theta$ in the interval

Obtain one correct answer, e.g. 0.47

Obtain second correct answer, e.g. 1.43, and no others in the interval

Use a correct method to form an equation for $l_1$

Obtain $\mathbf{r}=(2\mathbf{i}+4\mathbf{k})+\lambda (3\mathbf{i}+\mathbf{j}+2\mathbf{k})$

Express general point of a line in component form

Equate two pairs of components of $l_2$ and their $l_1$, and solve for $\lambda$ or $\mu$

Obtain e.g. $\lambda =-\frac{1}{5},\mu =-\frac{3}{5}$

Show that this does not fit the third component and hence the lines do not intersect.

Carry out the correct process for evaluating the scalar product of the direction vector of $l_1$ and $l_2$

Using the correct process for the moduli, divide their scalar product by the product of the moduli of their vectors and evaluate the inverse cosine of the result

Obtain AWRT ${38.2}^{\circ }$ or 0.667 radians

Commence integration and reach $p{x}^2\ln x+q\int x\text{d}x$

Obtain $3x^2\ln x-3\int x\text{d}x$

Complete integration and obtain $3x^2\ln x-\frac{3}{2}{x}^2$

Use limits correctly in an expression of the form $a{x}^2\ln x-b{x}^2$ and equate to 4, having integrated twice

Obtain $a=\exp \left(\frac{1}{6}(\frac{5}{a^2}+3)\right)$ correctly

Calculate the values of a relevant expression or pair of expressions at $a=2$ and $a=2.1$

Justify the given statement with correct calculated values

Use the iterative process $a_{n+1}=\exp \left(\frac{1}{6}(\frac{5}{a_n^2}+3)\right)$ correctly at least once.

Obtain final answer 2.02

Show sufficient iterations to at least 4 d.p. to justify 2.02 to 2 d.p. or show that there is a sign change in $(2.015,2.025)$

Divide by $(x^2-3)$ to obtain $x+k$ ($k\mathrm{\ne }0$)

Obtain quotient $x+5$

Obtain remainder $x$

Separate variables correctly and obtain $\int 6y\text{d}y=3y^2$

Obtain $\frac{1}{2}{x}^2+5x$

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

Use $y=2,x=2$ to evaluate the constant of integration in an integral containing $k\ln (x^2-3)$

Obtain $y^2=\frac{1}{6}{x}^2+\frac{5}{3}x+\frac{1}{6}\ln (x^2-3)$

$\frac{\text{d}}{\text{d}x}\sqrt{\sin 2x}=\frac{2\cos 2x}{2\sqrt{\sin 2x}}$ Accept $k\frac{\cos 2x}{\sqrt{\sin 2x}}$

Use correct product rule

Obtain correct derivative in any form

Equate the derivative to zero and obtain a horizontal equation

Use correct trig formulae to obtain an equation in one trig function

Obtain $a=\frac{1}{6}\pi$

Use $[\pi ]\int y^2\text{d}x$

Use double angle formula to obtain a form that can be integrated directly, e.g. $\int {\cos }^{2}x\sin 2x\text{d}x=\int 2{\cos }^{3}x\sin x\text{d}x$

Obtain $-(\pi )\times \frac{1}{2}{\cos }^{4}x$

Use correct limits correctly

Obtain final answer $\frac{1}{2}\pi$