A Level Mathematics 9709 Paper 31, May/June 2023

$3({\text{e}}^{2x}{)}^2-5({\text{e}}^{2x})-4=0$

${\text{e}}^{2x}=\frac{5\pm \sqrt{73}}{6},x=\frac{1}{2}\ln \left(\frac{5+\sqrt{73}}{6}\right)$

$x=0.407$

Show a recognizable sketch graph of $y=\mathrm{\mid }2x+3\mathrm{\mid }$

Find $x$-coordinate of intersection with $y=3x+8$

Obtain $x=-\frac{11}{5}$

State final answer $x>-\frac{11}{5}$ only

State unsimplified term in $x^3$, or its coefficient, in the expansion of $(1+4x{)}^{\frac{1}{2}}$

State unsimplified term in $x^2$, or its coefficient, in the expansion of $(1+4x{)}^{\frac{1}{2}}$

Multiply by $(3+x)$ and combine terms in $x^3$, or their coefficients

Obtain answer 10

Use correct double angle formulae

Obtain ${\cos }^2\theta +2\sin \theta \cos \theta -3{\sin }^2\theta =0$ from full and correct working

Factorise to obtain $(\cos \theta -\sin \theta )(\cos \theta +3\sin \theta )=0$

Solve a quadratic in $\sin \theta$ and $\cos \theta$ to obtain a value for $\theta$.

Obtain one correct value e.g. ${45}^{\circ }$

Obtain a second correct value e.g. ${161.6}^{\circ }$ and no others in the interval

State or imply $2xy+x^2\frac{\text{d}y}{\text{d}x}$ as derivative of $x^{2}y$

State or imply $2ay\frac{\text{d}y}{\text{d}x}$ as derivative of $a{y}^2$

Equate attempted derivative to zero and solve for $\frac{\text{d}y}{\text{d}x}$

Obtain answer $\frac{\text{d}y}{\text{d}x}=\frac{2xy}{2ay-x^2}$ from correct working

State or imply $2ay-x^2=0$

Substitute into equation of curve to obtain equation in $x$ and $a$ or in $y$ and $a$

Obtain one correct point

Obtain second correct point and no others

Obtain a vector for one side of the parallelogram

Correct method to obtain $\pm \overrightarrow{OD}$

Obtain $\overrightarrow{OD}=\mathbf{i}-4\mathbf{j}-3\mathbf{k}$

Using the correct process, evaluate the scalar product $\overrightarrow{BA}\cdot \overrightarrow{BC}$

Using the correct process for the moduli, divide the scalar product by the product of the moduli.

Obtain answer $\frac{2}{\sqrt{62}}$

State or imply $\sin \theta =\sqrt{\frac{58}{62}}$

Use correct method to find the area of $ABCD$

Correct unsimplified expression for the area

Obtain answer $3\sqrt{58}$

Correct separation of variables

Integrate to obtain $k\sec 2x$

Obtain $2\sec 2x$

Use double angle formula and integrate to obtain $py+q\sin 6y$

Obtain $\frac{1}{2}y-\frac{1}{12}\sin 6y$

Use $y=0,x=\frac{\pi }{6}$ in a solution containing terms $\lambda \sec 2x$ and $\mu \sin 6y$ to find the constant of integration

Obtain $\frac{1}{2}y-\frac{1}{12}\sin 6y=2\sec 2x-4$

Obtain $x=0.541$

State or imply the form $\frac{A}{2x+1}+\frac{B}{x+2}+\frac{C}{(x+2{)}^2}$

Use a correct method for finding a constant

Obtain one of $A=1,B=-2,C=3$

Obtain a second value

Obtain the third value

Integrate and obtain one of $\frac{1}{2}\ln (2x+1),-2\ln (x+2),\frac{-3}{x+2}$

Obtain a second term

Obtain the third term

Substitute limits correctly in an integral with at least two terms of the form $\frac{1}{2}\ln (2x+1),-2\ln (x+2)$ and $\frac{-3}{x+2}$ and subtract in correct order

Obtain $1-\ln 3$

Commence integration and reach $px{\text{e}}^{-2x}+q\int {\text{e}}^{-2x}\text{d}x$

Obtain $-\frac{1}{2}x{\text{e}}^{-2x}+\frac{1}{2}\int {\text{e}}^{-2x}\text{d}x$

Complete integration and obtain $-\frac{1}{2}x{\text{e}}^{-2x}-\frac{1}{4}{\text{e}}^{-2x}$

Use limits correctly and equate to $\frac{1}{8}$, having integrated twice

Obtain $a=\frac{1}{2}\ln (4a+2)$ correctly

Calculate the values of a relevant expression or pair of expressions at $a=0.5$ and $a=1$

Justify the given statement with correct calculated values

Use the iterative process $a_{n+1}=\frac{1}{2}\ln (4a_n+2)$ correctly at least once.

Obtain final answer 0.84

Show sufficient iterations to at least 4 d.p. to justify 0.84 to 2 d.p. or show that there is a sign change in (0.835, 0.845)

Substitute $x=-3$ to obtain value of $\text{p}(-3)$

Obtain $\text{p}(-3)=0$ and hence given result

Substitute $z=-1+2\sqrt{6}\text{i}$ and attempt expansions of $z^2$ and $z^3$

Use ${\text{i}}^2=-1$

Obtain $\text{p}(z)=0$ and hence given result

State $z_1=\sqrt{3}\text{i}$ and $z_2=-\sqrt{3}\text{i}$

Expand $(x+\text{i}y{)}^2=-1+2\sqrt{6}\text{i}$ and compare real and imaginary parts

Obtain $x^2-y^2=-1$ and $xy=\sqrt{6}$

Solve to obtain $x$ and $y$

Obtain $z_3=\sqrt{2}+\sqrt{3}\text{i}$ and $z_4=-\sqrt{2}-\sqrt{3}\text{i}$

Use $z^2=-1-2\sqrt{6}\text{i}$ to obtain $z_5$ and $z_6$

Obtain $z_5=\sqrt{2}-\sqrt{3}\text{i}$ and $z_6=-\sqrt{2}+\sqrt{3}\text{i}$