A Level Mathematics 9709 Paper 31, Oct/Nov 2024

Substitute $x=-\frac{1}{2}$ and equate the result to zero

Obtain a correct equation, e.g. $-\frac{4}{8}+\frac{a}{4}-\frac{5}{2}+b=0$

Substitute $x=2$ and $x=4$ and use $\mathrm{p}(4)=3\mathrm{p}(2)$

Obtain a correct equation, e.g. $3(32+4a+10+b)=256+16a+20+b$

Obtain $a=-32$ and $b=11$

Integrate to obtain $p{x}^3\ln 3x+q\int x^2\mathrm{d}x$

Obtain $\frac{1}{3}{x}^3\ln 3x-\frac{1}{3}\int x^2\mathrm{d}x$

Complete integration to obtain $\frac{1}{3}{x}^3\ln 3x-\frac{1}{9}{x}^3$

Use correct limits correctly in an expression of the form $r{x}^3\ln 3x+s{x}^3$

Obtain $\frac{53}{3}\ln 3-\frac{26}{9}$

State or imply $\frac{1+\frac{\mathrm{d}y}{\mathrm{d}x}}{x+y}$ as the derivative of $\ln (x+y)$

State or imply $6xy+3x^2\frac{\mathrm{d}y}{\mathrm{d}x}$ as the derivative of $3x^{2}y$

Substitute $(1,0)$ and solve for $\frac{\mathrm{d}y}{\mathrm{d}x}$

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2}$ or $0.5$

Factorise Or obtain an expression in ${\cos }^2\theta$ or ${\sec }^2\theta$

Use of $1+{\tan }^2\theta ={\sec }^2\theta$ (anywhere)

Obtain $1\times (1+{\tan }^2\theta +{\tan }^2\theta )=1+2{\tan }^2\theta$

Form an equation in $\tan 2\alpha$. Or multiply through by ${\cos }^{4}2\alpha$ to form an equation in $\sin 2\alpha$ or $\cos 2\alpha$

Solve for $\tan 2\alpha$ or equivalent

Obtain one correct solution for $\alpha$, e.g. $20.0(\mathrm{30..}{)}^{\circ }$

Obtain a second correct value for $\alpha$, e.g. ${70.0}^{\circ }$ ($\mathrm{69.9698.}{\mathrm{.}}^{\circ }$)

Obtain solutions ${110}^{\circ }$ ($110.0$) and ${160}^{\circ }$ ($160.0$) for $\alpha$, and no others in range

Sketch a relevant graph, e.g. $y=2+{\mathrm{e}}^{-0.2x}$ For the sketches: Correct curvature Intersections with the y-axis approximately correct Horizontal asymptote approximately correct – need not draw in Allow scale not marked and implied by their sketch

Sketch a second relevant graph, e.g. $y=\ln (1+x)$ and justify the given statement

Calculate the value of a relevant expression or values of a relevant pair of expressions at $x=7$ and $x=9$

Complete the argument correctly with correct calculated values

Use the iterative process correctly at least once

Obtain final answer 8.03

Show sufficient iterations to at least 4 decimal places to justify 8.03 to 2 decimal places, or show that there is a sign change in the interval $(8.025,8.035)$

Correct use of product rule to differentiate

Obtain $2\cos 2x+4\cos 2x\sin 2x$

Equate derivative to zero and solve for $2x$ or $x$

Obtain $x=\frac{7}{12}\pi$, $y=-\frac{1}{4}$

Use correct double angle formula to integrate Or use integration by parts and correct double angle formula

Obtain $-\frac{1}{2}\cos 2x+\frac{x}{2}-\frac{1}{8}\sin 4x(+C)$

Use limits 0 and $\frac{1}{2}\pi$ correctly in a solution containing $p\cos 2x$ and $q\sin 4x$

Obtain $\frac{1}{4}\pi +1$

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

Use a correct method to find a constant

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

Obtain a second value

Obtain the third value

State $\frac{-1.-2.-3}{3!}(2x{)}^3$ or $-8$

Use a correct method to obtain the coefficient of $x^2$ in the expansion of $(2+x^2{)}^{-1}$ or the coefficient of $x^2$ in the expansion of $(1+\frac{x^2}{2}{)}^{-1}$.

Obtain $(Bx+C)\times \frac{1}{2}x-\frac{1}{2}{x}^2$ or $-\frac{B}{4}{x}^3$ or $-\frac{B}{4}$

Obtain final answer $-8\frac{1}{2}$ or $-8\frac{1}{2}{x}^3$

Multiply numerator and denominator by $1-y\mathrm{i}$

Obtain $\frac{1}{1+y^2}+\frac{-y}{1+y^2}\mathrm{i}$

Express ${(a-\frac{1}{2})}^2+b^2$ in terms of $y$ and expand the bracket

Obtain $(\frac{1}{(1+y^2{)}^2}-\frac{2}{2(1+y^2)}+\frac{1}{4})+\frac{y^2}{(1+y^2{)}^2}$

Obtain $\frac{1}{4}$ from full and correct working

Show a vertical straight line through $1+0\mathrm{i}$

Show a circle centre $\frac{1}{2}+0\mathrm{i}$

Show a circle with radius $\frac{1}{2}$ and centre not at the origin

circle centre $\frac{1}{2}+0\mathrm{i}$ with radius $\frac{1}{2}$

Use a correct method to form a vector equation

Obtain $\mathbf{r}=8\mathbf{i}-5\mathbf{j}+6\mathbf{k}+\lambda (2\mathbf{i}+\mathbf{j}+4\mathbf{k})$

State the position vector of a point on $l$ in component form Or at least 2 correct components seen

Equate to $-t\mathbf{i}+4t\mathbf{j}+3t\mathbf{k}$ and solve for $t$

Obtain $t=-2$

Evaluate the scalar product of a pair of relevant vectors

Complete the process for finding the cosine of $\theta$

Obtain $\frac{2a+11}{\sqrt{21}\sqrt{10+a^2}}=\pm \frac{1}{\sqrt{6}}$

Form a 3-term quadratic equation in $a$ and solve for $a$

Obtain $a=-2,a=-86$

$\frac{\mathrm{d}V}{\mathrm{d}t}=\pm k\sqrt{V}$ or $\frac{\mathrm{d}V}{\mathrm{d}t}=16\pi \frac{\mathrm{d}h}{\mathrm{d}t}$

Correct use of chain rule and $V=16\pi h$

Obtain $\frac{\mathrm{d}h}{\mathrm{d}t}=\frac{\mathrm{d}V}{\mathrm{d}t}÷\frac{\mathrm{d}V}{\mathrm{d}h}=\frac{-k\sqrt{16\pi h}}{16\pi }$

$=-\left(\frac{k}{4\sqrt{\pi }}\right)\sqrt{h}=-\lambda \sqrt{h}$ since $\frac{k}{4\sqrt{\pi }}$ is constant

Separate variables correctly and commence integration

Obtain $-\lambda t=2\sqrt{h}(+C)$

Use the boundary conditions in an equation containing $pt$ and $q\sqrt{h}$ to form an equation in $\lambda$ and/or $C$

Use the boundary conditions in an equation containing $pt$ and $q\sqrt{h}$ to form a second equation in $\lambda$ and/or $C$ and solve

Hence $t=80-40\sqrt{h}$

Time to empty the tank is 80 minutes