A Level Mathematics 9709 Paper 32, Oct/Nov 2023

Show a recognizable sketch graph of $y=\mathrm{\mid }4x-2\mathrm{\mid }$. Roughly symmetrical. Should extend into the second quadrant. Ignore $y=4x-2$ below the axis if intention is clear e.g. dashed or the required lines are clearly bolder. Some indication of scale on both axes – accept dashes. Must go beyond $(0,2)$ and $(1,2)$. Ignore any attempt to sketch $y=1+3x$.

Obtain critical value $x=3$

Solve the linear equation $1+3x=2-4x$

Obtain critical value $\frac{1}{7}$

Obtain final answer $x<\frac{1}{7}$ [or] $x>3$

Obtain $\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{2}{t}\ln t$

Obtain $\frac{\mathrm{d}y}{\mathrm{d}t}=-2t{\mathrm{e}}^{2-t^2}$

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}t}÷\frac{\mathrm{d}x}{\mathrm{d}t}$ and substitute $t=\mathrm{e}$

Obtain $-{\mathrm{e}}^{4-{\mathrm{e}}^2}$

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

Obtain a correct equation with powers evaluated

Substitute $x=-1$ and equate result to 12

Obtain a correct equation with powers evaluated

Obtain $a=-3,b=6$

Show a circle with centre $4+3\mathrm{i}$. Accept a curved shape with correct point roughly in the middle.

Show a circle with radius 2 and centre not at the origin. The shape should be consistent with their scales

Show correct vertical line. Enough to meet correct circle twice or complete line for any other circle.

Shade the correct region on a correct diagram Any other shading must be accompanied by words to explain which region is required

Carry out a complete method for finding the greatest value of $\arg (z)$ e.g ${\tan }^{-1}\frac{3}{4}+{\sin }^{-1}\frac{2}{5}(0.6435+0.4115)$

Obtain answer 1.06 (accept 1.055 or 1.056) radians or ${60.45}^{\circ }$ (accept ${60.4}^{\circ }$ or ${60.5}^{\circ }$)

Split fraction to obtain $1+\frac{x-4}{x^2+4}$

Attempt integration and obtain $p\ln (x^2+4)$ or $q{\tan }^{-1}\left(\frac{x}{2}\right)$ from correct working

Obtain $\frac{1}{2}\ln (x^2+4)$

Obtain $-2{\tan }^{-1}\left(\frac{x}{2}\right)$

Correct use of correct limits 0 and 6 in an expression involving $p\ln (x^2+4)$, $q{\tan }^{-1}\left(\frac{x}{2}\right)$ and no incorrect terms.

Obtain $6+\frac{1}{2}\ln 10-2{\tan }^{-1}3$

Sketch a relevant graph. e.g. $y=\cot x$: $x$ intercept should be correct. Not touching the $y$-axis. No incorrect curvature. Ignore anything outside $0<x⩽\frac{1}{2}\pi$.

Sketch a second relevant graph and justify the given statement e.g. $y=2-\cos x$: Condone if looks almost straight, but not if drawn with a ruler and not incorrect curvature. Correct $y$ intercept. Needs to be drawn for $0<x⩽\frac{1}{2}\pi$. Ignore outside this.

Calculate the value of a relevant expression or values of a pair of expressions at $x=0.6$ and $x=0.8$ Must be working in radians. Values correct to at least 2 significant figures. Need all relevant values but only one (pair) needs to be correct to award M1. Complete set of values for their expression. If not comparing with 0 or 1 then the pairing must be clear, not just embedded values.

Complete the argument correctly with correct calculated values (awrt 2 s.f.). Clear comparison for their expression. Allow work on a smaller interval.

Use the iterative process correctly at least once. Must be working in radians

Obtain final answer 0.68

Show sufficient iterations to at least 4 d.p. to justify 0.68 to 2 d.p. or show there is a sign change in the interval $(0.675,0.685)$. Allow recovery. Allow truncation. Allow small differences in the $4^{\text{th}}$ s.f.

Use correct expansion for $\cos (2\theta +\theta )$

Use correct double angle formulae to express $\cos 3\theta$ in terms of $\cos \theta$ and $\sin \theta$

Show sufficient working to confirm $\cos 3\theta \equiv 4{\cos }^3\theta -3\cos \theta$

Use the identity and correct double angle formula to obtain an equation in $\cos \theta$ only. Must come from using all three terms in the given equation.

Obtain $\theta ={90}^{\circ }$

Solve a 3-term quadratic in $\cos \theta$ to obtain a value of $\theta$

Obtain one value e.g. ${25.3}^{\circ }$

Obtain a second value e.g. ${137.5}^{\circ }$ and no extras in range

Multiply both sides by $a+2\mathrm{i}$ and attempt expansion of right-hand side

Use of ${\mathrm{i}}^2=-1$ seen at least once (or implied)

Compare real and imaginary parts to obtain an equation in $a$ only $[2=\lambda (2a+2),3a=\lambda (-a+4)]$

Obtain $3a^2+4a-4=0$ from correct working

Solve given quadratic to obtain a value of $a$ and use this to form an equation in $\lambda$ only (based on an equation seen in *their* working in (a) or (b))

Obtain $a=-2,\lambda =-1$ or $a=\frac{2}{3},\lambda =\frac{3}{5}$

Obtain second correct pair of values

Use correct product rule

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}=\cos x\cos 2x-2\sin x\sin 2x$

Equate derivative to zero and use correct double angle formulae

Obtain $\cos x(1-6{\sin }^{2}x)=0$ or equivalent

Obtain $a=0.42$

Use double angle formula and obtain $p{\cos }^{3}x+q\cos x$ correctly

Obtain $\pm (-\frac{2}{3}{\cos }^{3}x+\cos x)$

Correct use of limits $\frac{1}{4}\pi$ and $\frac{3}{4}\pi$ (or use double the integral from $\frac{1}{4}\pi$ to $\frac{1}{2}\pi$)

Obtain $\frac{2\sqrt{2}}{3}$

Use the correct process to calculate the scalar product of the direction vectors

Divide the scalar product by the product of the moduli and equate the result to $\cos {60}^{\circ }$

Obtain correct equation in $c$

Obtain $c=2$

Calling $(6,-3,6)$ $A$, find $\overrightarrow{AP}$ for a general point $P$ on $l$

Equate the scalar product of *their* $\overrightarrow{AP}$ and a direction vector for $l$ to zero and obtain an equation in $\lambda$

Solve and obtain $\lambda =2$

Carry out a method to calculate $\mathrm{\mid }\overrightarrow{AP}\mathrm{\mid }$

Obtain $\sqrt{11}$ from correct working

Correct separation of variables.

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

Express $\frac{1}{y^2+y}$ in partial fractions or express the denominator of the fraction as a difference of two squares

Obtain $\frac{1}{y}-\frac{1}{y+1}$ or $\frac{1}{{(y+\frac{1}{2})}^2-{\left(\frac{1}{2}\right)}^2}$

Obtain $\ln y-\ln (y+1)$

Use $x=1,y=1$ to find constant of integration or as limits in a definite integral in an expression containing terms of the form $\frac{p}{x},q\ln y$ and $r\ln (1+y)$

Correct equation in $x$ and $y$

Obtain $y=\frac{{\mathrm{e}}^{\frac{1}{x}-1}}{2-{\mathrm{e}}^{\frac{1}{x}-1}}$

State that $y$ approaches $\frac{1}{2\mathrm{e}-1}$