A Level Mathematics 9709 Paper 23, May/June 2023

Attempt to express equation in terms of $\sec \theta$ only

Obtain $6{\sec }^2\theta -17\sec \theta -14(=0)$

Attempt solution of 3-term quadratic equation to find one value of $\theta$, from $\cos \theta =\mathrm{.}\mathrm{.}\mathrm{.}$

Obtain 73.4

Obtain 286.6

State or imply equation $\ln y=\ln A+(A-B)x$

Equate $A-B$ to gradient of line

Obtain $A-B=4.2$

Substitute appropriate values to find value of $\ln A$

Obtain $\ln A=1.92$ and hence $A=6.82$ and $B=2.62$

Integrate to obtain the form $k\ln(2x + 3)

Obtain correct $3\ln (2x+3)$

Apply limits 0 and 6 correctly to obtain $k\ln 15-k\ln 3$

Apply relevant logarithm properties correctly to obtain form $\ln b$

Obtain $12-\ln 125$

Draw (more or less) correct sketch with vertex on positive $x$-axis

Indicate in some way the two roots

Consider sign of $3-{\text{e}}^{-\frac{1}{2}x}+5x-4$ or of $3-{\text{e}}^{-\frac{1}{2}x}-\mathrm{\mid }5x-4\mathrm{\mid }$ for 0.36 and 0.37

Obtain $-\mathrm{0.035...}$ and $\mathrm{0.018...}$, or equivalents, and justify conclusion

Use iteration process correctly at least once

Obtain final answer 1.295

Show sufficient iterations to 6 sf to justify answer or show sign change in interval [1.2945, 1.2955]

Attempt use of product rule to find first derivative

Obtain $-\frac{1}{2}{\text{e}}^{-\frac{1}{2}x}(x^2-5x+4)+{\text{e}}^{-\frac{1}{2}x}(2x-5)$

Obtain $x=4$ for point $B$

Substitute $x=4$ to find the value of the derivative

Obtain $3{\text{e}}^{-2}$

Equate their first derivative to zero and simplify as far as quadratic equation

Obtain at least $x^2-9x+14=0$

Solve to find relevant $x$ value and substitute to find the value of $y$

Obtain $x=7$ and $y=18{\text{e}}^{-\frac{7}{2}}$

Obtain at least either $(\frac{1}{2}\sin \theta +\frac{1}{2}\sqrt{3}\cos \theta )$ or $(\frac{1}{2}\cos \theta +\frac{1}{2}\sqrt{3}\sin \theta )$

Expand and simplify with correct use of ${\sin }^2\theta +{\cos }^2\theta =1$

Use $\sin \theta \cos \theta =\frac{1}{2}\sin 2\theta$

Confirm given result $\sqrt{3}+2\sin 2\theta$

Identify value of $\theta$ is $\frac{3}{8}\pi$

Obtain $\sqrt{3}+2\sin \frac{3}{4}\pi$ and conclude $\sqrt{3}+\sqrt{2}$

Identify integrand as $\sqrt{3}+2\sin 4x$

Integrate to obtain form $k_{1}x+k_2\cos 4x$

Obtain correct $\sqrt{3}x-\frac{1}{2}\cos 4x$

Obtain $\frac{1}{8}\pi \sqrt{3}+\frac{1}{2}$

Attempt use of quotient rule (or equivalent) to find $\frac{\mathrm{d}x}{\mathrm{d}t}$

Obtain $\frac{1}{(t+2{)}^2}$

Equate $\frac{\mathrm{d}y}{\mathrm{d}x}$ to 1

Obtain $(2t+a)(t+2{)}^2=1$ and expand to confirm given result

Substitute $t=-1$, equate to zero and attempt solution for $a$

Obtain $a=3$

Divide their cubic by $t+1$ at least as far as the $x$ term

Obtain $2t^2+9t+11$

Calculate discriminant, obtain $-7$ and conclude no further value of $t$