A Level Mathematics 9709 Paper 21, May/June 2025

Attempt use of product rule

Obtain $6\cos (x^2+1)-12x^2\sin (x^2+1)$

Apply logarithms to both sides and use relevant logarithm property

Obtain $x<\frac{\ln 0.05}{\ln 4}$ or equivalent and hence $x<-2.16$

Attempt solution of equations $3x+8=\pm 9$ or of quadratic equation $(3x+8{)}^2=9^2$

Obtain values $-\frac{17}{3}$ and $\frac{1}{3}$

Conclude $-\frac{17}{3}<x<\frac{1}{3}$

State $-5,-4,-3$ only

Sketch decreasing positive curve for $y=3{\text{e}}^{-2x}$

Sketch curve for $y=\sec x$

Equate and arrange at least as far as ${\text{e}}^{-2x}=\dots$ or ${\text{e}}^{2x}=\dots$ with $\cos x$ present

Confirm $x=\frac{1}{2}\ln (3\cos x)$

Use iterative process correctly at least once

Obtain final answer 0.487

Show sufficient iterations to justify answer, or show a sign change in the interval $[0.4865,0.4875]$

Differentiate to obtain $12{\text{e}}^{2x}-3{\text{e}}^{3x}$

Attempt solution for $x$ of equation of form $k_1{\text{e}}^{2x}+k_2{\text{e}}^{3x}=0$ (but not for $y=0$), to obtain ${\text{e}}^{\pm x}=k$

Obtain $x=\ln 4$ or $x=2\ln 2$

Identify $x=\ln 6$ as point where curve meets x-axis

Integrate to obtain the form $k_3{\text{e}}^{2x}+k_4{\text{e}}^{3x}$

Obtain $3{\text{e}}^{2x}-\frac{1}{3}{\text{e}}^{3x}$

Apply limits to obtain answer $\frac{100}{3}$ only

Substitute $x=\frac{3}{2}$ equating to 0, and $x=-1$ equating to $-15$

Obtain $\frac{27}{8}a+\frac{9}{4}b-\frac{3}{2}a-24=0$

Obtain $-a+b+a-24=-15$

Solve to obtain $a=2$ and $b=9$

Divide $\text{p}(x)$ by $2x-3$

Obtain quotient $x^2+6x+8$

Conclude $(2x-3)(x+2)(x+4)$

Use factorised form to determine value of $\sin \theta$, with $-1<\sin \theta <1$

Obtain finally $\sin \theta =-\frac{3}{4}$ only and hence angle 228.6

Attempt use of quotient rule (or equivalent) for derivative of $x$

Obtain $\frac{5}{(3t+4{)}^2}$ or unsimplified equivalent

Obtain $\frac{6}{3t+4}$ for derivative of $y$

Attempt to express $\frac{\text{d}y}{\text{d}x}$ in terms of $t$

Obtain $\frac{6}{5}(3t+4)$ stating or implying that $c=\frac{6}{5}$

Attempt solution of $2\ln (3t+4)=\ln 100$

Obtain $3t+4=10$ (or equivalent) and hence $t=2$ (only)

Substitute to obtain $a=\frac{1}{2}$

Substitute to obtain $m=12$

State increasing, with reference to positive gradient

Use correct identities to express in terms of $\sin x$ and $\cos x$

Obtain $4{\sin }^{2}x{\cos }^{2}x+4{\cos }^{2}x({\cos }^{2}x-{\sin }^{2}x)$

Confirm $4{\cos }^{4}x$

Use identity from part (a) to obtain $k<5$ or $k>9$

Obtain $k<5,k>9$

State or imply integrand is $2{\cos }^2\frac{1}{2}t$ or $\sqrt{4{\cos }^4\frac{t}{2}}$

Use double-angle identity to express in terms of $\cos t$

Obtain $1+\cos t$ and integrate to obtain $t+\sin t$

Use limits to obtain $\frac{2}{3}\pi +\sqrt{3}$