Rewrite: $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$. This is a homogeneous ODE. Let $y=vx$:

$\frac{1-v^2}{2v}dx + x\,dv/... $

After solving, the general solution is $x^2 + y^2 = Cy$ (circles with centres on the $y$-axis).

Substituting $(1,1)$: $1+1=C\cdot1 \Rightarrow C=2$, giving $x^2+y^2=2y$, i.e. $x^2+(y-1)^2=1$.

This is a circle centred on the $y$-axis at $(0,1)$.

Total outcomes $= 36$.

$P(E_1)=\frac{6}{36}=\frac{1}{6}$, $P(E_2)=\frac{1}{6}$, $P(E_3)=\frac{18}{36}=\frac{1}{2}$

$E_1$ and $E_2$: $P(E_1\cap E_2)=\frac{1}{36}=\frac{1}{6}\cdot\frac{1}{6}$ ✓ Independent.

$E_1$ and $E_3$: For sum odd with die A showing 4 (even), die B must be odd: 3 outcomes. $P(E_1\cap E_3)=\frac{3}{36}=\frac{1}{12}=\frac{1}{6}\cdot\frac{1}{2}$ ✓ Independent.

$E_2$ and $E_3$: For sum odd with die B showing 2 (even), die A must be odd: 3 outcomes. $P(E_2\cap E_3)=\frac{3}{36}=\frac{1}{12}=\frac{1}{6}\cdot\frac{1}{2}$ ✓ Independent.

$E_1\cap E_2\cap E_3$: Die A=4, Die B=2, sum=6 (even) — impossible! $P(E_1\cap E_2\cap E_3)=0$.

But $P(E_1)\cdot P(E_2)\cdot P(E_3)=\frac{1}{6}\cdot\frac{1}{6}\cdot\frac{1}{2}=\frac{1}{72}\neq 0$.

So $E_1,E_2,E_3$ are not mutually independent (as a triple), even though pairwise independent.

The sentence sets up a contrast: the filmmaker is known for aesthetic (artistic/visual) qualities, not for some other quality described in the blank. Since she does not speak out on political matters, the missing quality must be related to politics or controversy.

• Polemical — means controversial, argumentative, or politically charged. This directly fits the contrast: aesthetic vs. polemical qualities.
• Cinematic — essentially synonymous with aesthetic in the context of film; not a contrast.
• Narrative — refers to storytelling style; not linked to the political theme.
• Commercial — relates to box-office value; does not address the political angle.

Correct Answer: A — Polemical. The sentence contrasts aesthetic with polemical (politically contentious) qualities.