i. Prove that $G(X, Y, Z) = Y^3Z - X^4$ has no factors of degree 1 and intersects every line through the origin at least thrice at the origin.
ii. Let $G(X, Y, Z)$ be a homogeneous polynomial of degree 4. Assume that there is a point $Q$ such that every line through $Q$ intersects $G$ at least thrice at $Q$. Prove that $Q$ is the only singular point of $G$.
iii. Find a homogeneous polynomial $H(X, Y, Z)$ of degree 2 such that the curve $H$ is singular at one point and contains no other point.
iv. Let $H(X, Y, Z)$ be a homogeneous polynomial of degree 2. Prove that the curve $H$ in the projective plane is singular at a point $R$ and contains at least one other point if and only if $H = MN$ where $M$ and $N$ are lines that contain $R$.
