Projective geometry

In $\mathbb{R}P^2$, where do the lines $y = 3x + 1$ and $y = 3x - 2$ (in the chart $z = 1$) meet? (Choose all correct descriptions of the meeting point.)

How many points does the projective plane $\mathbb{F}_3P^2$ over the field with three elements have?

Four distinct collinear points have cross-ratio $(a, b; c, d) = 3$. Over all $24$ orderings of the four points, how many different values does the cross-ratio take?

Which properties are preserved by every projective transformation of $\mathbb{R}P^2$? (Choose all that apply.)

In the proof of Theorem 2 (four points), we write $v_4 = \lambda_1 v_1 + \lambda_2 v_2 + \lambda_3 v_3$ and claim every $\lambda_i \neq 0$. Why?

The non-degenerate conic $x^2 - 4xy + 4y^2 + x = 0$ meets the line at infinity in how many points, and what kind of affine conic is it?

The Lean theorem line_unique has hypotheses (hpq : p ⨯₃ q ≠ 0) (hp : ℓ ⬝ᵥ p = 0) (hq : ℓ ⬝ᵥ q = 0) and concludes ∃ c : K, c • (p ⨯₃ q) = ℓ. What goes wrong if hpq is dropped?

Which statements about the theorems of Desargues and Pappus are true? (Choose all that apply.)

Write join(p, q) returning the cross product of two triples (the line through two points, or the point on two lines) and collinear(p, q, r) returning True when the three points of $\mathbb{R}P^2$ lie on a line (use the triple product $p \cdot (q \times r)$, with exact integer or fraction arithmetic). Then confirm Pappus' theorem on an example.