This question concerns the following subsets of R²: A = {(x, y) = R² : (x + 1)² + y² ≤ 4}, B = {(x, y) = R² :y> −2x}, C = {(x, y) = R²: y ≥ (x − 1)²}. (a) Sketch, on separate diagrams, the sets A, B, AUB and B-A. (b) Prove that C is a proper subset of B.

2. Let a,, az,., a, be numbers. Show that there is a single number r such that foo: (z-a) +(2 – a,)° +..+ (x – a,) is a minimum and find this number.

3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2. (a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z² r< R. Express this in the cylindrical and spherical coordinate systems. - R² assuming (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.

2) Prove by using the pigeonhole principle that among any 17 points in a square of side one there is at least a pair whose distance is smaller than or equal to 1/(2v2).