Mathematics Paper 3
1. Prove by contradiction that √3 is irrational.
4𝑥 2 −3𝑥+5
2. Express (𝑥−1)2 (𝑥+2) in partial fractions.
3. A conical tank with height 6 m and base radius 2 m is filled with water at 0.5 m³/min.
Find the rate at which the water level is rising when the depth is 3 m.
, 4. Solve 2sin(2𝑥) + cos(𝑥) = 0 for 0 ≤ 𝑥 ≤ 2𝜋.
5. Plane 𝛱1 has equation 𝑥 + 2𝑦 − 𝑧 = 4, and plane 𝛱2 has equation 3𝑥 − 𝑦 + 2𝑧 = 1.
(a) Find the line of intersection of 𝛱1 and 𝛱2 .
(b) Calculate the acute angle between 𝛱1 and 𝛱2 .
1. Prove by contradiction that √3 is irrational.
4𝑥 2 −3𝑥+5
2. Express (𝑥−1)2 (𝑥+2) in partial fractions.
3. A conical tank with height 6 m and base radius 2 m is filled with water at 0.5 m³/min.
Find the rate at which the water level is rising when the depth is 3 m.
, 4. Solve 2sin(2𝑥) + cos(𝑥) = 0 for 0 ≤ 𝑥 ≤ 2𝜋.
5. Plane 𝛱1 has equation 𝑥 + 2𝑦 − 𝑧 = 4, and plane 𝛱2 has equation 3𝑥 − 𝑦 + 2𝑧 = 1.
(a) Find the line of intersection of 𝛱1 and 𝛱2 .
(b) Calculate the acute angle between 𝛱1 and 𝛱2 .
