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Q1. (a) A cruise line contracts to purchase two identical ships at a cost of £364 M each.

During construction of the first ship the costs of labour and materials are in the ratio of 3:4 and the ship builders make a profit of 4%.

Determine EACH of the following:

(i) the cost of the materials for the first ship; (5)

(ii) the percentage profit made on the second ship if the labour costs have increased by 2% and the material costs have decreased by 5%. (5)

(b) The ratio of the volumes of two solid cubes is 729:64.

Determine the side length of the larger cube if the surface area of the smaller cube is 384 cm². (6)

Q2. (a) Solve the following system of equations for x and y: (10)

3x² + 2y² + y = 13

3x + 2y - 7 = 0

(b) Fully factorise EACH of the following:

(i) 30ab² + 39ab – 126a ; (3)

(ii) (2x - 5y)² - 9y². (3)

Q4. (a) The force, F, produced on a ship’s rudder is proportional to the area, A, of the rudder, the square of the ship’s speed, V, in knots and the sine ratio of the rudder angle, α.

For a ship travelling at 10 knots, with a rudder area of 24 m² operating at an angle of 23°, the rudder force is 144 kN.

Calculate the force on a similar rudder of area 29 m2 operating at an angle of 15° when the ship’s speed is 14 knots. (8)

(b) Y = (3 + 9/(x - 2))/(x – 3/(x - 2))

Express as a single algebraic fraction in its simplest form. Y (8)

Q7. (a) The blade efficiency E of a particular turbine is given by:

E = (2u(V-u)/V² )        Where u = the speed of the blade

V = the constant velocity of the jet.

Determine EACH of the following:

(i) the value of u for maximum efficiency; (6)

(ii) the maximum percentage efficiency. (2)

(b) Differentiate EACH of the following functions:

(i) 3/x³ -4/x² +2/√x-√x (4)

(ii) 2sin t – cos t – t + ln t. (4)

Q8. (a) Evaluate ∫_π/6^5π/6(Q2sinθ-cosθ)dθ (6)

(b) Determine the volume of solid of revolution when the shaded area shown in Fig Q8(b)

is rotated about the x-axis through one complete revolution. (10)

Q9. A solid metal cylinder has diameter 12 cm and length 25 cm.

Nine holes of diameter 2 cm are drilled through the cylinder, parallel to its axis, as shown in Fig Q9.

Calculate EACH of the following:

(a) the percentage decrease in the volume of the cylinder; (6)

(b) the total surface area of the original cylinder; (4)

(c) the percentage increase in the total surface area after the nine holes have been drilled. (6)