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Q1. (a) Two ships heading on reciprocal courses pass each other. One ship has a speed of 6.5 knots greater than the other.

Three hours after passing each other the ships are 111.3 nautical miles apart.

Calculate the speed of EACH ship. (8)

(b) A ship’s fuel consumption varies inversely as the calorific value of the fuel and directly as the square of the ship’s speed.

The ship burns 28 tonnes of fuel per day of calorific value 42 MJ/kg when sailing at 14 knots.

Determine the daily consumption when the ship is burning fuel of calorific value 44 MJ/kg and sailing at 17 knots. (8)

Q2. (a) Y = (5x - 10)/(2x² + 3x - 2) + x/(2x - 1)-4/(x + 2)

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

(b) Solve the following system of equations for a and b: (8)

(a - 1)/4 + b/3 = 8

5a – 3b = 20

Q3. (a) The value, V, in thousands of pounds, of a propulsion unit after t years is given by:

V = 275e^{- 0.085t}

Calculate EACH of the following for the propulsion unit:

(i) the initial value; (2)

(ii) the number of complete years before its value is less than £100k. (6)

(b) Use laws of indices to simplify EACH of the following:

(i) ((75 a^3/8)/(25 a^1/4)) (4)

(ii) (x²/y⁴ )^-1/2 × (y³/x^-6)^1/3

Q4. (a) Solve the following equation , for x > 0, correct to 3 decimal places: (8)

8 x = (50+3x²)/(15+x)

(b) Transpose the following formula to make a the subject: (8)

T =2π√((a²+b²)/gh)