DieselShip's Online Content Library

Q1) a) Solve the following complex equation for a and b, where a and b are real numbers:

(3 + ja)/(3 + jb) = (2a + b)/(2b + j3) (8)

(b) Two impedances, Z₁ = 3 + j8 and Z₂ = 5 - j4 are connected in series to a supply voltage, v, of 250 volts.

Calculate the current, i amperes, as a complex number in polar form, given that

i = v/Z where Z = Z₁ + Z₂. (8)

Q2. a) The wind force, ω Newtons, on a vertical surface varies directly with the area of the surface, A m², and the square of the wind velocity, v knots. When the wind velocity is 20 knots, the force on a ship’s vertical bridge window, of area 2 m² is 130 N.

Calculate the force on a similar window of area 1.8 m² when the wind velocity is 60 knots. (6)

(b) Factorize fully EACH of the following:

(i) 12x⁵ - 17 x⁴ + 6 x³ (4)

(ii) 200 - 50 x - 32 x2 + 8 x3 (6)

Q3. a) A hydrofoil, on a ferry run , takes 10 minutes to overtake a vessel on parallel course 4 nautical miles ahead.

On the return journey, the hydrofoil takes 5 minutes to pass the same vessel which is still on the same course and speed, again from a distance of 4 nautical miles.

Calculate the speed of the hydrofoil and the vessel. (8)

(b) Solve the following system of equations for x and y:

2 x² - 3y² = 5

2 x - 3y = 1 (8)

Q4) (a) The pressure in a particular life raft falls according to the formula P_t = P_o e^-kt

Where P_o is the initial pressure, P_t is the pressure t hours after inflation and k is a constant.

When t = 0, the initial pressure is 100 units.

After 15 hours the pressure was 70 units.

If the pressure falls to 50 units or less, the raft is deemed unsafe.

Calculate EACH of the following for the raft:

(i) the value of k, correct to 3 decimal places; (5)

(ii) the safe lifetime in hours, to the nearest hour, assuming the pressure is not topped up. (5)

(b) Calculate the value of x, correct to 3 decimal places, such that

log_e (2 - 1/x) = - 0.3 (6)

Q5) (a) On the same set of axes plot the graphs, in intervals of 1, of y = x² + 2x - 3 in the range - 4 ≤ x ≤ 3 and y = 4 - x² in the range - 3 ≤ x ≤ 3 (12)

Suggested scales: horizontal axis 2cm = 1

vertical axis 1cm = 1

(b) Using the graphs drawn in Q5(a), solve the system of quadratic equations:

y = 4 - x²

y = x² + 2x – 3 (4)

Q7) The cost, £C per 10 metres, of laying a pipeline of cross - sectional area, x m², is given by

C = 2000 x + 500/x

Calculate EACH of the following for the pipeline:

(i) the cross - sectional area which minimises the cost per 10 metres; (8)

(ii) the minimum cost per 10 metres. (2)

(b) When a flywheel rotates through an angle of Ɵ radians in t seconds, its angular velocity is given by dƟ/dt radian/s, and its angular acceleration is given by (d^2 Ɵ)/dt² radian/s².

For a certain flywheel θ = 21t - 1.5t².

Determine EACH of the following for this flywheel:

(i) the angular velocity when t = 5; (3)

(ii) the time that elapses before the angular velocity is zero; (2)

(iii) the angular acceleration. (1)

Q8) (a) A gas expands in a cylinder according to the relationship PV^1.2 = 1480. The initial volume of the gas is 0.05 m³ and the final volume of the gas is 0.075 m³.

Calculate the work done by the gas as it expands from 0.05 m³ to 0.075 m³. (8)

Note: the work done by the gas as it expands from V₁ to V₂ units of volume is W.

where

(b) Given (d² H)/dt² = 8 + sin⁡ t - 2 cos ⁡t  and dH/dt = - 2 when t = π/2 and H = 0 when t = 0.

Express H as a function of t . (8)

Q9) The logic circuit shown in fig Q9(a) has three inputs A,B and C, and one output X.

Produce EACH of the following for this circuit:

(i) an unsimplified Boolean expression for X (4)

(ii) the truth table, by simplifying as fully as possible the Boolean expression obtained for X in Q9a) (i). or otherwise (4)

(iii) the equivalent circuit with the minimum number of gates. (2)

(b) Determine EACH of the following, without the use of a calculator conversion function:

(i) the conversion of 101110₂ to decimal (1)

(ii) the conversion of DABE₁₆ to decimal (2)

(iii) the value, in hexadecimal form, of the operation AB₁₆ ÷ 10011₂. (3)