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Q1. (a) Given Z = ((1 - j)(5 + j3))/(3 - j5), determine EACH of the following:

(i) Z in the form a + jb, where a and b are real numbers; (5)

(ii) the value of Z³ in its simplest form.   (3)

(b) The impedance, Z, of an electronic circuit is given by Z = 4 - j7 ohms and the current, I, is given by I = 8 + j3 amps.

Determine, in polar form, the voltage, V, across the impedance, given V = iZ. (8)

Q2. (a) The force produced on a ship’s rudder, FkN, varies directly as the area of the rudder,

A m², the square of the ship’s speed, V knots, and the sine ratio od the rudder angle, q°.

For a ship travelling at 15 knots, with a rudder area 22m² operating at an angle of 28° when the ship’s speed is 280kN.

Calculate the force on a similar rudder of area 25 m² operating at an angle of 28° when the ship’s speed is 20 knots:   (8)

(b) fully factories EACH of the following:

(i) 12x² - 28x + 15;                   (2)

(ii) 36x³ - 4xy²;    (3)

(iii) 3x³ + 2x² - 15x - 10 (3)

Q3. (a) A particular white medal bearing for marine use is composed of, by mass, 89% tin, 7.5% antimony and the remainder is copper.

Determine the masses of antimony and copper required to combine with 200 kg of tin to make this white metal.   (6)

(b) The bending moment in Newton meters at a point in a beam is given by:

M = (5x(15 - x))/4

Where x meters is the distance from the point of support.

Evaluate x, correct to two decimal places, when the bending moment is 50 Nm.       (6)

(c) Solve the following equation for x:

(x + 3)/4 - 5x/6 + (2 - x)/8 = 7/24

Q4. (a) The amount, A_t, micrograms, of a certain radioactive substance remaining after t years decreases according to the formula:

A_t = A₀ e^{- 1.2} × 10^{- 4} t

Where A₀ is the amount present initially.

(i) Determine the amount of this substance present initially if 400 micrograms remain after 1000 years.          (3)

(ii) The half - life of a substance is the time taken for the amount to decrease to half

Its original amount.

Determine the half - life of this substance.          (5)

(b) Transpose the following formula to make t the subject:

i = Ie^{- 1/CR}

(c) Simplify the following as fully as possible:

(8a⁹ b³ c⁶)^2/3/(4(a³ b² c)²)

Q5. During a particular tidal the depth of water, d meters, in a harbor t hours after midnight, can be modelled approximately by the function d = cos 30°.

(a) Plot the graph of d = 4 + cos 30°t, 0 £ t £ 10, in intervals of 1 hour.

Suggested scales with landscape orientation: horizontal axis 2 cm = 1

vertical axis 2 cm = 1 (10)

(b) Use the graph drawn in (a) to determine EACH of the following during this period:

(i) the minimum ground clearance of a vessel in the harbor of draught 2.5m;  (2)

(ii) the length of time a vessel of draught 3.8 m may be expected to be around: (3)

(iii) the expected depth of water in the harbor at 0730 hours.  (1)

Q6. A vertical aerial, AB, is 18m high, standing on ground which is inclined 10° to the horizontal.

A stay connects the top of aerial A to a point C on the ground 12 m downhill from B, the foot of the aerial.

Calculate EACH of the following:

(i) the length of stay AC;     (5)

(ii) the angle the stay AC makes with the ground;          (3)

(iii) the length of a second stay which connects to a point D, 6m from the top of the aerial to point C.          (2)

(b) A current, i amps, is given by I = 5cos(100pt + 25), where t is the time in seconds.

Calculate the earliest time, t > 0, for which the current I = 2amps.       (6)

Q8. (a) Calculate the total shaded area shown in fig Q8(a). (10)

(b) Evaluate ∫_(π/4)^(π/3)(2cosθ - sinθ) dq correct to 3 decimal places. (6)