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Q1. (a) Given Z₁ = 3a + j5a and Z₂ = 11b + j2b. Solve the follow complex equation for a and b, where a and b are real numbers: Z₁ + Z₂ = - 7 + j21 (8)

(b) Three mooring lines exert horizontal forces on a bollard, positioned at O, as follows:

20 kN at 30°

16 kN at 70°

10 kN at 85°

The angles are those that the forces make with the real axis Ox.

Determine using complex numbers the magnitude and direction of the resultant force on the bollard. (8)

Q2 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, θ.

For a ship travelling at 12 knots, with a rudder area 30 m² operating at an angle of 30°, the rudder loss is 320 kN.

Calculate the force on a similar rudder of area 25 m² operating at an angle of 22° when the ship’s speed is 16 knots. (8)

b) W = (4 - (8/(x + 1)))/(x - (2/(x + 1)) )

i. Express W as a single algebraic fraction it its simplest form (6)

ii. Solve for x when W = 3 (2)

Q3a. Solve the following system of equations for x and y 12x² + 16y² = 13;

4y = 2x + 3; (10)

b. Transpose the following formula to make A the subject T = √((2ghDA²)/d(S² - A² ) ) (6)

Q4. The tension in the tight side of a belt, T Newtons passing round a particular sheave and in contact with the sheave for an angle of θ is given by:

T = 63.5 e^0.13θ

Determine the value of θ when T is 84.2 N. (6)

b. Solve the following equation for x: (3x²  = 81^{2x - 4} )(6)

c. Simplify the following as fully as possible: (2a³ b² c⁴ )²/(a⁹ b⁶ c³ )^2/3 (4)

Q5.a) On the same set of axes plot the graphs, in intervals of 0.5 of y = (x - 1)² - 2

in the range - 1.5 ≤ x ≤ 3 and y = 2 - x² in the range - 2 ≤ x ≤ 2.5.

Suggested scales: horizontal axis 2 cm = 1

vertical axis 2 cm = 1.(12)

b) Use the graph drawn in Q5 (a), solve the system of quadratic equations;

y - (x - 1)² - 2

y = 2 - x².(4)

Q6 A 14 - meter - tall building has dimensions as shown in the end elevation in Fig Q6.

The building has a horizontal base and the walls are vertical.

Calculate the size of EACH of the angles x⁰,y⁰ and z⁰ (16)

Q7. A drip tray is to be fabricated from a thin rectangular steel plate, 36cm × 24cm, by cutting squares of side v cm from each cornet and folding up the edges which are then welded up along the corner joints. The fold lines are shown by the dotted lines in Fig Q7.

a. Determine, using differential calculus, the value of x which maximizes the capacity of the tray.(14)

b. Calculate the maximum capacity of the tray. (2)

Q8. A gas expands in a cylinder according to the relationship PV^1.3 = 1362. The initial volume of the gas is0.06 m^3 and the final volume of gas is 0.095 m³.

Calculate the work done by the gas during the expansion. (8)

Note: The work done by gas as it expands from V₁ to V₂ units of volume in W, where

B. Given dy/dx = 6x² + 2x - 1/3 - 3/x² and y = 60 when x = 3. Determine the value of y when x = 1. (8)

Q9. (a) The logic circuit in Fig Q9(a) has inputs A, B, and output X.

(i) Produce the truth table for this circuit; (3)

(ii) State a Boolean expression for X; (1)

(iii) State the type of logic gate which produces the same output as the circuit. (1)

(b) Simplify, as fully as possible, the following Boolean expression:

(c) Determine EACH of the following, without using a calculator conversion function:

(i) the value, in hexadecimal form, of FC₁₆ ÷ 10101₂  (5)

(ii) the conversion of BDAC₁₆ to decimal form. (2)