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Q1. (a) solve the following complex equation for u and b, where a and b, are real numbers:

a + jb = (9∠(π/3))/(3∠(π/4) × 2∠(- π/2))

(b) The resultant impedance, Z, of TWO circuits of impedances Z1 and Z2, connected in

parallel, is given by the formula:

1/Z = 1/Z₁ + 1/Z₂

Calculate the resultant impedance, expressing it in polar form, when

Z₁ = 3 + j5 and Z₂ = 2 - j3.

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

(3x + 2)/4 - (4y + 1)/2 = 19/4

(1 - 2x)/3 + (1 - 3y)/5 = 16/15

b) Factorise fully EACH of the following:

(i) 15Z² + 14Z - 8

(ii) 48m³ - 27mn²

(iii) 5a² - ab - 10a + 2b.

Q3. TWO ferries leave the same port at the same time and arrive at the same destination EIGHT minutes apart.

The faster ferry averaged 2 knots more than the other.

The distance between the TWO ports is 24 nautical miles.

Calculate the average speed of each ferry.

b) Determine the mass of brass alloy, containing 45% copper, Which would be required to combine with 12 kg of copper to form a brass alloy containing 60% copper.

Q4. Solve for x in EACH of the following equations:

a) 7^{2(x - 3)} = 2401 × 3^{(5 - x)}

b)2 ln ⁡x = in (2x + 3) + In (x - 2), x > 0.

c) e^{(2x + 1)} = 25.

Q5. The quantity, Q (m³/s), of water discharged through an orifice under different pressure heads.

H(m), are recorded in Table QS.

(a) Draw a straight line graph to verify that the relationship between Hand Q is a law of the: form Q = aHⁿ, where a and n are constants.

Table Q5

Suggested scales will landscape orientation:

Horizontal axis 2 cm = 0. I

vertical axis 2 cm = 0. I

(b) Use the graph plo11ed in Q5(a) to estimate the value of a and n.

Q6. a) An idler gear, 12 cm in diameter, is to be fitted between a 40 cm diameter driving gear and a 60 cm diameter driven gear as shown in Fig Q6 (a).

The difference between the centres A and B is 55 cm.
Calculate the size of angle ABC and CAB.

(b) An engine unit is slung from a horizontal beam by TWO chains 3.0 metres apart.

The lengths of the chains are 2.2 metres and 2.5 metres, and both are hooked to the same lifting eye of the engine unit.

Calculate the angles made by the chains with the beam. (8)

Q7. A small open rectangular box is to be fabricated from a square sheet of steel of side 20 cm, by cutting congruent squares from each comer and bending the edges up which are then welded along the comer joints.

Given that the volume of the box has to be maximised, calculate EACH of the following:

(a) the side length, using differential calculus, of the square to be cut from each comer; (12)

(b) the volume of the box; (2)

(c) the external surface area of the box. (2)

Q8. a) Calculate the shaded area between the curve y = 8 - x² and the line y = - 2x, as shown in Fig Q8 (a). (8)

b) Given ds/dt = (2t³ - 9t² + 4t)/(2t² - t), and s = 3m when t = 10 seconds, express s as a function of t. (8)