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Q1. Given: z₁ = 1 + j2, z₂ = 2 - j, z₃ = 1.5 - j4.5 and Z = (Z₁ Z₂)/(Z₁ + Z₂ ) + Z₃ , determine EACH of the following:

(a) Z in Cartesian form; (8)

(b) Q, in polar form. given Q = E²/Z when E = 15∠120° (8)

Q2. a. Solve the following system of equation for x and y: (10)

5/x - 3/y = 13/8; 3/x - 5/y = 49/24

b. Solve the following equation for x: (2x - 3)(4x + 5) = 9 (6)

Q3. a. Express the following function of x as a single algebraic fraction in its simplest form:(7)

2/x + 1/(x - 1) - 3/(x - 1) + 2/x(x + 1)(x - 1)

b. Factorise fully EACH of the following:

i. a³ - ab⁴ (3)

ii. 12x³ + x² - 6x (3)

iii. 12p² + 36pq + 27q² (3)

Q4. a. Solve for x in the following equation: 32^3x = 8^{2x + 6} (6)

b. Express the following in its simplest form: [a^5/2/b^{- 5/3}]^1/5×[b²/a^3/2 ]^1/3 (6)

c. Transpose the following formula to make t the subject: i = I e^{- t/CR} (4)

Q5. Table Q5 shows corresponding measurements of the diameter (d cm) and the breaking load (m kg) of a particular type of rope.

a) Draw a straight line graph to verify that the diameter and breaking load are related by a law of form m = kdⁿ where k and n are constants. (10)

Table Q5

Suggested scales: horizontal axis 2cm = 0.1

vertical axis 2cm = 0.1

b) Use the graph drawn in Q5 (a)to estimate the value of k and n. (6)

Q6. At 0800 hours a patrol boat is set on a course to intercept a vessel as soon as possible.

The vessel is 30 nautical miles due east of the patrol boat and is sailing on a steady course of 030° at a speed of 15 knots. The best speed of the patrol is 25 knots.

Calculate the earliest possible interception time, to the nearest minute. (16)

Q7. a. A company produces particular components in batch sizes which are in multiples of a hundred components.

The production cost, C, of x hundred components is given by: C = 2x³ - 12x² - 72x + 500

Determine, using differential calculus, the batch size which minimizes the cost of production (8)

b. Determine the first and second derivatives of EACH of the following functions:

i. y = x⁵/5 + 3/x (4)

ii. S = cost + ln⁡t  (4)

Q8. a. Determine the volume of solid of revolution when the shaded area shown in Fig Q8(a) is rotated through one complete revolution about the x axis. (10)

Q9 (a) Determine EACH of the following, without using a calculator conversion function:

(i) the conversion of 1275₁₀ to hexadecimal; (2)

(ii) the hexadecimal operation EB4C – A5DF. (2)

(b) The truth table for a logic system with inputs P, Q and R, and output X, is shown in Table Q9(a)

Table Q9(b)

Produce EACH of the following for the logic system:

(i) a Boolean expression in its simplest form; (4)

(ii) the equivalent logic circuit with the minimum number of gates; (3)

(iii) the equivalent logic circuit using only NAND gates, (crossing out any redundant gates). (5)