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Q1. a. Given: z = x + jy, where x and y are real, solve the following equation for x and y.

z/(1 - j) + z/j = 20/(30 - j) (8)

b. Given: z₁ = 5∠20°, z₂ = 4∠30° and z₃ = 2∠15°, express (z₁ + z₂)/z₃ as a complex number in polar form. (8)

Q2. a. Calculate the mass of a metal containing 55% nickel which would be required to combine with 10 mg of pure nickel to form an alloy containing 85% nickel. (6)

b. Solve EACH Of the following equations for x:

i) 3/(x + 3) + 2/(x - 3) = 5/(x - 1) (6)

ii) x² - 13x + 40 = 0 (4)

Q3. Solve for x in EACH of the following equations:

i) 3^{1 - x} = 2^{x + 1} (8)

ii) ln⁡((3 + x)/(3 - x)) = 1.25 (8)

Q4. a. Solve the following system of equations for a, b and c:

3a + b - 2c = 4

a - 2b + c = 6

7a - 6b - c = 10 (10)

b. The sag, y meters, in a cable of length L meters stretched between two supports, x meter apart, as illustrated in Fig Q4(b), is given by the formula L = (8y²)/3x + x

Calculate the distance x when L is 75m and y is 2.4m (6)

Q5. Variables P and V are thought to be related by a law of the form: PVⁿ = C where n and C are constants. Observation of P and V are recorded in Table Q5.

a) Draw a straight line graph to verify this relationship. (10)

Table Q5
Suggested scales: horizontal axis 2 cm = 0.1
vertical axis 2 cm = 0.1
b) Use the graph drawn in Q5 (a)to estimate the value of n and C. (6)

Q6. In the diagram shown in Fig Q6, B and D represent the centers of two gear wheels.

An idler is to be placed at C.

Calculate the values of x and y. (16)

Q7. a. The rate R, in tonnes per hour, at which a particular vessel consumes fuel is given by:

R = 15 + 0.00048V³, where V is the speed of the vessel in knots.

Determine EACH of the following for this vessel when it embarks on a passage of 500 nautical miles:

i. the speed of the vessel which minimizes the amount of fuel consumed; (10)

ii. The minimum amount of fuel consumed. (2)

b. Determine ds/dt given s = 2(t² - t)/√t (4)

Q8. a. The work done during an adiabatic expansion follows the law PVⁿ = C, where C and n are constant, as the volume increases from V₁ to V₂.

The work done represented by the area in Fig Q8(a).

An amount of steam expands so as to the law PV^1.13 = C.

Calculate the work done, in Joules , when the steam expands from a volume of 0.2 m³ at a pressure of 850 kN/m² to a volume of O.5m³.(12)

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

(i) the binary operation 11011 x 1011. (2)

(ii) the hexadecimal operation BC7E – 9ADF. (1)

(iii) the conversion of DC4B₁₆ to decimal; (2)

(iv) the conversion of 1110111111₂ to hexadecimal. (1)

(b) A logic circuit behaves according to the Boolean expression:

(i) without simplification, draw the circuit diagram for the expression using only XOR, NAND and NOR gates. (3)

(ii) simplify the expression as fully as possible. (4)

(c) Simplify the following Boolean expression as fully as possible: