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Q1. (a) Given Z = x + jy, where x and y are real, solve the following complex equation for x and y: 2Z/(1 + j) - Z/(2 - j) = 9 + j3 (8)

(b) Solve the following complex equation for r and θ, where r and θ are real numbers:

θ° = (2 < 40° × 4 < 30°) + (10 < 65°)/(5 < 20°) (8)

Q2. (a) Three currents i₁ , i₂ and i₃ in an electrical network are given by the following equations:

i₁ + 2 i₂ - 5 i₃ = - 6.53

2 i₁ + 3 i₂ - i₃ = - 0.55

-3 i₁ + i₂ + 4 i₃ = 7.42

Solve the system for i₁ , i₂ and i₃. (10)

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

3/(x - 1) - 2/(x + 2) - 6/(x² + x - 2) (6)

Q3. (a) The crippling load P for a solid steel rod varies directly as the fourth power of its diameter, d, and indirectly as the square of its length, l.

A steel rod, 3 m long and 5 cm diameter, used as a strut, fixed at both ends, has a crippling load of 265.4 kN.

Determine the crippling load of a similar strut of, 2.6 m long and 4.5 cm in diameter. (8)

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

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

where x in metres is the distance from the point of support.

Evaluate x, correct to two decimal places, when the bending moment is 60 Nm. (8)

Q4. (a) The value £ V of a standby generator after t years is given by:

V = 95000 e^{- 0.226t}

The generator is to be replaced when its value falls to its scrap value of £ 10000.

Determine EACH of the following for the generator:

(i) It’s cost when new; (2)

(ii) Its value after 5 years; (2)

(iii) Its expected lifetime to the nearest year. (6)

(b) Solve the following equation for x:

2x² = 8^{2x - 3} (6)

Q5. (a) During a gas engine test, of a given mass of gas contained in a cylinder, the values of pressure P (kPa) and the volume V (m³) were recorder as shown in table Q5.

Plot a straight - line graph to verify that the relationship between P and V is the gas law PV_n = C, where n and C are constants. (10)

Suggested scales horizontal axis 2 cm = 0.1

Vertical axis 2 cm = 0.1

(b) Use the graph plotted in Q5(a) to estimate the value of n and C (6)

Q6. Fig Q6 shows a double crank mechanism where AB is the frame.

The distance between the centres A and B is 16 cm.

The crank BC is 30 cm, the crank AD is 24 cm and link CD is 25 cm.

In the position shown angle BAD is 135°.

Calculate the size of the angle ABC for this position. (16)

Note: angle BCD is not a right angle.

Q7. (a) Bar staff in a pub are paid £ 8 an hour. If the bar is understaffed the lost profit is £ {512/x} per hour, where x is the number of staff on duty. Determine the staffing level which will minimize the total staffing cost per hour. (8)

(b) Given P = 20 x - 50 sin⁡ (x, solve EACH of the following for x in the range 0 ≤ x  ≤  π):

(i) dP/dx = 0 ; (5)

(ii) (d² P)/(dx² ) = 0. (3)

Q8. The region in the first quadrant enclosed by the parabola y = 1/4 x², the x - axis and the line x = 2 represented by the shaded region in Fig Q8. Determine EACH of the following for the shaded region:

(a) Its area; (4)

(b) The volume of solid of revolution formed by rotating it through one revolution about the x - axis. (6)

(c) The volume of solid of revolution formed by rotating it one revolution about the x - axis. (6)

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

Produce EACH of the following for this circuit

(i) An unsimplified Boolean expression for the output of EACH gate in terms of the inputs A, B and C (4)

(ii) The truth table, including columns for A, B, C, D, E, F and X. (4)

(iii) The equivalent logic circuit using only NAND gates (crossing out any redundant gates) (4)

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

(i) The conversion operation of 7882₁₀ to hexadecimal (2)

(ii0,) The hexadecimal operation FAB7 - AC2B. (2)