DieselShip's Online Content Library

Q1. (a) Solve the following complex equation for r and q:

r∠θ° = (5∠20° × 2∠40°) + (8∠85°)/(2∠55°) - 5∠10° (8)

(b) Two impedances, Z₁ = 5∠60° and Z₂ = 2∠ - 30° are connected in series to a supply voltage, v, of 100 volts.

Calculate the current, I amperes, as a complex number in Cartesian form, given that:

Q2. (a) A hydrofoil on a ferry run takes 6 minutes to overtake a vessel on a parallel course 3 nautical miles ahead.

On the return journey the hydrofoil takes 4 minutes to pass the same vessel which is still on the same course and speed, again from a distance of 3 nautical miles.

Calculate the speed of the hydrofoil and the vessel. (8)

(b) Make p the subject of the following formula:

t = r(√(f + r)/(f - p) - 1) (8)

Q3. (a) Solve the following system of equations for x and y:

3x² - 2y² = - 5

3x - 2y = 1 (8)

(b) Solve the following equation for x > 0, correct to 3 decimal places:

1/(x + 2) - 2x/(x - 1) = 2 (8)

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 recorded as shown in Table Q5.

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

Table Q5

Suggested scales: horizontal axis 2 cm = 0.1

Vertical axis 2 cm = 0.1 (10)

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

Q7. (a) A company specializes in producing water pump control panels.

The cost of producing x control panels per day is £(x² + 50x + 15).

The selling price of each control panel £ (300 - 4x)

Use differential calculus to determine EACH of the following:

(i) the daily output for maximum profit;     (8)

(ii) the maximum daily profit;         (2)

(iii) the production cost of each panel when operating at maximum profit.      (2)

(b) Given T = 5 + 3cos q - 2sin q, calculate correct to 2 decimal places, the value of

dt/dq when q = 2p/3 radians.       (4)

Q8. (a) A gas expands in a cylinder according to the relationship PV^1.25 = 1294.

The initial volume of the gas is 0.05m³ and the final volume of the gas is 0.105m³.

Calculate the work done by the gas during the expansion.

Note: the work done by the gas as it expands from V₁ to V₂ units of volume is

W Joules, where W = ∫_V1^V2)PdV.   (8)

(b) Given dy/dx = 3x² + 4x + 1 - 3/x² and y = 19 when x = 2,

Determine the value of y when x = - 1.     (8)

Q9. (i) Many new cars have systems to switch on windscreen wipers, cither automatically

When moisture is Detected. Or when they are Switched on manually by the driver. The system only operates if the ignition switch is on. A typical logic system for this operates as follows:

1.Ignition switch A outputs a logic I when it is switched on.

2.Moisture sensor B outputs a logic I when rain is detected.

3.Wiper switch C outputs a logic I when it is switched on by the driver.

4.The windscreen wiper motor operates when output X is at logic I.

Produce EACH of the following for this logic system with inputs A, B,C and output X:

(i) the truth table, including columns for A, B, C and X: (3)

(ii) a Boolean expression for output X in its simplest form: (5)

(iii) the logic circuit for the Boolean expression obtained in (ii) (2)

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

(i) the hexadecimal operation CF9F + EDAF:       (2)

(ii) the binary operation 10110     (2)

(iii) the conversion of DBCA to decimal.     (2)