DieselShip's Online Content Library

Q1. (a) Given Z = (10 ∠ 15^o + 4 ∠ 50^o)/(5 ∠ 25^o), determine Z as a single complex number in polar form. (8)

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

(3a + 2b)/(2a + j) = (3 + jb)/(1 + ja) (8)

Q2. (a) The hypotenuse of a right - angled triangle is I cm longer than twice the length of the Shortest side.

The other side is 1cm shorter than twice the length of the shortest side.

Calculate the lengths of the three sides of this tringle. (8)

Q3. (a) Solve the following system of equations for a, b and c:

3a + 2b - 3c = 4

2a - b + 6c = - 1

a + 3b + 2c = 16

Q4. (a) The tension in the tight side of belt, T newtons, passing round a sheave and in

contact with the sheave for an angle of θ radians are given by:

T = 42.3e^0.28θ

Determine the value of θ when T is 68.5N.

(b) Solve for x in the following equation:

16^0.25x² = 32^{x - 1.2} (6)

(c) Make t the subject of the following formula:

i = Ie - t/CR   (4)

5. Table Q5 indicates the deflection, d mm, of a beam under loads, W newtons. The deflection is related to the load by the formula W = kd^a where k and n are constants.

Q6. The bearing of a tower from a ship is due east and the angle of elevation from sea level to the of elevation to the top of the tower is 4°.

After sailing 600 meters in a straight line the bearing of the tower is due north and the angle of elevation to the top of the tower is 7°.

Calculate the height of the tower above sea level. (16)

Q7. A cylindrical container has to have a volume of 250πcm³ and is to be made of metal

weighing 0.2∕cm².

Determine EACH of the following for the container:

(a) the dimensions which minimize its weight; (12)

(b) its minimum weight. (4)

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

The work done can be represented by the shaded area in fig Q8(a)

Calculate the work done, in joules, when an amount of steam expands, following the law PV^1.3 = C, from a volume of 0.25 m³ at a pressure of 820 kN/m² to a volume of 0.6m³. (12)

Q9. (a) The truth table for a logic system with inputs A, B and C, and output X. is shown in Table Q9(a)

Table Q9(a)

Produce EACH of the following for the logic system:

(i) a Boolean expression in its simplest form; (3)

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

(iii) the logic circuit using only NAND gates (crossing out any redundant gates). (4)

(b) Determine, without using calculator conversion function the value of DC₁₆ ÷ 10100₂.

Giving the answer in the tree forms: binary, hexadecimal and decimal.