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Q1. (a) THREE mooring lines exert horizontal forces on a bollard, positioned at O, as follows:

32 kN at 40°

18 kN at 65°

25 kN at 110°

The angles are those that the forces make with the real axis Ox.

Determine, using complex numbers, the magnitude and direction of the resultant force on the bollard.          (8)

(b) Given Z = √(2∠45°), determine Z + Z - 1 as a complex number in polar form. (8)

Q2. (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, 2.5m long and 4.5cm in diameter, used as a strut, fixed at both ends, has a crippling load of 235.2 kN.

Determine the crippling load of a similar strut, 3m long and 5cm in diameter. (8)

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

(14x - 6)/(2x² - 11x + 12) + 4x/(2x - 3) - 1/(x - 4) (8)

Q3. The sag d metres, in a cable of length, L metres, when suspended between two points and subject to a tension, T Newtons, may be determined from the formula:

d² - T/w d + 1/4 L² = 0

Where w is the weight in Newtons per metres run.

Calculate, correct to two decimal places, the sag in the cable when T = 2 200 N, L = 40 m and the total weight of the cable = 220 N, (a) (8)

(b) Solve for x in the following equation:

3/(x² - 9) + 2/(x - 3) = 3/(x² - 2x - 3) (8)

Q4. (a) Solve EACH of the following for x, correct to 3 decimal places:

(i) log_e⁡(e^x + 10) = 3; (4)

(ii) 7 = 5e^-0.4x (4)

(b) Transpose the following formula to make V the subject:

t = CR log_e⁡(E/V) (4)

(C) Express the following in its simplest form:

(a³ b⁶)^2/3/√(a² b⁴) (4)

Q5. (a) On the same set of axes plot the graphs, in intervals of 1, of

y₁ = 2x² + 5x - 6 and y₂ = - x² - 2x + 8 in the range - 4 ≤ x ≤ 2.

Additional intermediate points may be plotted to aid the drawing of smooth curves.

Suggested: horizontal axis 2 cm = 1

Vertical axis 1 cm = 1 (12)

(b) Using the graphs plotted in Q5(a), for x and y, the system of equations:

y₁ = 2x² + 5x - 6

y = - x² - 2x + 8 (4)

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

The distance between the centres A and B is 18 cm.

The crank BC is 27 cm, the crank AD is 36 cm, and the link CD is 33 cm.

In the position shown angle ABC for this position.  (16)

Q7. (a) The volume of the square - based glass display case shown in Fig Q7(a) is 864 cm³.

The length of the base is x cm and the base is not made of glass.

Determine EACH of the following for the display case:

(i) an expression, in terms of x, for the area of glass, A cm², in the case; (4)

(ii) the dimensions of the case that minimize the amount of glass used;         (6)

(iii) the minimum area of glass used.        (2)

(b) Determine the first and second derivatives of the following function:

u = 3sinθ + (1 - 4 sin^2⁡θ)/(1 + 2sinθ) (4)

Q8. (a) The shape of a rugby ball may be represented by the rotation of the shaded area in Fig Q8(a), about the x axis, through one complete revolution.

(i) Determine, using integral calculus, a formula for the volume of a rugby ball in terms of the constants A and B.       (10)

(ii) Use the result in Q8(a)(i) to calculate the volume of an adult size rugby ball which has a = 15 cm and b = 9.8 cm.(2)

(b) Evaluate: ∫₀^π/6(2sinθ/tanθ) dθ. (4)