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Q1. (a) In a three cylinder engine the power developed in the No. 1 cylinder is 5% less than in the No. 2 cylinder.

The power developed in the No. 3 cylinder is 20% more than in the No.1 cylinder.

Express the power developed by EACH of the cylinders as a percentage of the total power of the engine. (8)

(b) The mass of a square bar varies as its length and the square of its side.

The length of bar A is 34 of the length of bar B and the side of bar A is 23 of that of bar B.

The mass of bar A is 14 kg and both bars are composed of the same material.

Calculate the mass of bar B. (8)

Q2. (a) Solve for x, x > 0, in the following equation: (8)

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

(b) For a particular ship, at deadweight displacement, the power of the main engine is given by:

P = v (av² + b) where v is the speed of the ship in knots, and a and b are positive constants.

The powers are 3456 kW and 7632 kW when the speeds are 9 and 12 knots respectively.

Calculate the power when the speed of the ship is 14 knots. (8)

Q3. (a) A formula associated with the magnetic field strength of a solenoid is given by:

B = (μ₀ N r² I)/(2(r²+x² )^3/2

Calculate the value of x when B = 0.01, μ₀ = 4π × 10^-7, N = 1000, r = 0.1 and I = 10. (8)

(b) Solve the following system of equations for A and B in the range of

0 ≤ A ≤ π/2 and 0 ≤ B ≤ π/2 radians.

3sin A + 4cos B = 1.77

2sin A – cos B = 0.52 (8)

Q4. Solve for x in EACH of the following equations:

(a) 7^2x-1 = 4^x+1 (6)

(b) log((5 - x)/(3 - x)) = 0.5 (6)

(c) √(x⁵ ) = 7 (4)

Q6. A parallelogram has sides of 25 cm and 15 cm.

The two acute angles between the sides are 30°.

Calculate EACH of the following for the parallelogram:

(a) the lengths of the diagonals; (12)

(b) the area. (4)

Q7. (a) The displacement s metres of a body from a fixed point is given by the equation:

s = 10/3 t³ – 33/2 t² + 20t + 6 where t is the time in seconds.

Determine EACH of the following for this body:

(i) its initial velocity; (3)

(ii) the times when it is at rest; (4)

(iii) the time when its acceleration is 7 ms^-2. (3)

(b) Given h = 4 – 3 sin t + 5cos t, where t is the time in seconds, evaluate.

h – dh/dt when t = 1 second (6)

Q8. A solid of revolution is formed when the area bounded by the curve y = 2x² + 3 and the lines y = 1, x = -1 and x = 2, as shown by the shaded area in Fig Q8, is rotated about the

x-axis through one complete revolution.

The dimensions are in centimetres.

Calculate EACH of the following for this solid of revolution:

(a) its volume; (12)

(b) its mass, if its density is 2720 kg m^-3. (4)