Q1. The characteristic impedance, Z, of a transmission line, may be determined from the complex formula Z2 = (R + jωL)/(G + jωC)
Determine Z2 in the polar form, r ∠θ, for a transmission line where
R = 2.1 ohms, ω = 104 radians/second, L = 0.2 × 10- 3 henrys, G = 4.2 x 10-6 - ohms and C = 1.2 x 10-9 farads,
b) Given Z = x + jy, where x and y are real, solve the following equation for x and y:
2Z/(1 + j) - Z/(1 - j) = 15/(2 + j)
Q2. (a) Solve the following system of equations for x and y:
2x2 - 3xy + y2 = 35
3x - 2y = 12
Make T the subject of the following formula:
d2 = 16/πf(M + √(M2 + T2)
Q3. (a) A box - shaped vessel, floating on an even keel, has a water - plane area of 96 m².
The beam of the vessel is 15.2 m less than its length.
Calculate the length and beam of the vessel
(b) Express the following in its simplest form:
3x/(x + 1) + 2/(x - 3) + (x - 11)/(x2 - 2x - 3)
Q4. a) A tyre has a slow puncture causing the tyre pressure to drop.
The pressure in the tyre, P units, t hours after inflation to P0 units is given by:
P = P0e-kt, Where k is a constant.
Given that the tyre was inflated to a pressure of 40 units and that after 32 hours it had dropped to 15 units, determine the length of time it would have taken for the pressure to have dropped from 40 units to 30 units.
b) Solve for x in the following equation:
ex - e- x = 3
Q6. A diesel engine unit has a vertical stroke of 320 mm and a connecting rod AB of length 570 mm as shown in Fig Q6.
Angle CAB is the angle between the vertical and the position of the connecting rod.
Twice during each down stroke of the piston the angle CAB equals 8 degrees.
Calculate the piston travel between these TWO positions. (16)
Q7. (a) The displacement, s metres, of a body from a fixed point is given by the equation:
s = 48t + 8t2 - 4/3 t3, where t is the time in seconds.
Determine EACH of the following for the body:
(i) the time (t > 0) when its velocity is zero. (5)
(ii) the time when its acceleration is zero. (2)
(iii) its acceleration after FIVE seconds. (1)
(b) Given P = 3cos x - 1/√x + In x3, determine dP/dx and (d2 P)/dx2 .(8)
Q8 (a) The shape of an inflatable polyurethane marker buoy may be represented by the rotation of the shaded area in Fig 08(a). about the x axis. through one complete revolution.
Determine, using integral calculus. the volume of the buoy, given that the unit of length is the metre.
(b) Evaluate ∫0π cos θ tanθ dθ.
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