DieselShip's Online Content Library

Q1. Solve the following complex equation for a and b, where a and b are real:

11a + 3b + j(3a + 2b) - 7 + j(a - 3b + 2I) (8)

(b) Given Z = (Z₁ - Z₂)/(Z₃ - 2Z₁) Where Z₁ = 2 + j5, Z₂ = 1 + j3 and Z₃ = 9 + j5

Express Z as a complex number in polar form. (8)

Q2. (a) The angular velocity, Ω of a connecting rod may be derived from the formula:

Ω = ωcosθ/√(n² - sin² θ)

(i) Transpose the formula to make n the subject. (6)

(ii) Evaluate n when Ω = 2.2 radians per seconds, ω = 10 radians per second and θ = π/4 radians. (4)

(b) Solve the following system of equations for x and y. (6)

x² + y² = 25

y = 1/2x + 1

Q3. (a) The heat generated by an electric current following through a wire varies directly as the lime, I seconds. the square of the voltage, V volts, and indirectly as the resistance, R ohms.

When the voltage is 40 volts and the resistance is 70 ohms the heat generated after 15 seconds is 480 units.

Determine the heat generated in 20 seconds when the voltage is 30 volts and the resistance is 42 ohms. (6)

(b) A propulsion problem causes a reduction in a ship's speed of 4 knots throughout a passage of 360 nautical miles, resulting in the ship arriving at its destination S hours behind schedule.

Calculate the normal service speed of the ship.

Q4. (a) Solve the following equation for x:

log_e⁡(2 - 3x²) = - 0.6〗

(b) Express the following in its simplest form:

6∜(16a⁴ b⁸) + 2b∛(27a⁶ b³) - 4√(9a² b⁴)

(c) Given y = ⁡(4 log 9)/log⁡27 - log⁡3,  determine the value of y without the use of mathematical tables or calculator.

Q6. The depth of water, h metres, over a sandbar at the mouth of a river on a particular day, is given by:

h = 5 + 3 cos⁡ πt/6

Where t is the number of hours after local high - water.

Calculate EACH of the following for that day:

(i) the minimum depth of water over the sandbar.

(ii) the time when the minimum depth occurs:

(iii) the latest time, after high - water, when a vessel of draught 4.5 metres may sail over the sandbar with a clearance of 1.3 metres.

b) TWO radar targets are observed simultaneously at ranges of 5 and 9 nautical miles when the difference in their bearings is 32°.

Q7. a) Use differential Calculus to determine the coordinates and nature of the stationary points for the function: (12)

y = 2x³ - 9x² + 12x.

b) Given u = 1 - 2/t + 3/t² determine du/dt and (d² u)/(dt²). (4)

Q8. A watertight bulkhead can be represented by the area enclosed by the curves:

y₁ = 8 - 001x₂, - 10  ≤  x  ≤ 10

y₂ = - 0.007x₃, - 10  ≤  x  ≤  10

y₃ = 0.007x₃, - 10  ≤ x  ≤ 10

as shown by the shaded area in Fig Q8 (a).

Calculate the area of the bulkhead, given that the units of length are metres.

b) Evaluate ∫_0.8¹(2 sinθ - cosθ)dθ.