DieselShip's Online Content Library

Q1. (a) A ship is scheduled to complete a journey of 540 nautical miles at an average speed of 12 knots. It covers the first quarter of the journey at an average speed of 13.5 knots. However due to a series of factors its speed in the last three quarters of the journey is reduced and it arrives at its destination 1 hour 40 minutes late.

Calculate the average speed of the ship over the second part of its journey. (8)

(b) The wind force W on a vertical surface varies directly as the area, A m², of the surface and directly as the square of the wind velocity, v km/hr. When the wind speed is 36 km/hr the force on an area of 2.5 m² is 320 Newtons.

Calculate the force on a surface of area 4 m² when the wind speed is 60 km/hr. (8)

Q2. (a) Solve for x in the following equation: (6)

(6x + 1)/(2x - 3)=(3x - 1)/(2x - 3)

(b) Solve the system of equations for A in the range 0 ≤ A ≤ π/2       (6)

4sinA + 5 cosA = 6.353

13sinA - 8cosA = 3.752

(c) Factorise completely: (4)

2x³ y² + x² y³ - 6y⁴ x

Q3. (a) The bending moment, M, at a point on a beam is given by:

M = 3x(20 - x)/2

where x metres is the distance from the point of support to the end of the beam.

Calculate the value of x when the bending moment is 50 Nm. (8)

(b) Transpose the terms in the following equation to make C the subject: (8)

f = 1/2π √(1/LC - R²/L² )

Q4. (a) Determine the value of x correct to three decimal places that satisfies the following equation: (6)

0.027^x-1 = 3.26

(b) The tension, T Newtons, in the tight side of a belt passing round a pulley wheel and in contact with the pulley for an angle θ radians is given by the equation: (5)

T = 43.8 e^0.32θ

Determine the value of θ when T is 75 Newtons.

(c) The life expectancy, N years, of a certain machine costing £C, and its value £V after n years are related by the formula:

n = (lnV – lnC)/ln(1-2/N)

Calculate the age of the machine which cost £75000 with a life expectancy of 6 years that has depreciated to a value of £10000. (5)

Q5. Table Q5 shows the values of the resistance, R ohms, and the voltage, V volts, recorded in an experiment.

(a) Draw a straight line graph to show that R and V are related by a law of the form R = a/V + b where a and b are constants. (10)

(b) Determine approximate values of a and b. (6)

Suggested scales:      horizontal axis 2 cm = 0.5

vertical axis 2 cm = 2

Q6. (a) A tower 55 metres high stands on the top of a hill which has a 10° incline. The angle of depression from the top of the tower to a point A on the slope is 72°. B is a point further down the slope from A. The angle of depression from the top of the tower to B is 48°. Points A an B and the top of the tower all lie in the same vertical plane.

Calculate the distance between the points A and B. (12)

(b) Given: 2 tanθ = tanα + tanβ

Solve for θ in the range 0° ≤ θ ≤ 360° when α = 30° and β = 45°. (4)

Q7. (a) The rate at which a particular ship’s engine consumes fuel is given by:

rate = 30 + 0.002v³ tonnes per hour (where v is the speed of the ship in km/hr).

Calculate EACH of the following:

(i) the speed at which the minimum amount of fuel is used on a voyage of 1500 km; (10)

(ii) the minimum amount of fuel for the journey. (2)

(b) Determine the first derivative of the following function: (4)

R = h + 4√h - 3/(2√h) + 1/h

Q8. (a) Fig Q8(a) shows the graph of the function

y = -1/2 [x⁴+ x³- 2x² ]

Determine the area enclosed by the function and the x axis. (10)

(b) Evaluate: ∫₁³((2x²+1)/x) dx     (6)

Q9. (a) (i) Determine the mass of a hemispherical copper container whose external and internal diameters are 28 cm and 26 cm respectively. (4)

Note: Copper weighs 8.9 × 10³ kg per m³.

(ii) The entire surface of the container is given a protective coating.

Determine the total surface area to be covered. (4)

(b) Fig Q9(b) represents a regular pentagon of side 200 mm.

Calculate the shaded area. (8)