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Q1. (a) A dealer buys 45 portable welding machines at £260 each.

One third of the machines were sold at a profit of 40% and the remainder were sold at a profit of 20%.

Calculate the overall percentage profit. (6)

(b) The fuel consumption per unit of time, C, of a ship, is directly proportional to the cube of its speed v.

On a particular day the ship’s speed is increased by 8% above normal for 5 hours, decreased by 12% for 10 hours and is normal for the remaining 9 hours.

Calculate the percentage decrease in the fuel consumption below normal for that day. (10)

Q2. (a) A communications mast stands vertically on a horizontal base area.

An anchor point on the base area is 20 m from the foot of the mast.

The distance from the anchor point to the top of the mast is 5 m more than the height of the mast.

Calculate the height of the mast. (6)

(b) Solve the following equation for x: (6)

((3x + 1) (2x - 1))/3x(x - 1) -2 = 0

(c) Fully factorise: (4)

a² b – 3a² - 4b + 12

Q3. The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where g, f and c are constants.

The points (1, 8), (-2, -1) and (2, 1) lie on the same circle.

Determine EACH of the following for this circle:

(a) the values of g, f and c; (13)

(b) the radius, r, given that r = √(g²+ f²- c) (3)

Q4. (a) The pressure, P units, in a defective tyre, t hours after being inflated to a pressure of 50 units, is given by:

P = 50e- 0.01kt where k is a constant.

After 24 hours the tyre pressure falls to 40 units.

Calculate EACH of the following:

(i) the value of k, correct to 3 decimal places; (4)

(ii) the tyre pressure after 48 hours; (2)

(iii) the tyre pressure after 1 week. (2)

(b) Solve for x, x > 0, in the following equation: (8)

3^x2 = 27^x+1

Q7. (a) The drag, D, acting on an aircraft operating under certain conditions, is given by:

D = aV² + b/V²

where V is the airspeed of the aircraft and a and b are positive constants.

Determine the value of V, in terms of a and b, for minimum drag. (8)

(b) Differentiate EACH of the following functions:

(i) y = x² + 2x - 1/x + 1/√x (4)

(ii) v = 10 – t + 5sin t – 3cos t. (4)

Q9. (a) The debris from excavating a tunnel is estimated to be 174720 m³.

This debris is to be stacked in the form of a frustum of a cone such that the vertical height is 18 m and the area of the base is four times the area of the top.

Calculate the base area of the stack. (10)

(b) Metal washers have an outside diameter of 30 mm, an inside diameter of 10 mm and a thickness of 2.5 mm.

The density of the metal is 7500 kg m^-3.

Calculate, to the nearest kilogram, the mass of a batch of 20000 such washers. (6)