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Q1. (a) In manufacturing an engineering component the costs of labour and materials are in the ratio 7:3.

The component is sold at £5568 producing a profit of 16%.

Determine EACH of the following:

(i) the cost of the materials; (4)

(ii) the percentage increase in material costs when labour costs increase by 5%, the percentage profit is 12 ½ % and the selling price is £5832. (6)

(b) A cable 200 m long has to be cut into four lengths.

Three of the lengths are to be equal and the fourth length must be 10 m shorter than the sum of the equal lengths.

Calculate the length of the longer piece. (6)

Q2. (a) Solve the following system of equations for a, b, and c: (10)

4a + 6b – 5c = –3

5a + 2b + 3c = 13

15a + 4b – 8c = –16

(b) Factorise EACH of the following as fully as possible:

(i) 12r² + r – 6; (3)

(ii) 3x³ – 12 xy² (3)

Q3. (a) The lengths of the sides of a right-angled triangle are 2x – 3, 5x and 5x – 1 cm.

Determine the value of x. (8)

(b) Solve the following equation for x, x > 0, correct to 2 decimal places: (8)

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

Q4. (a) A formula associated with a thermodynamic process is given by:

T₁/T₂ = (p₁/p₂ )n-1/n

Calculate the value of n when T₁ = 740, T₂ = 296, p₁ = 30, and p₂ = 1.2. (8)

(b) Solve for s in the following logarithmic equation: (8)

Ln((4 - s)/(3 - s)) = 0.75

Q5. (a) Draw the graph of the function y = tan x for the range 1.2 ≤ x ≤ 1.45 radians, in intervals of 0.05 radians. (10)

Suggested scales: horizontal axis 2 cm = 0.05

vertical axis 2 cm = 1

(b) By plotting a suitable straight line on the graph drawn in Q5(a), solve the equation:

4x = tan x (6)

Q6. A patrol boat is due north of a vessel at a distance of 40 nautical miles.

The vessel is making good a steady course of 120° at 15 knots.

The patrol boat intercepts the vessel after 3 hours.

Calculate the course and speed made good by the patrol boat. (16)

Q7. (a) A stainless steel tank is to be fabricated in the shape of a triangular prism with a regular tetrahedron at each end, as shown in Fig Q7(a).

The length of each edge of the tetrahedron is x metres.

The external surface area, A, of the tank is given by:

A = (3√3)/2 (x²+16/x)

Determine EACH of the following for the tank:

(i) dA/dx; (3)

(ii) the value of x which minimises the external surface area; (5)

Verify that the result obtained gives minimum surface area.

(iii) the minimum external surface area. (2)

(b) When a flywheel rotates through an angle of ϴ radians in t seconds, its angular velocity is given by dϴ/dt rads/s, and its angular acceleration is given by (d² ϴ)/(dt² ) rads/s².

For a certain flywheel ϴ = 27t – 3t² .

Determine EACH of the following for this flywheel:

(i) the angular velocity when t = 4; (3)

(ii) the angular acceleration; (1)

(iii) the time that elapses before the angular velocity is zero. (2)

Q8. An architectural feature of a grey sandstone building is a rectangular wall, 6 m × 5 m, with an arched window.

The curved edge of the window is part of the parabola with equation y = 3x – 1/2 x², as shown in Fig Q8.

Determine EACH of the following:

(a) the area of the window; (10)

(b) the area of the sandstone; (2)

(c) the distance of the top of the window from the base. (4)