Q1. (a) An alloy is made by combining metal A with metal B such that the ratio of their volumes is 7:5 respectively. The relative density of A is 8.9 and that of B is 7.1.
Determine the percentage mass of EACH of the metals. (8)
(b) A rod 5.2 metres long is cut into 4 lengths A, B, C and D.
A is 10% longer than B. B is 1.6 m longer than C. D is 50 % longer than C.
Calculate the lengths of A, B, C and D. (8)
Q2. (a) Determine the values of x, for x ≥ 0, which satisfy the following equation: (8)
10√x-2 = 5 x
(b) Factorise completely: (4x - 3)3-4(4x - 3) (2x + 1)2 (8)
Q3. (a) Solve for a and b in the following system of equations: (8)
(4b - 1)/2 + (2a + 1)/5 = 5/2
(2b - 3)/5 + (3a- 1)/7 = -32/35
(b) Transpose the terms in the following equation to make A the subject: (8)
T = √(2ghDA2/d(S2- A2))
Q4. (a) Given T1/T2 = [p1/p2 ]n-1)/n
Calculate the value of n when T1 = 645, T2 = 300, p1 = 19.2 and p2 = 1.2 (8)
(b) Determine the values of t, for t > 0, which satisfy the following equation: (8)
log 3t - 52 -log(2t) = log [8/t]
Q5. The intensity of radiation, R, from certain radioactive materials at a particular time, t, is considered to follow the law:
R = ktn where k and n are constants.
A test produced the values shown in Table Q5.
(a) Using the values in Table Q5, draw a graph to verify the law. (10)
Suggested scale: horizontal axis 2 cm = 0.1
vertical axis 2 cm = 0.1
(b) Use the graph drawn in Q5(a) to determine approximate values for k and n. (6)
Q6. (a) A ship travels 25 km on a bearing 130°.
It then travels 40 km on a bearing 200°.
Calculate EACH of the following:
(i) the distance of the ship from its starting position; (5)
(ii) the bearing the ship must take in order to return in a straight line to its starting position. (5)
(b) Determine the values of β in the range which satisfy the equation: 0° ≤ β≤ 180o
sin2 3β/2 = 0.5 (6)
Q7. (a) The efficiency, η, of a steam turbine is given by:
η = 4(nρ cosα -n2 ρ2 ) where n and α are constants.
Determine EACH of the following:
(i) the value of ρ such that η is a maximum; (8)
(ii) the maximum value of η. (2)
(b) Determine the first and second derivatives of the following function: (6)
y = 5 sin x+ 6x3 - 2x√x
Q8. (a) Fig Q8(a) shows the graphs of the function y1 = 81 - x4 and y2 = x2- 9
Determine the shaded area enclosed by the two functions. (10)
(b) Evaluate ∫π/2π(10 + 8sin ∝ - 3cos∝) dα (6)
Q9. Fig Q9 shows the area of an aluminium plate which is 5 mm thick.
It has the form of a trapezium with the major segment of a circle removed.
AB = CD = 300 mm, BC = EF = 800 mm
Angle AEF = Angle EFD = 110° and the maximum depth of the major segment is 700 mm.
(a) the area of the plate; (14)
(b) the mass of the plate. (2)
Note: density of the aluminium is 2700 kg/m3
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