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Q1. (a) Given Z = (1 - J)/(1 + J2),determine Z - Z^{- 1} as a complex number in polar form.    (10)

(b) solve the following complex equation for r and θ:

r∠θ = 14.5  - 0.67 -  5∠60⁰ . (6)

Q2. (a) solve the following system of equations for x and y.

4x²  + xy + 9y² = 91

3y - 2^x = 13

(b) Transpose the following formula to make the subject:

T = A(1 - 1/r^{n - 1})

Q3. (a) TWO positive integers differ by 6.

The sum of the squares of the two numbers is 818.

Determine the value of EACH of the two numbers. (8)

(b) solve for x in the following equation:

(x + 3)/(x + 2) = (3x - 7)/(2x - 3) (6)

(c) Factories the following as fully as possible:

x² + 4xy + 4y² (2)

Q4. (a) The value, £C, of a piece of machinery after t years is given by: C = 92000 e^0.205t

Calculate the number of complete years before its value falls to less than £18.000. (6)

(b) solve for x in the following equation:

log4x⁵ - log x = log 16x + log2 (5)

(c) Evaluate the following expression for A without using tables or calculator:

A = (4 log 9  -  2 log 27  +  log 243)/(7 log 3) (5)

Q5. The stress, o MN\/m², in an alloy steel plate was recorded at different, x cm, from one end

The results are shown in Table Q5.

(a) Draw a straight - line graph to verify that the relationship between σ and x is a law of the form σ = a^x2 + bx where a and b are constants.

Table Q5

Suggested scales: horizontal axis 2 cm = 1

Vertical axis 2 cm = 2

(b) use the graph drawn in Q5(a) to estimate the value of a and b. (6)

Q6. At 0900 hours a customs cutter is set on a course to intercept a vessel as soon as possible.

The vessel is 15 nautical miles distant, bearing 035⁰ and is saiing on a steady course of 095⁰ at a speed of 20 knots.

The best speed of the cutter is 32 knots.

Calculate the earliest possible interception time, to the nearest minute. (16)

Q8. (a) A tunnel entrance has a stone walled arch.

Determine, using integral calculus, the area of the stonework represented by the shaded area in the rectangular cross section of the tunnel entrance, shown in Fig Q8(a), given that the unit of length is the meter. (8)

(b) dv/dt = (4t² + 4t - 15)/(2t - 3), and v = 5ms^-1when  t = 1s, Determine v as a function of t.