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Q1. a. Solve the following complex equation for x and y: 2 + j = ((1 - j)/(1 + j))² + 1/(x + jy)  (8)

b. Three mooring lines exert horizontal forces on a bollard, positioned at O, as follows:

35 kN at 50°, 44 kN at 190°, 29 kN at 305°

The angles are those that the forces make with the real axis Ox.

Determine, using complex numbers, the magnitude and direction of the resultant force on the bollard. (8)

Q2. a. Simplify the following as fully as possible:  (3a² bc⁴)²/(a³ b⁶ c⁹)^1/3  (4)

b. Solve the following equation for x: 4^2x-1 = 8^{2x + 1}  (4)

c. The value of a vessel at any age can be determined from the formula:

R_V= C[1 - r/100]ⁿ  where n = age of the vessel in years

r/100 = The annual rate of depreciation as a percentage

C = value of the vessel when new in £M

R_V  = Residual value of the vessel after n years in £M

Use the formula to determine how long it will take a vessel, which cost £23 M to build and depreciates in value by 16 % each year, to fall to its scrap value or £O.92 M. (8)

Q3. a. S = (6x - 17)/(2x² + 5x - 3) + 4/(2x - 1) + x/(x + 3)

Express S as a single fraction in its simplest form: (8)

b. Solve the following system of equation for x, y and z:

4x + 5y - 2z = -3

3x + 4y - 3z = -7

2x - 2y - z = -3(8)

Q4. a. A box-shaped vessel, floating on an even keel, has a water-plane area of 72 m². The length of the vessel is 11.5 m greater than its beam. Calculate the length and beam of the vessel. (8)

b. Factories EACH of the following as fully as possible:

(i) x² - y² + 2y - 1 (4)

(ii) 4a³ b -  6a² b - 4ab (4)

Q5. a) Draw the graph of y = 2sinA + 3cosA in the range ,0° ≤ A ≤ 180°inintervals of 20°. (10)

b) Using the graph drawn in Q5(a), determine EACH of the following:

i) the value of A when y = 0;(2)

ii) the maximum value of y;(2)

iii) the value of A at the maximum value of y. (2)

Q6 a. A vessel travels 64.5 miles on a bearing 075°.

It then travels 48 miles on a bearing 130°.

Calculate EACH of the following:

i. The distance of the vessel from its starting position; (6)

ii. The bearing of the vessel from its starting position. (6)

b. Determine the value of θ in the range 0° < θ < 90° which satisfies the equation:

3sin²⁡θ-cos²⁡θ = 1  (4)

Note: sin²⁡ θ + cos²⁡ θ = 1

Q7. a. The displacement s meters of a body from a fixed point is given by the equation:

S = 2/3 t³-13/2 t² + 15t + 8 where t is the time in seconds

Determine EACH or the following for the body:

i. Its initial velocity; (3)

ii. The times when it is at rest; (4)

iii. The time when its acceleration is 3 ms^-2. (3)

b. Determine the first and second differential coefficients of the expression:

y = 6√x+3 ln⁡ x + 5 sin⁡x  (6)

Q8. a. A watertight bulkhead can be represented by the area enclosed by the curves.

y₁= 6-0.01x², -10 ≤ x ≤ 10

y₂= -0.005x³, -10 ≤ x ≤ 0

y₃= 0.005x³, 0 ≤ x ≤ 10 as shown by the shaded are in Fig Q8(a) Calculate the area of bulkhead. (10)

Q9. (a) The truth table for a logic system with inputs A, B and C, and output X, is shown in Table Q9(a):

Table Q9(a)

Produce EACH of the following for the logic system:

(i) a Boolean expression in its simplest form; (4)

(ii) the logic circuit, with the minimum number of parts (3)

(iii) the logic circuit, using only NAND gates. (3)

(b) Determine EACH of the following, without using a calculator conversion function:

(i) the binary operation 10111011 – 1010111; (1)

(ii) the hexagonal operation CD4B + EA7;     (2)

(iii) the conversion of B5EA₁₆ to decimal ; (2)

(iv) the conversion of 111001100111₂ to hexadecimal. (1)