DieselShip's Online Content Library

Q1) (a) THREE mooring lines exert horizontal forces on a bollard, positioned at O, as follows:

18 kN at 55°

35 kN at 85°

42 kN at 140°

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)

(b) Given Z₁ = 2a + j7a and Z₂ = 9b + j5b, solve the following complex equation for a and b, where a and b are real numbers:

Z₁ + Z₂ = 30 - j. (8)

Q2) (a) Express the following in its simplest possible form:

3/(2x - 1) - 2/(x + 1) - (x + 4)/(2x² + x - 1) (8)

(b) Fully factorise EACH of the following:

(i) x⁶ - 25 x⁴ ; (2)

(ii) x^3 - x^2 - x + 1; (4)

(iii) 18 x^2 - 23 x - 6 . (2)

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

Given that the points ( - 2,4), (1,1) and (5,5) lie on the same circle, determine the values of g, f, and c for this circle. (10)

(b) Make u the subject of the following formula:

a = T/√(ku² - lv² ) (6)

Q4) (a) Solve for x in EACH of the following equations:

(i) 10²x²  = 2^{x + 1}; (8)

(ii) √(x³ - 19) = 18. (4)

(b) Evaluate the following without using mathematical tables or

a calculator:

2 (log 64 - log⁡8/log⁡4 ). (4)

Q5) (a)Draw the graph of y = 4 tan⁡ 0.8x, for the range 0 ≤ x ≤ 1.4 in intervals of 0.2. (10)

Note that the angles x is in radian measure.

Suggested scales: horizontal axis 2 cm = 0.2

Vertical axis 2 cm = 1

(b) Using the same axes and scales, draw the graph of

y = 5/2 π - 4x , for the range 0 ≤ x ≤ 1.4, on the graph produced in Q5(a). (3)

(c) Use the graph plotted in Q5(a) and Q(5)(b) to solve the equation.

4(x + tan⁡ 0.8 ) = 5/2 π , for 0 ≤ x ≤ 1.4 . (3)

Q6) AB is a link 68 cm long which has a block pivoted to EACH end.

The blocks can slide in grooves as shown in Fig,(Q6)

The point of intersection of the line of centres is at C.

Initially, BC = 34.8 cm and AC = 40.6 cm.

Calculate EACH of the following:

(a) the angles between the line of centres ( i.e angle < BCA); (3)

(b) the inclination of AB to AC; (3)

(c) the distance A moves if block B moves 20 cm towards C from the given position. (9)

Q7) An open rectangular dive training tank, with square ends of side x metres an a volume of 392 m³, is shown in Fig Q7.

The tank was constructed at a cost of £75 per square metre for the base and £150 per square metre for the vertical sides and ends.

Determine EACH of the following for the tank:

(a) the length L, in terms of x; (2)

(b) the total construction cost in terms of x; (7)

(c) the dimensions, given that the construction cost was minimised. (7)

Q8) (a) A body starts with an initial velocity, v₀ ms^{- 1}, and its acceleration, a ms^{- 2}, is given by a = 2 + 6t, where t is the time, in seconds, from the start.

Given a = dv/dt and velocity v = dS/dt, determine EACH of the following for the body:

(i) v as a function of t , given that v = 7 ms^{- 1} when t = 1 second. (4)

(ii) the initial velocity v₀. (1)

(iii) the distance from the start position, S metres, as a function of t. (4)

(iv) the distance travelled in 2 seconds. (1)

Q9) (a) The logic circuit shown in Fig Q9 (a) has three inputs A, B and C, and one output X.

Produce EACH of the following for this circuit:

(i) an unsimplified Bollean expression for the outputs D and X in terms of the inputs A, B and C. (2)

(ii) the truth table, including columns for A, B, C, D and X. (3)

(iii) the equivalent logic circuit using only NAND gates (crossing out any redundant gates). (5)

(b) Determine, without using a calculator conversion function, the value of EA₁₆ ÷ 10010, giving the answer in three forms: binary, hexadecimal and decimal. (6)