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Q1. (a) Impedances Z₁ = 10∠60°ohmsand Z₂ = 2∠ - 30°ohmsareconnected in parallel to a voltage supply, v, of 200 volts.

Calculate the current, I amperes, as a complex number in cartesian form,

Given that i = v/Z when Z = (Z₁ Z₂)/(Z₁ + Z₂)     (10)

(b) Solve the following complex equation for x and y:

1/(0.2x - jy) = (3 - j)/(2 + j5)                  (6)

Q2. (a) A ship’s fuel consumption varies directly as the square of the ship’s speed and indirectly as the calorific value of the fuel.

If the ship burns 90 tons of fuel per day of calorific value 48 MJ/kg when sailing at 20 knots, determine the daily consumption of the ship when using fuel of calorific value 44MJ/kg and sailing at 22knots.       (8)

(b) fully factorise EACH of the following:

(i) 2x³ - 3x² + 16x - 24; (3)

(ii) 12x³ - 3xy²;    (2)

(iii) 9x² - 10x - 16. (3)

Q3. (a) Solve for x in the following equation:

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

(b) Express the following as a single algebraic fraction in its simplest form:

(2a + 4b)/(a + 3b)² × (a² + ab - 6b²)/(a² - 4b²) (8)

Q4. (a) Given the formula:

T₁/T₂ = [P₁/P₂ ]^{n - 1}/n)

Calculate the value of n when T₁ = 690, T₂ = 345,  P₁ = 30  and  P₂ = 1.5.      (8)

(b) Solve EACH of the following for x:

(i) 4e^{- 0.5x} = 3    (4)

(ii) log_e⁡(e^x + 12) = 3.8

Q5. (a) Draw the graph of y = 3sin q - cos q, in the range 0 ≤ θ ≤ 4, in intervals of 0.5 radians.

Suggested scales: horizontal axis 2 cm = 0.5

Vertical axis 2 cm = 0.5

(b) Using the graph drawn in Q5(a), estimate the solution of the following equation for q,

In the range 0 ≤  q  ≤  4: 3tan q = 1

Q6. (a) Fig Q6(a) shows a right - angled triangle with its sides tangential to an inscribed circle.

Given AB = 60 cm and BC cm, calculate the diameter of the circle.     (10)

(b) An alternating current, I milliamps, is given by:

i = 20 sin⁡(100π - 0.4) where t is the time in seconds.

Calculate the least value of t, t > 0, for which the current i = 12 milliamps.      (6)

Q7. (a) A car wash operates daily from 1000 hours to 1730 hours.

The average queuing time, q minutes, for customers arriving x hours after opening time is given by the function q = 0.2 (2.5 + 7.5x² - x³).

Calculate EACH of the following for this car wash:

(i) the busiest time of day;   (8)

(ii) the expected queuing time at the busiest time.         (2)

(b) Given u = (1 - sin² θ)/(1 - sinθ) - cosθ, determine  du/dq and (d² u)/(dq²)      (6)

Q8. (a) A body starts with an initial velocity of 2 ms^-1 and its acceleration is 3 + 4t ms^-2 where t is the time in seconds, from the start.

Given acceleration a = dv/dt and velocity v = ds/dt, where s is the displacement of the body in meters, from the starting point, determine EACH of following for the body:

(i) v as a function of t;         (4).

(ii) its velocity after 5 seconds;      (1)

(iii) its displacement, s, as a function of t;  (4)

(iv) its displacement after 6 seconds.        (1)

(b) Evaluate ∫₁⁴(√(x + 1/√x))dx         (6)

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

Produce EACH of the following for this circuit:

(i) a simplified Boolean expression for the outputs D, E and X in terms of the

Inputs A, B and C: (3)

(ii) the truth table, including collum’s for A, B, C, D, E and X: (3)

(iii) the simplest expression for X obtained in (i), by using Boolean algebra.

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

(i) the binary operation 1101101 – 1011010: (1)

(ii) the binary operation 101110 Χ 101: (2)

(iii) the conversion of 1011101101₂ to hexadecimal: (2)

(iv) the conversion of 2766₁₀  to hexadecimal. (2)