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Q1. (a) Given (2Z₁ - Z₂)/(Z₁ - Z₃ ), where Z₁ = 4 + j5, Z₂ = 3 + j5, Z₃ = 3 + j3, express Z as a complex number in polar form. (8)

(b) Two impedances, Z₁ = 4∠30° and Z₂ = 3∠- 10° are connected in series to a supply voltage, n, of 150 volts.

Calculate the current, i amperes, as a complex number in Cartesian form,

given that i = v/Z  , where Z = Z₁ + Z₂. (8)

Q2. (a) On a particular planet, the height, h metres, of a projectile, t seconds after launching is derived from the formula h = at + bt² where a and b are constants.

Given that a projectile on this planet has h = 296m when t = 2 and h = 725m when t = 5, determine EACH of the following:

(i) the values of a and b; (7)

(ii) the duration of the flight. (5)

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

T = √((2ghDA²)/(d(S² - A²))) (4)

Q3. (a) The lengths, in centimeter's, of the sides of a right - angled triangle are 2x - 1, 3x + 1 and 3x - 4.

Determine the lengths of the THREE sides. (8)

(b) Given y = 1 - (x³ - x² - 6x )/(x³ - x² - 12x), express y in its simplest form. (8)

Q4. (a) The time, t hours, to charge a new tablet to a level C (expressed as a decimal fraction of the battery’s full charge) is given by;

t = 2.2In (1 - C), 0 < C < 1.

Determine EACH of the following for the tablet:

(i) the percentage of the full charge achieved after charging for 51/2 hours; (6)

(ii) the total time taken, to the nearest ¼ hour, to reach 95% of full charge. (4)

(b) Solve the following for equation x:

log₁₀ (3/2x - 1/(2x² )) = 0 (6)

Q5. Draw the graph of the function y = 0.5x² - 2x - 10 = 0 for the range, - 4 ≤ x ≤ 8, in intervals of 1. (8)

(b) Use the graph drawn in Q5(a) to determine EACH of the following:

(i) the minimum value of y; (2)

(ii) the solutions of the equation 0.5x² - 2x - 10 = 0; (3)

(iii) the solutions of the equation 0.5x² - 2x - 7 = 0; (3)

Q6. (a) A quadrilateral shaped metal plate has dimensions as shown in Fig Q6(a).

The angle CDA is 75°.

Calculate EACH of the following for the plate:

(i) the shortest distance from A to C; (4)

(ii) the size of the angle DAB. (8)

Q7. A ship sailing at v knots consumes fuel at a rate of (1.5 + 0.001v³) tonnes per hour.

Determine EACH of the following for this ship when it completes a passage of 2000 nautical miles at a speed of v knots:

(i) the speed which gives the greatest fuel economy; (10)

(ii) the minimum amount of fuel consumed. (3)

(b) Given y = √x/2 - 2/√x determine dy/dx. (3)

Q8. A dam is to be built to contain water in a new reservoir.

Relative to axes, as shown in Fig Q8, the sides of the walls can be represented by parts of the graphs of y = 4x, 0 ≤ x ≤ 5, and y = 36 - 1/4 x², 8 ≤ x ≤ 12.

The shaded area represents the constant cross–section of the dam wall.

Calculate EACH of the following for the dam wall, given that the dimensions are in metres:

(a) its height; (2)

(b) its cross - sectional area, using integral calculus; (12)

(c) its volume, given that it is 360m in length. (2)

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

For this logic system:

(i) produce an unsimplified Boolean expression for output X; (2)

(ii) use a Karnaugh map or Boolean algebra to simplify as fully as possible, the expression for X obtained in Q9(a)(i); (5)

(iii) use the expression for X obtained in Q9(a)(ii) to produce the logic circuit using only NAND gates, crossing out any redundant gates. (3)

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

(i) the binary operation 10111 X 1101; (1)

(ii) the conversion of 1101101₂ to decimal form; (1)

(iii) the conversion of 3AB₁₆ to binary form; (2)

(iv) the hexadecimal operation AD8F + DECC. (2)