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Q1) (a) Given Z = √50<45°, determine EACH of the following as a complex number in polar form:

(i) Z + Z^{- 1} ; (8)

(ii) Z/Z^{- 1} . (3)

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

1.34 + j 12.32 + 10 < 30° = r < θ . (5)

Q2) (a) Solve for x in the following equation:

4x/(3x - 5) + 3/(x + 5) = 2 (8)

(b) Solve the following equations for a, b and c :

a - 2b - 6c = 1

2a + 3b - 3c = 4

3a + b + 2c = 16 (8)

Q3) (a) The resistance, R ohms, of a copper wire at t℃ is given by R = R^' [1 + α (t - 20)] where R^' is the resistance of the wire at 20℃ and α is the temperature coefficient of resistance of copper at 20℃ .

Given R = 16.86 ohms when t = 40℃ and R = 19.07 ohms when t = 75℃, determine the values of R^' and α. (10)

(b) Factorise fully EACH of the following

(i) 24 x⁴ y - 4x³ y² - 4x² y³ ; (3)

(ii) 3a³ b - 27 ab³. (3)

Q4) (a) Make T the subject of the following formula:

I = GR² e^{- C/T} (6)

(b) Solve the following equation for x :

ln⁡(2 - 2 x²) = - 0.5〗 (5)

(c) Express the following in its simplest form:

5 ∛(a³ b⁶ ) + 7 b √(a² b² ) - 4 ∜(a⁴ b⁸ ) (5)

Q5) (a) On the same set of axes plot the graphs, in intervals of 0.5, of y₁ = 2 x² - 3x - 5 in the range - 1  ≤ x  ≤ 3.5 and y₂ = 5 - (2 - x)² in the range - 1 ≤ x  ≤ 4.5.               (12)

Suggested scale: horizontal axis 2 cm = 1

Vertical axis 1 cm = 1

(b) Using the graphs plotted in Q(5), solve the system of equations:

y = 2 x² - 3x - 5

y = 5 - (2 - x)² (4)

Q6) TWO spheres of diameters 25 mm and 50 mm fit into an oil funnel spout as shown in Fig Q6.

Calculate EACH of the following for this spout:

(a) The internal taper angle; (8)

(b) The diameter D at the top. (8)

Q7) (a) Use differential calculus to determine EACH of the following for the function

y = 8x³ - 6 x² = 36 x + 41

(i) the coordinates of the stationary points ; (8)

(ii)the nature of the stationary points. (4)

(b) Given y = √x - ln⁡ x² ,

determine dy/dx . (4)

Q8) The plain nylon sheave of a pulley - block may be considered as being formed by rotating the area bounded by the curve y = x² + 3 and the lines x = - 1, x = 1 and y = 0.8, about the x - axis through one complete revolution.

(a) Sketch the bounded area. (4)

(b) State the bore of the sheave. (2)

(c) Calculate the volume of the nylon comprising the sheave. (10)

Note : that the unit of length is the centimetre.

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 Boolean expression for the outputs D, E and X in terms of the inputs A, B and C; (3)

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

(b) Simplify, as fully as possible, the following Boolean expression:

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

(i) The binary operation; 11001 x 1011; (2)

(ii) The hexadecimal operation; DF8A + BCE9; (2)

(iii) The conversion of FA2E₁₆ to decimal. (2)