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Q1. a) Given Z = (Z₁ Z₂)/Z₁ + Z₂) + Z₃, determine Z in Cartesian form when Z₁ = 1 + j,

Z₂ = 1 - j2 and Z₃ = 0.6 - j1.2.

b) Solve the following complex equation for r and θ, where r and θ are real numbers:

r ∠ θ° = (6 ∠ 30°)/(1.2 ∠ 85°) + (3 ∠ 10° × 2 ∠ 25°)

Q2. a) Solve the following system of equations which model the crrents

flowing in THREE branches of an electrical network

1.2i₁ - i₂ + i₃ = 2.4

2.1i₁ - 01i₂ - i₃ = 3.1

0.1i₁ - 2.1i₂ + i₃ = - 0.9

b) Solve for x in the following equation:

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

Q3. (a) The deflection, y, at the centre of a rod under constant load, varies directly as the cube of the length, L, and indirectly as the fourth power of the diameter, d, of the rod.

Calculate the percentage change in y if L decreases by 1 % and d increases by 2%.

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

f = 1/2π √(1/LC - R²/L²)

(c) Fully factorise the following expression:

9x⁴ - 4x²

Q4. (a) A block of metal, initially at 0°C, is placed in a pre - heated oven set at 250°C.

The temperature of the metal, T°C, is given by the function T = 250(1^-ekt) where t is the time in minutes the metal has been in the oven and k is a constant.

Given that when t = 10, the temperature of the metal is 131.9℃, determine EACH of the following:

(i) The value of k

(ii) the temperature of the metal when t = 15.

b) Solve the following equation for x, stating the result correct to 3 decimal places:

In (1 - x³) = - 0.45

c) Transpose the following equation to make C the subject:

t = 1/M In (1/(1 + C))

Q5. a) Draw the graph of the function y = 1/π(2θ – sin θ), for the range

0  ≤  θ  ≤  6.4 radians, in intervals of 0.8 radians

Suggested scales: horizontal axis   2 cm = 1

vertical axis   5 cm = 1

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

(i) the value of θ such that y = 3.1.

(ii) the value of y when θ = 2 radians.

Q7. a) An open tank is to be in the shape of a triangular prism, as shown in Fig Q7(a).

The triangular cross - section of the tank is right - angled and isosceles, with equal sides of length x cm.

The tank has a length of I cm and when full of water it is to have a wetted surface area of 10800 cm2

Determine EACH of the following for the tank:

(i) an expression, in terms of x, of its capacity in cm3; (5)

(ii) the value of x which maximises its capacity; (5)

(iii) the maximum capacity in litres. (2)

b) Determine the first and second derivatives of the function:

C = 2r + 5/r (4)

Q8. A church has a dray sandstone wall with THREE arched windows as shown in Fig Q8(b).

EACH of the THREE windows are as shown in Fig Q8(a).

The curved edge of each window is part of the parabola with equation y = 8x - 2x²

a) the area of each window. (7)

b) the area of the sandstone (5)

c) the height of the bottom of each window from the ground. (4)