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Q1). (a) The resultant impedance, Z, of two parallel circuits of impedances Z₁ and Z₂ is given by the formula  1/Z =  1/Z₁ +  1/Z₂ .

Calculate the resultant impedance, expressing it in polar form, when Z₁= 4 + j3 and Z₂ = 3-j4. (8)

(b) Given Z =  (2 + j)(5 - j3)/(6 - j7), determine EACH of the following:

(i) Z in the form of Z = a + jb, where a and b real numbers; (5)

(ii) the value of Z₄ in its simplest form (3)

Q2). (a) The sag, d metres, in a cable of length, L metres, when suspended between two points and subject to a tension, T Newtons, may be determined by the formula:

d²-T/w d +  1/4 L²= 0

where w is the weight in Newtons per metre run.

Calculate, correct to two decimal places, the sag in the cable when T = 2000 N, L = 50 m and the total weight of the cable = 250 N.(8)

(b)The current I amperes in an A.C. circuit is given by the formula

I=  V/√(R²+ (X_L- X_C)² )

(i) Determine I when V = 100, R = 12 and XL - XC = 16.             (2)

(ii) Transpose the formula to make XC the subject. (6)

Q3 (a) An increase of 3 knots in the normal service speed of a ship, throughout a passage of 750 nautical miles, would reduce thew passage time by 5 hours.

Calculate the normal service speed of the ship. (10)

(b) Solve the following system of equations for x and y: (6)

x² + y² = 25

y = x + 1

Q4). (a) The temperature, T℃, of a cooling liquid after t minutes is given by

T = 120 e^-0.04t

Calculate the time taken, to the nearest minute, for the temperature of the liquid to fall to 60℃. (6)

(b) Solve the following equation for x:

log⁡(x²+ 26) - log⁡(x + 2) = log⁡7    (6)

(c) Simplify the following as fully as possible:

(16 a⁴ b⁸ c²)^3/4/(4b² c)² (4)

Q5) (a) Plot the graph of the function y = 0.5 x²- x - 2 for the range -3 ≤ x ≤ 5 in intervals of 1. (8)

Suggested scales:

horizontal axis 2 cm = 1

vertical axis 2 cm = 1

(b) Use the graph plotted in Q5(a) to determine the solution of EACH of the following equations:

(i) 0.5 x² – x - 2 = 0 (2)

(ii) 0.5 x² – x = 6 (3)

(iii) x² –2 x = 9. (3)

Q6) A radio mast stands at the top of a 15° incline.

From a point A, on the incline, the angle of elevation to the top of the tower is 35^o from the horizontal.

At a point B, which is 100 metres further up the incline in the same vertical plane, the angle of elevation to the top of the mast is 45° from the horizontal.

Calculate the height of the mast. (16)

Q7) (a) The velocity, v, of waves of length, x, in a canal, may be determined from the formula v²=  gx/2π + 2πT/ρx

where T is the surface tension of the water and ρ is its density.

Determine EACH of the following:

(i) the value of x for minimum value of v²; (8)

(ii) the minimum value of v.   (4)

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

u = sin⁡ t-log_et (4)

Q8) (a) The work done during an adiabatic expansion follow the law PVⁿ = C, where C and n are constants, as the volume increases from V₁ to V₂.

The work done can be represented by the shaded area in Fig Q8(a).

An amount of steam expands so as to satisfy the law PV^1.33 = C.

Calculate the work done, in Joules, when the steam expands from a volume of 0.3 m³ at a pressure of 750 kN/m² to a volume of 0.6 m³. (12)

Q9) (a) Determine EACH of the following, without using a calculator conversion function:

(i) The conversion of 101101₂ to decimal (1)

(ii) The binary operation 1101010 – 111011;    (1)

(iii) The conversion of 11111011₂ to hexadecimal;  (2)

(iv) The binary operation 10001111  1011.    (2)

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