Scenario:

You are working as a research technician at national research laboratory where good analytical skills and understanding of mathematics and Statistical analysis are essential for everyday work within your research group. Your line manager has informed you that to have a successful promotion within your research team during the forthcoming year you need improve and update your present mathematical and Statistical skills so that you can deal with a variety of problems where use of algebraic, trigonometric equations and differentiation and integration on variety of engineering, scientific analysis and calculations are all essential for the new position. In addition, your line manager has also highlighted the importance sound understanding of statistical data collection and analysis that are required to enable you to make positive contributions to your research team in particular with collection, and analysis of the data.

As preparatory task for your forthcoming promotion, your line manager has presented with variety of mathematical and statistical problems and requested from you to present your model solutions to his attention within 2 months to enable him to consider your promotion within the company.

Assignment 1

LO1: Task 1 (PASS)

1. a) Explain Systematic Errors and Random Errors with suitable a case study for each error. Briefly comment on your answer based on your explanation.

1. b) A computer engineer created a program in which the input signals are accepted from the sensors in the form of denary numbers and the output device requires the signal to be in binary form, but the processor works only based on the signals in the form of octal number. If the input is given to be 5613.90625, then what will be the binary number for the output signal? Give a brief comment based on your answer.

1. c) An electrical engineer while checking the electric circuit of an electronic device connects an Alternating voltage of 240 V, 50Hz across impedance of (60 - j 100) Ω. Determine the resistance, the capacitance, the magnitude of the impedance and its phase angle and the current flowing. Give a short comment on your answer.

1. d) A complex signal encoder sends signals through the radio waves in the form of complex equation (x - j2y) + (y - j3x) = 2 + j3. The decoder on the other end decodes the complex signal to reveal the message. Solve the above complex equation to decode the signal. Briefly give a comment on your answer.

LO2: Task 2 (PASS)

2. a) Two electrical forces are represented in the form of complex numbers. The electrical forces are Z1 and Z2and they are represented as (2+ j4) and (3- j) respectively. Determine the value for Z1+ Z2, Z1- Z2and Z2- Z1. Show the results by drawing an Argand Diagram.

2. b) A compound wall is constructed in front of a house in a curved style. The wall was designed with the help of the function 1/ 1+sinx between the limits 0 to π/2. Use the trapezoidal rule and find the area between the compound wall and the house using 6 or more intervals. Give the answer correct to 4 significant figures. Use graph paper to plot curve of the compound wall. Briefly comment on the result obtained based on the above scenario.

2. c) In a complex electric circuit, the resistance of one of its positive resistor is expressed by the equation x2 - 2x - 13 = 0, where x is the resistance of the resistor. Determine the value of the resistance correct to three decimal places by using Newton-Raphson method. Briefly comment on your answer.

Task 3 (MERIT)

3. a) In an overhanging bridge 3 suspension cables are inspected randomly to check whether the bridge design is safe. The tensions on the three cables are given in the form of three equations. Briefly comment on the results obtained.

T1 + T2 + T3 = 4

2T1 - 3T2 + 4T3 = 33

3T1 - 2T2 -2T3 = 2

Determine the values of T1, T2 and T3 using Cramer's rule.

3. b) An alarm indicator in a grinding mill complex should be activated if (i) the power supply to all mills is off and (ii) the hopper feeding mills is less than 10% full, and (iii) if less than two of the three grinding mills are in action. Devise a logic system to meet these requirements. Briefly give a comment based on your answer.

Task 4 (DISTINCTION)

4. a) An experiment is conducted on a simple pendulum to analyze the wave form obtained. The oscillations of a heavily damped pendulum satisfy the differential equation d2x/dt2 + 6dx/dt + 8x = 0, where  x cm is the displacement of the bob at time t seconds. The initial displacement is equal to +4cm and the initial velocity in 8 cm/s. Solve the equation for x. Briefly comment on your answer.

4. b) In an aerodynamic test conducted in a wind tunnel, the test results of some of the aerodynamic properties were given in the form of equations. Use the method of bisection to find the positive root of the equation 5x2 + 11x - 17 = 0 correct to 3 significant figures. Briefly give a comment based on your result obtained.