Download Integrals - Mathematics - Past Exam Paper and more Exams Mathematics in PDF only on Docsity! Cork Institute of Technology Higher Certificate in Engineering in Electrical Engineering – Stage 1 (National Certificate in Engineering in Electrical Engineering – Stage 1) (NFQ – Level 6) Autumn 2005 Mathematics (Time: 3 Hours) Instructions Answer FIVE questions. All questions carry equal marks. Examiners: Mr. J.C. Calnan Mr. M. Ahern Mr. K. O Connell Q1. (a) Simplify 79 9189 1 13 x x n nn + ++ − (5 marks) (b) Show that log 21 8 + log 64 70 = log 10 – log 3 –3 log 2 (5 marks) (c) Make R the subject of the formula R 1 = 1 1 R + 2 1 R (5 marks) (d) Factorise f(x) = x3 + 5x2 – x – 5 and hence solve the equation f(x) = 0. (5 marks) Q2. (a) Express 3 Sinθ + 5Cosθ the form r Sin (θ + α), where r and α are constants and use it to solve the equation 3 Sin θ + 5 Cos θ = 2 for values of θ between 0˚ and 360˚. (10 marks) (b) An alternating current is given by i = 60 Sin(100t - 4 π ). Sketch roughly one cycle of the waveform and state the amplitude, frequency, periodic time and the phase angle. (10 marks) 2 Q3. (a) In a network the currents i, i2, i3 are related by the following equations: i1 + 7i2 – 3i3 = 4 2i1 + 5i2 – 4i3 = 3 3i1 – 4i2 + 2i3 = 9 Solve the equations to find values for i, i2, and i3. (10 marks) (b) Two circles have diameters which differ by 4cm. If the sum of the areas of the circles is 146cm2, determine their radii. (10 marks) Q4. (a) If Z1 = 2 – j5 and Z2 = 4 + j3 express 2 21 Z ZZ − in both Cartesian and polar forms. (10 marks) (b) In the triangle ABC, A = 105˚, C = 60˚ and b = 4 find angle B and sides a and c. (10 marks) Q5. (a) In each of the following cases state the variable you would plot to obtain a straight line graph: (i) S = bt + dt2, b and d are constants (ii) W = n a + b a and b are constants (iii) R = R0eat R0 and a are constants (iv) P = RIn R and N are constants (4 x 2 marks) (b) The values of resistance R and voltage V were measured in an experiment and the results are shown below: R ohms 59.5 74.2 84.1 94.8 152.2 v millivolts 135 107 94 83 51 Resistance and voltage are thought to be connected by a law of the form R = v a + b, where a and b are constants. Verify that this is so, find values for the constants. Check the law for one value of the independent variable. (12 marks)
