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)
Summer 2005
Mathematics
(Time: 3 Hours)
Examiners: Mr. J. Calnan
Mr. M. Ahern
Mr. K O Connell
Instructions
Answer FIVE questions.
All questions carry equal marks.
Q1. (a)
Simplify
7n-3 + 14 x 7n-5
(4 marks)
7n-4 x 9
4
+ log
45
= log 6 − log 13
(4 marks)
(b)
Show that log
(c)
Make x the subject of the formula Z =
(d)
Factorise f (x) = x3 - 4x2 + x + 6 and hence solve the equation f(x) = 0.
(4 marks)
(e)
Draw a rough graph of the functions y = x2 – 2 and y = 2-x2.
(4 marks)
Q2. (a)
15
26
25 x 2 − 49
(4 marks)
In a network the currents i1, i2 and i3 are related by the equations:
3 i1 + 2i2 + 5i3 = 2
5i1 + 3i2 – 2i3 = 4
2i1 – 5i2 – 3i3 = 14
Solve the equations and check your solutions.
(7 marks)
(b) Two resistors when connected in series have a total resistance of 50ohms. When connected in
parallel their resistance is 10.5ohms. If one of the resistors has a resistance R1 ohms:
(i)
Show that R12 – 50R1 + 525 = 0
(ii)
Calculate the resistance of each resistor
(7 marks)
(c) If a = 5, b = 8, C = 103o15’, solve ═∆ ABC, i.e. determine the unknown side length and
angles.
(6 marks)
Q3. (a)
If Z1 = 2 + j5 and Z2 = 3 – j2 express Z1 + Z2,
Z1
1
1
in both cartesian
and +
Z2
Z1 Z 2
and polar forms
(b)
(10 marks)
The current flowing in a circuit at any time t seconds is given by
I = 75 sin (100 π t + 0.32) amperes. Find:
(c)
Q4. (a)
(i)
the amplitude, periodic time, frequency and phase angle.
(ii)
the value of the current when t = 0.
(iii)
the value of the current when t = 0.006s
(iv)
the time when the current first reaches 50 amps.
(v)
the time when the current is a maximum.
Sketch (roughly on plain paper) one cycle of the oscillation.
(10 marks)
In each of the following cases, state the variables you would plot to obtain a straight line
graph:
a
− b , where a and b are constants.
x
(i)
y=
(ii)
w = af + bf 2, where a and b are constants.
(iii)
(iv)
(2 marks)
(2 marks)
n
(2 marks)
at
(2 marks)
q = ah , where a and n are constants.
y = ke , where k and a are constants.
2
(b)
The following table gives values of the current i amperes in a circuit after time t seconds:
t
1
2
3
4
5
6
i
72.00
30.50
14.25
6.50
2.75
1.25
Verify, graphically, that these values are related by a law of the form i = I e –kt where I
and k are constants. Use your graph to determine probable values for I and k. Check the
law for one value of the independent variable.
(12 marks)
Express 3 sin θ + 5 cos θ = 4 for values of θ between 0o and 360o.
(7 marks)
(b)
Write down the first four terms in the expansion of (2x – 3y)7.
(6 marks)
(c)
The frequency of a tuned circuit is given by f =
Q5. (a)
1
2π LC
where the L is the inductance
and C the capacitance. If L increases by 0.5% and C increases by 1% find the
approximate % change in f.
Q6. (a)
(7 marks)
Differentiate the following with respect to the variable in each case:
2
−
x4
7
− 3 e - 5x + 9
x
(i)
7 x5 −
(ii)
e2x sin3 (5x – 7)
(4 marks)
(iii)
ℓn5x
(4 marks)
3
(4 marks)
1 – 2x
(b)
An open rectangular box, with square ends, is fitted with an overlapping lid that covers
the open top, the front face and the whole of the square ends. Find the maximum volume
of the box if 8m2 of sheet material is used altogether in constructing the box and lid.
(8 marks)
3
Q7. (a)
(i)
Evaluate the integrals:
∫[x
4
−
7 6
+ + 8] dx
x5 x
(4 marks)
3
(ii)
(1 − x) 3 − 1
∫0 x dx
(4 marks)
π
2
(iii)
∫ (cos 7 x − sin 5 x)dx
(4 marks)
0
(b)
Find the area enclosed between the curve y = x3 – 2x2 – 5x + 6 and the x-axis between the
limits x = -2 and x = 3.
(8 marks)
4