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Cork Institute of Technology
Higher Certificate in Engineering in Electronic Engineering – Stage 1
(NFQ Level 6)
Summer 2006
Mathematics
(Time: 3 Hours)
Instructions Answer any FIVE questions. Graph paper and Log tables are available All questions carry equal marks.
Examiners: Dr. W. P. O Connor Mr. J. Berry Dr. R O’Dubhghaill
Q1. (a) Use the laws of indices to simplify each of the following, giving answers with positive
indices only.
(i) (^264)
6 2 4 36
AB C
A BC −
(ii) (^) ( 3 ) ( 2 1 )
( 3 4 ) ( 2 )
- 4
x x (6 Marks)
(b) Solve for X in each the following equations (i) log 3 ( x − 4 )+log 3 ( 2 x + 1 )= 1
(ii) ln(x^5 ) – ln(x^3 ) = 2. (iii) e x^ − e −^ x = 4 (10 Marks)
(c) Make C the subject of the following formula
= ^2 +( −^1 )^2
C
Z R L
ω (4 Marks)
Q2 (a) Solve for x , y and z given that :
2x – y + z = 9 –2x + 7y – z = – 3x + 4y – 2z = –5 (6 Marks) (b) Solve for all possible values of x and comment on your answers. x^3 – 3x^2 + 4x – 12 = 0 (6 Marks) (c) The following table gives values of power, P and current, I which are related by the law P = RI^ n , where R and n are constants. I(amps) 1.4 4.7 6.8 9.1 11.2 13. P(watts) 49 552 1156 2070 3136 4290
Verify the law exists. Find the approximate values of R and n. Hence state the law. (8 Marks)
Q3. (a) The instantaneous value of voltage in an ac circuit at any time t seconds is given by v(t) = 175 sin(55 π t + 0.48) volts. Determine the
(i) amplitude, periodic time, frequency, phase angle in degrees and phase time. (ii) value of the voltage when t = 0 (iii) value of the voltage when t = 5 ms (iv) time when the voltage is first a maximum. (v) time when the voltage first reaches 100 volts.
Draw a sketch of v(t) over one cycle, showing relevant points. (14 Marks)
(b) Two waves x 1 = a sin ( ω t )and x 2 = a sin ( ω t − φ)meet. Find the equation for the
resultant wave (x 1 – x 2 ) and discuss the following special cases.
(i) φ = 0 (ii) φ = π 2 (iii) φ = π
(6 Marks)
Q6. (a) Show that (3 + j) is a root of the equation
z^2 − 5 z +( 7 − j )= 0 and find the other root of this equation.
(b) Write ( − 3 + j )in the form r (cos φ + j sin φ)and hence evaluate
( 3 )^4
− + j
(9 Marks)
(c) A series circuit consists of a 12 Ω resistor, a coil of inductance 0.1 H and a
capacitance of 160 μ F. The supply voltage is 240 V, 50 Hz, determine
(i) The inductive reactance (ii) The capacitive reactance (iii) The impedance (iv) The current flowing and its phase angle relative to the supply voltage. (v) The voltage across the resistance (vR) (vi) The voltage across the coil (v (^) L). (vii) The voltage across the capacitor (vC) (viii) Sketch a voltage diagram illustrating vR, vL and vC. (11 Marks)
Q7. (a) Differentiate each of the following with respect to the variable in each case.
(i) x x x
y x 6 3
2
= −^2 + + −
(ii) if (^)
x
y x 1
ln 1 , show that dx x ( x )
dy −
(iii) y = 7e-2t^ .cos5t
(iv) y = ( 9 x^3 + 5 x^2 + 2 ) (12 Marks)
(b) Find the local maximum and local minimum of the curve y = 2 x^3 + 3 x^2 − 12 x − 8 and distinguish between them. (4 Marks)
(c) A body moves x metres in t seconds, where x = t^3 – 7t^2 + 3. Find (i) the velocity at t = 6 seconds. (ii) The acceleration at t = 8 seconds (4 Marks)
