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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2009
Module Title: Technological Mathematics 2
Module Code: MATH
School: Electrical & Electronic Engineering
Programme Title: Bachelor of Engineering in Electronic Engineering – Year 1
Programme Code: EELXE_7_Y
External Examiner(s): Dr. B. O’Regan Internal Examiner(s): Dr. P. O’Connor
Instructions: Answer QUESTION 1 (worth 40 marks) and TWO other questions (worth 30 marks each)
Duration: 2 HOURS
Sitting: Autumn 2009
Requirements for this examination: Graph paper, Log Tables
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
- (i) Given z 1 (^) = 12 + j 5 and z 2 (^) = 4 − j 3 determine 1 2
z z
in Cartesian form
and in polar form. Verify your answers are the same. (5 marks)
(ii) Simplify the following complex equation
o o
o o I ∠ + ∠
∠φ =. (5 marks)
(iii) The voltage (V) volts in a circuit at time (t) is given by
) 4
() 100 sin( 30
V t = π + t.
Find an expression for
dV dt
. Hence evaluate
dV dt
at
(a) t = 0 and (b) t = 10ms. (5 marks)
(iv) Show that y = ( 2 x − 10 ) is a solution to the differential equation
dx
dy .
(5 marks)
(v) Find the equation of the tangent line to the curve y at the point (2,21). y = 3 x^2 + 2 x + 5 (5 marks)
(vi) Evaluate dx x
x
2
( 1 )^2
. (5 marks)
(vii) Evaluate the integral dx x
x
( 1 + )^2
. (5 marks)
(viii) The acceleration a ms −^2 of an object is given by a = 9. 81. Derive expressions for the velocity v ms −^1 and the displacement x m if the initial velocity and displacement are both zero. (5 marks)
- (a) Evaluate the following integrals:
(i) dx x x
∫ x −^ x + − − )
(ii) dx x
x x
4
(iii) (^) ∫ e tan x sec^2 x. dx
(iv) ∫ x xdx
4
0
cos 4 cos 2.
π
(v) dx x
x
3
0 2 1
(20 marks)
(b) Sketch the curve y = x^2 − 1 and find the area enclosed between it and the x -axis, where − 1 ≤ x ≤ 1. (10 marks)
- (a) Differentiate y = 4 x^2 + 2 x + 11 from first principles.
(10 marks)
(b) Use Newton’s method to show that x^3 − 4 x^2 + 5 x − 8 = 0 has a root between 3 and 4. Find this root correct to one decimal place. (10 marks)
(c) Find the particular solution of the second order differential equation
2 12 6
2 = x − dx
d y given that y ( 0 )=− 4 and y^1^ ( 0 )=− 12.
(10 marks)
