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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2011
Module Title: Technological Mathematics 2
Module Code: MATH 6046
School: Electrical & Electronic Engineering
Programme Title: Bachelor of Engineering in Electronics – Year 1
Programme Code: EELXE_7_Y
External Examiner(s): Dr. P. Kirwan 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 2011
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.
- (a) Given z (^) 1 7 j and z 2 (^) 1 j 2 determine^1 2
z z
in Cartesian form and in
polar form. Verify your answers are the same. (5 marks)
(b) If (^) z (^) 3 8 (cos 60 ^ j sin 60 )^3 and (^) z (^) 4 cos 45 ^ j sin 45 , express 4
3 z
z in
polar form. (5 marks)
(c) The voltage (V) volts in a circuit at time (t) is given by V t ( ) 32 ( 1 e ^0.^8 t ).
Find an expression for
dV dt
. Hence evaluate
dV dt
at (a) t 0 and (b) t = 7
seconds. (5 marks)
(d) Show that y e^3 t is a solution to the differential equation
y "^ ( t ) 6 y '( t ) 9 y ( t ) 0 (5 marks)
(e) Determine the equation of the tangent line to the curve y at the point ( 2 , 5 ). y 1 x x^2 (5 marks)
(f) Evaluate
3
1
( x^2 2 x 3 ) dx. (5 marks)
(g) Evaluate the integral.
dx x
x x x ) 2
5 3 (5 marks)
(h) The acceleration a ms ^2 of an object is given by a 6 t 6 where t is the time in seconds. Derive expressions for the velocity v ms ^1 and the displacement x m if the initial velocity and displacement are both zero. (5 marks)
(b) Locate the turning-point on the following curve and determine whether it is a maximum or minimum point.
y 4 x e x (14 marks)
- (a) Evaluate the following integrals:
(i) dx x
x x 3 )
(^6 5 ^4 4
(ii) dx x
x x )
(iii) [ 8 x^5 e (^2 x^ )] dx
6
(iv)
1
0
5 sin( 2 x ) dx
(v)
2
0 9 2
dx x
(20 marks)
(b) Sketch the curve y x^2 2 x 3 and find the area enclosed between it and the x-axis. (10 marks)
- (a) Differentiate y 3 x^2 4 x 2 from first principles. ( 11 marks)
(b) Use Newton’s method to determine the positive root of the quadratic equation x^2 2 x 2 0 , correct to four significant figures. (11 marks)
(c) Find the particular solution of the second order differential equation
2 6 4 0
2 x dx
d y given that y ( 0 ) 29 and y '^ ( 0 ) 42.
( 8 marks)
