TM2401

NATIONAL UNIVERSITY OF SINGAPORE

EXAMINATION FOR

TM2401 – ENGINEERING MATHEMATICS II

(Semester II: 2008/2009)

Time Allowed: 2 Hours

________________________________________________________________________ INSTRUCTIONS TO CANDIDATES 1. This examination paper contains FOUR (4) questions and comprises FOUR (4)

printed pages. 2. Answer ALL questions. 3. All questions carry equal marks. 4. This is a Closed Book examination. 5. The candidate is allowed to bring into the examination hall a single A4 size sheet of

notes/formulae written on both sides.

2 TM2401

Question 1 (a) A periodic function f(x) of period 2 is defined as follows:

⎩⎨⎧

<<<<

=21,210,2

)(xxx

xf

Sketch f(x). Find the Fourier coefficients a0 and bn. Its other Fourier coefficient is given as follows:

⎪⎩

⎪⎨⎧

=−

== oddn

n

evennan ,4

,0

22π

Write down the Fourier series of f(x).

By considering the value of f(x) at x = 0, find the sum of the series ∑∞

=12

1n n

.

(13 marks) (b) Given z is complex number, show that

θniz

z nn sin21

=⎟⎠⎞

⎜⎝⎛ − (4 marks)

By considering the expansion of 51⎟⎠⎞

⎜⎝⎛ −

zz or otherwise, express θ5sin in terms of

sines of multiples of .θ (8 marks)

[Hint: consider the equality ( )5

5 1sin2 ⎟⎠⎞

⎜⎝⎛ −=

zzi θ . ]

3 TM2401

Question 2

(a) State the range of k and give the general solution of u for the differential equation,

02

22

2

2

=∂∂

+∂∂

∂+

∂∂

yuk

yxu

xu , if it is hyperbolic.

(5 marks)

(b) Solve the Laplace equation ayaxyu

xu

≤≤≤≤=∂∂

+∂∂ 0,0,02

2

2

2

given the

boundary conditions,

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=⎟⎠⎞

⎜⎝⎛

===

=

ayax

yax

x

u

at ,sin

0at ,0at ,0

0at ,0

π .

(20 marks)

Question 3 (a) Determine the non-parametric form of the curve C1, given by jir )1( 2 ++= tt for

40 ≤≤ t . Sketch C1 and determine the unit tangent vector to the curve at

⎟⎠⎞

⎜⎝⎛ 0,

411,

21 .

(9 marks)

(b) Sketch the curve C2, given by kjir ttt +++= )1( 2 for 40 ≤≤ t . Find

rkji dtC

∫ ⋅⎟⎠⎞

⎜⎝⎛ ++

2

31 and dszC∫ +

2

242

(16 marks)

4 TM2401

Question 4 (a) The equation of a flat plane, S, is given by 1=++ CzByAx where A,B and C are

constants (Fig. 1). Determine the values of A,B and C given that the plane passes through the points (1,0,0), (0,1,0) and (0,0,1).

(5 marks)

(b) Determine the equation of the 3 lines where the surface S cuts the Cartesian co-ordinate planes.

(5 marks)

(c) Determine the volume integral dVxIV∫= where V is the volume in the first octant

bounded by the plane S given in part (a) as shown in Fig. 1.

(15 marks)

Fig. 1.

- END OF PAPER -

x

y

z

(1,0,0) (0,1,0)

(0,0,1)

S