8/12/2019 0910SEM1-MA1505

1/33

Matriculation Number:

NATIONAL UNIVERSITY OF SINGAPORE

FACULTY OF SCIENCE

SEMESTER 1 EXAMINATION 2009-2010

MA1505 MATHEMATICS I

November 2009 Time allowed: 2 hours

INSTRUCTIONS TO CANDIDATES

1. Write down your matriculation number neatly inthe space provided above. This booklet (and only thisbooklet) will be collected at the end of the examination. Donot insert any loose pages in the booklet.

2. This examination paper consists of EIGHT (8)questions

and comprisesTHIRTY THREE (33)printed pages.3. AnswerALLquestions. For each question, write your answer

in the box and your working in the space provided inside thebooklet following that question.

4. The marks for each question are indicated at the beginning ofthe question.

5. Candidates may use calculators. However, they should lay outsystematically the various steps in the calculations.

For official use only. Do not write below this line.

Question 1 2 3 4 5 6 7 8

Marks

8/12/2019 0910SEM1-MA1505

2/33

MA1505 Examination

Question 1 (a) [5 marks]

Find the slope of the tangent to the curve

y= (35x 69)43

whenx= 2.

Answer1(a)

(Show your working below and on the next page.)

2

8/12/2019 0910SEM1-MA1505

3/33

MA1505 Examination

(More working space for Question 1(a))

3

8/12/2019 0910SEM1-MA1505

4/33

MA1505 Examination

Question 1 (b) [5 marks]

Let

f(x) =ax3 +bx2

be a function defined on (,), where a and b are non-zeroconstants. Given that fhas a point of inflection at (1, 2), find the

value of the product ab.

Answer1(b)

(Show your working below and on the next page.)

4

8/12/2019 0910SEM1-MA1505

5/33

MA1505 Examination

(More working space for Question 1(b))

5

8/12/2019 0910SEM1-MA1505

6/33

MA1505 Examination

Question 2 (a) [5 marks]

Let

f(x) =23 4x

7 2xand let

n=0

cn(x 2)n

be the Taylor series for f at x = 2. Find the exact valueofc0+c2009.

Answer2(a)

(Show your working below and on the next page.)

6

8/12/2019 0910SEM1-MA1505

7/33

MA1505 Examination

(More working space for Question 2(a))

7

8/12/2019 0910SEM1-MA1505

8/33

MA1505 Examination

Question 2 (b) [5 marks]

Use the method of separation of variables to find u (x, y) that

satisfies the partial differential equation

2uxy= [sin (x+y) + sin (x y)] u,given that u (0, 0) = 1 and u (, ) =e2.

Answer2(b)

(Show your working below and on the next page.)

8

8/12/2019 0910SEM1-MA1505

9/33

MA1505 Examination

(More working space for Question 2(b))

9

8/12/2019 0910SEM1-MA1505

10/33

MA1505 Examination

Question 3 (a) [5 marks]

Let

f(x) =x2, x ,andf(x+ 2) =f(x) for all x. Let

a0+

n=1

(ancos nx+bnsin nx)

be the Fourier Series which represents f(x). Find the exact

valueofa2010+b2010.

Answer3(a)

(Show your working below and on the next page.)

10

8/12/2019 0910SEM1-MA1505

11/33

MA1505 Examination

(More working space for Question 3(a))

11

8/12/2019 0910SEM1-MA1505

12/33

MA1505 Examination

Question 3 (b) [5 marks]

Let

f(x) = 1, 0< x

8/12/2019 0910SEM1-MA1505

13/33

MA1505 Examination

(More working space for Question 3(b))

13

8/12/2019 0910SEM1-MA1505

14/33

MA1505 Examination

Question 4 (a) [5 marks]

Let Sbe the plane which passes through the points (1, 0, 0), (2, 1, 0)

and (3, 2, 1). Let Lbe the line which passes through (0, 0, 0) and

is parallel to the vector 3i +j 265k. Find the coordinates of thepoint of intersection ofLandS.

Answer4(a)

(Show your working below and on the next page.)

14

8/12/2019 0910SEM1-MA1505

15/33

MA1505 Examination

(More working space for Question 4(a))

15

8/12/2019 0910SEM1-MA1505

16/33

MA1505 Examination

Question 4 (b) [5 marks]

A space curve Cis defined by the vector parametric equation

r (t) = 2t2

i +

t2

4tj + (3t 5) k.LetTdenote the tangent vector to Cat the point correspondingto t = 1. Find the length of the projection ofT onto the vectori + 2j + 2k.

Answer4(b)

(Show your working below and on the next page.)

16

8/12/2019 0910SEM1-MA1505

17/33

8/12/2019 0910SEM1-MA1505

18/33

MA1505 Examination

Question 5 (a) [5 marks]

Let

f(x , y , z ) =xy+yz+zx+ 1505.

Find the exact valueof the directional derivative of f at thepoint (2, 3, 4) in the direction of the vectoru= i j 2k.

Answer5(a)

(Show your working below and on the next page.)

18

8/12/2019 0910SEM1-MA1505

19/33

MA1505 Examination

(More working space for Question 5(a))

19

8/12/2019 0910SEM1-MA1505

20/33

8/12/2019 0910SEM1-MA1505

21/33

MA1505 Examination

(More working space for Question 5(b))

21

8/12/2019 0910SEM1-MA1505

22/33

MA1505 Examination

Question 6 (a) [5 marks]

Find theexact valueof the double integral

D

|x y|dxdy,whereD is the rectangular region: 0 x 1 and 0 y 2.

Answer6(a)

(Show your working below and on the next page.)

22

8/12/2019 0910SEM1-MA1505

23/33

MA1505 Examination

(More working space for Question 6(a))

23

8/12/2019 0910SEM1-MA1505

24/33

MA1505 Examination

Question 6 (b) [5 marks]

Find theexact valueof the iterated integral

60

6x

2xyln{(1 +y2)(1+x2)}

dy

dx.

Answer

6(b)

(Show your working below and on the next page.)

24

8/12/2019 0910SEM1-MA1505

25/33

MA1505 Examination

(More working space for Question 6(b))

25

8/12/2019 0910SEM1-MA1505

26/33

MA1505 Examination

Question 7 (a) [5 marks]

Find theexact valueof the volume of the solid enclosed laterallyby the circular cylinder about z-axis of radius 1, bounded on top

by the elliptic paraboloid

2x2 + 4y2 +z= 18 ,

and bounded below by the plane z= 0.

Answer

7(a)

(Show your working below and on the next page.)

26

8/12/2019 0910SEM1-MA1505

27/33

MA1505 Examination

(More working space for Question 7(a))

27

8/12/2019 0910SEM1-MA1505

28/33

MA1505 Examination

Question 7 (b) [5 marks]

Find theexact value of the line integral

C

(ex cos y) dx+ (2x ex sin y) dy ,whereCconsists of two line segments: C1from (ln 3, 0) to

0, 1

ln 36

,

andC2 from

0, 1ln36

to ( ln 2, 0).

Answer7(b)

(Show your working below and on the next page.)

28

8/12/2019 0910SEM1-MA1505

29/33

MA1505 Examination

(More working space for Question 7(b))

29

8/12/2019 0910SEM1-MA1505

30/33

8/12/2019 0910SEM1-MA1505

31/33

MA1505 Examination

(More working space for Question 8(a))

31

8/12/2019 0910SEM1-MA1505

32/33

MA1505 Examination

Question 8 (b) [5 marks]

Use Stokes Theorem to find theexact value of the line integral

C

(yzdx+xzdy+xydz) ,whereCis the curve of intersection of the plane

x+y+z= 2

and the cylinder

x2 +y2 = 1,

oriented in the counterclockwise sense when viewed from above.

Answer8(b)

(Show your working below and on the next page.)

32

8/12/2019 0910SEM1-MA1505

33/33

MA1505 Examination

(More working space for Question 8(b))

END OF PAPER