Final-ssce1693-20162017.docx

UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE ...................................................................... FINAL EXAMINATION SEMESTER I SESSION 2016/2017 COURSE CODE

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SSCE 1693

COURSE NAME

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ENGINEERING MATHEMATICS I

PROGRAMME

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1SKMM, 1SKMP, 1SKMB, 1SKMT, 1SKMO, 1SKMI, 1SKMV, 1SKTB, 1SKTK, 1SKTN, 1SKTG, 1SKTP, 1SKEM, 1SKEE, 1SKEL, 1SKAW, 1SMBE

LECTURER

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NIKI ANIS AB KARIM WAN ROHAIZAD WAN IBRAHIM ZUHAILA ISMAIL ZAITON MAT ISA NORASLINDA MOHAMED ISMAIL NUR ARINA BAZILAH AZIZ MOHD ARIFF ADMON HAZIMAH ABDUL HAMID AMIDORA IDRIS NOR’AINI ARIS IBRAHIM MOHD JAIS WAN RUKAIDA WAN ABDULLAH ZAKARIA DOLLAH

DATE

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26 DECEMBER 2016

DURATION

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3 HOURS

INSTRUCTION

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ANSWER ALL QUESTIONS

__________________________________________________________________ (THIS EXAMINATION BOOKLET CONSISTS OF 8 PRINTED PAGES)

SSCE 1693

QUESTION 1 (6 MARKS) By using only the identities of hyperbolic functions prove that

sinh  3x   4sinh3  x   3sinh  x 

QUESTION 2 (5 MARKS) dy , given dx

Find

ex tanh  x  y   sinh 1  tan  2 y    0 .

QUESTION 3 (6 MARKS) Solve

x

1 4 x4 1

dx

QUESTION 4 (7 MARKS) Determine if these integrations are convergent or divergent. (a)







3

0

xe3x dx (3 marks)

(b)

1

1 x2  4

(4 marks)

2

SSCE 1693

QUESTION 5 (8 MARKS) Given

f  x  e (a)

2 x  3

Prove that the Taylor series expansion for f (x) at x = 0 is given by 

2n x 2 n n! n 0

f  x  

Show your calculations up to the 4th derivative. (5 marks) (b)

By using the following equation 

f  x  

m0

f  m  0  m x m!

find the 102nd derivative of f (x) at x = 0, that is: f 

102 

 0 . (3 marks)

QUESTION 6 (8 MARKS) Given the following three points in 3D space: A(–3, –2,5) , B(0,1,7) and C(4, –1,2). B(0,1,7)

A(–3, –2,5) C(4, –1,2)

BCA .

(a)

Calculate the angle of

(b)

Find the equation of the plane containing points A, B, and C.

(c)

Given a point Q(5,8,10), find the shortest distance between Q and the plane containing A, B, and C.

(3 marks) (3 marks)

(2 marks)

3

SSCE 1693

QUESTION 7 (20 MARKS) Find P–1 of matrix P using the Elementary Row Operations.

(a)

 1 1 1 P   0 1 1  1 0 1 (7 marks) (b)

Determine the eigenvalues for the matrix A below and verify that the columns of P can be composed from the eigenvectors of A.

 1 2 2  A   2 1 2   2 2 1 (10 marks) Determine the eigenvalues of D = P–1AP.

(c)

(3 marks)

QUESTION 8 (20 MARKS) (a)

Given the equation r  8sin  , convert it into Cartesian coordinates and show it is a circle of radius 4 centered at (0,4). Hence, sketch the graph. (5 marks)

(b)

Test the symmetries of the curve r 2  4 cos  . (6 marks) On the empty grid at the final page of this examination script, sketch the graph r 2  4cos on the with the help of the following table. (Detach and include the sketch with your answer book.)

(c)

θ(o)

0



6



4



2  3

2



6



4

2  3



2

r (4 marks) (d)

Find the points of intersection of the curves r 2  4cos and r  1  cos . Sketch these points of intersection on a single diagram. (5 marks)

4

SSCE 1693

QUESTION 9 (20 MARKS) (a) Given u = 2 – i and v = i – 3. u Express 2 in the form of a + ib. u  v  Then, determine the modulus and argument of

u . u  v  2

(5 marks) (b) Given a complex number w  1  i (i) Calculate z  2  w2  2w   4 in the form a + ib . (ii) Express z in polar form. 1 3

(iii) Find all possible values of z and sketch them on an Argand Diagram. (8 marks) (c) Using De Moivre’s Theorem show:

cos 3   4cos3    3cos   Hence, obtain all solutions of x for the following equation: 3x  4 x3  1

(7 marks)

5

6

7

8