NATIONAL UNIVER Y OF SINGAPORE

THIRD (SUPPLEMENTARY) EXAMINATION HE DEGREE OF B.ENG HEMICAL)

CN301 'ing Mathematics

May/June 1985

INSTRUCTIONS:

1. This paper contains SEVEN(7) questions and comprises (This includes the Normal,

Tables).

2. Answer ONLY FIVE(5) clue! ns.

3. All questions are of equ... weioht.

4. Approved modE of hand-held lculators are allowe

Q1. Answer BOTH Part (a) and Pa

(a) Show that the errors involvu in the solution of a single nonlinear equation by the secant or Regula-Falsi method

are approximately related by en+1 en en _ 1 , where en

denotes the error or difference between the true solution

and the estimated solution after n iterations.

marks)

(b) Solve the equation: en+1 Aenen_ i . Determine p such that

e 1 /e tends to a finite limit as n tends to infinity; n+ n Calculate this limit. And hence, what is the order of the

secant method? marks)

Q2. Answer BOTH Part (a) and Part (b).

(a) Solution of the following nonlinear equation is required.

14200

5 (1.35 x

Initial estimate of V is 1000. What is the value of V after TWO iterations by the successive substitution method?

(5 marks)

Time Allowed : 3 hours

-Inc 0

F- and Chi-Square Disi

.../2

Q2.

r. a

7 77

x O. y 1.0

1,2,...,n.

4

1" 4V

Third (Supplement; LA6m. for B.Eng.(Chemical) - CN301

Q4. The amounts of a chemica. ompound y, which were diss: 100 grams of water at val us temperatures, x, were ri r E

as follows :

can y

8 8 7,333

15

12 10 14 12.0

30

25 21 24 23.33

45

31 33 28 30.67

60

44 39 42 41.67

75

48 51 44 47.67

zx . 675 y488 zx

y

17142

.37.5

27.11 Exy = 25005

(a) Fit a linear model to the above experimenta

marks)

(b) Prepare a complete table of analysis of variance.

(11 marks)

(c) Is the regression significant? (Use=

(2 marks

(d) Test whether the linear model is adequate. (Use= 0.05 in the appropriate statistical best; and residual plots are not required).

(2 marks)

Q5. Answer ALL the SEVEN parts.

(a) State a sufficient condition for the convergence of the solution of the equation x = g(x) by the successive sub Lion method.

marks)

.14

arks)

4,

(d) fr.(

ScJI4 U . 'Kt is

tion, show where

(e) erenti at hct

estimate and an interval

l parent boot, me , median and a

answe

are there any cplain how you

marks)

erences

ark

(g) Stat

:ference may be divided into two major what

and what are their differences?

(4 marks)

06. Answer

of +hie

he le' , one ENGINEERING, place them thoroughly for 10 minutes 7 letters, one at a time. drav, in the correct c

11 the won. an empty ball mill, and tumble Open the mill and draw out What is the probability of ?r, le letters of the word

7 marks)

(b) A balance' — fairly tossed. Let the event A denotes

the occur a number greater than 4, B the occurance

of a numb than 4, C the occurance of a )dd number, D the occ irice of the number 4, E the oco of an even nt

he probabilities of A,B, • E.

(ii) ihich of the ewent Are mutual isi e?

(iii) P of the e e. equally

(iv) Find the proba •1' ' "event A or event B.

(7 marks)

.../5

Third (Supplementary) Exam. for B.Eng.(Chemical) CN301

5.

Q6.

(c) Suppose that in a production rui ten units, three are defective. A sample of three is to be randomly drawn from the ten. Whet is the probability of receiving at least one deft 't if the samplee re drawn (i) with repi3cemer( out replaceme

(6 mar

Q7. Answer BOTH part (a) and Part (b).

(a) The fol:4in lata were obtained for the calibration of

the Ruska dea6 weight gauge used with our Burnett P-V-T apparatus. The weights corresponding to the 1000 psi loading had the following apparent masses:

2c n1r7n 26,03575

26.03551 26.03

26.03529 26.03588 26.03',;„

26.03573 26.03586

Can we say with 95 confidence inter apparent mass ,u does not exceed 26., with Appropriate statistical test assu i ns you have made.

average ify lur answer 3tP

(10 marks)

(b) Samples from a t ,rocess contain x% of a particular compound. It is found that over all samples x has a normal distribution, with mean 12,.6 and standard deviation 2.3. What is the probability tat a particular random sample shall have an x value between 14.0 and .0? What is the value of

x which i .eeded only once per 20 samples?

(10 marks)

.../6

• Exam. CN301

b.

na

./7

7.

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K ,S2S 14 r4r4r4r4r4

St,1M;

414 R» 2PAr$ ,r4 r4r4

NN'3?

.../8

Third Supplemontary Exam. For B.Eng.(Chemical) - CN301

0 33 8

0.9 3.0

0.95 1 0.975 2.7 099 .11 8

995 61 7 [0 U.999 3111

Ff

01 000000 0.00 0 6

0.26 0 26 0 26 0 7

0 54 0.54 0.54 01

0.88 0 87 017 0.9

1.36 1,16 1.35

095

1 80 1,78 1.77 0 975

220 2.18 2(6 0 99

2.72 2.68 2.65 0.995

3.11 '06 303 0.999

4.03 93 3.85

28 062 1 06 1 89

43 69 99

18 4 54 584

06 0.26 107 0.53 0 0,86 0.9 3.32

0.95 1,72 0 975 2,07 0.99 2.31 0.993 2.82 0.999 3.51

0 26 055

0

0 86 1,32

1,71

71 2.06 2,49

2.48 2.80

2.78 3,47

3,44

.00 0 26 0 53 0.85 1

Third (Supplementdry for E3.Eng(Chemicai)

Ex,w. CNS01

8.

Sti d t' t-Distriblition

■ees of rice:Joni, tf n r. 1 F(

er of Deg

027 I027 037 0.94 I 3.91

48 1.44

2

3.75 4 60

0.00 000 0.26 0.26 054 0 34 OK! 081 1,35 1,34

1 76 1.75 235 233 2.62 2.60 2,98 1,93 3.79 3,73

miOwi m AJegitinm ire

28 30 40

000 i -- 026 ' 026

0 53 013

086 (585

11 3,33

71) 1.68

2,03 2.04 702

2.47 2.46 2.42

2.76 2 2.70

3.41 3.39 3 31

11 V() 00 0 (X)0 026 026 0 2 U33 033 044 4 0.90 0.89 0 88 0. 1.42 3.40 1 38

1.86 383 I I 2.31 2 26 227

300 2.90 282 • 1,30 33r 325 :

000 000 0.26 0 26 0.26

53 0.53 0.33 3 0 86 (3 86 0 86 0 86 3.53 1 )3 1 33 1 33

'74 '.73 1.73 1 73 2.11 2 10 2,09 2.09 2.37 2.55 2.54 2,33 2.90 2,88 2 86 2.85 3.65 3.61 3.58 1,55

0 G.. -30 1)00 0 26 0.25 025 025 053 0 53 0 53 0.52 0 85 0.85 0 84 0 84 1.30 1.29 3.29 12!

1 68 3.66 (65 365 2.03 1.98 1 97 3.96 240 2 37 2)5 233 2,48 2.63 2.60 2.5* 3.26 3,17 3.33 3,09

3 37 4 03

0.00 0 26 0 54 0.87 1,34

1.75 2.32 2 58 2.92 3 69

Third (Suoolempntary) Exam. For 3.Eng.(Chemical) - CN3O1

9.

END OF PAPER.