btec hnc - analytical methods - algebraic methods

Algebraic MethodsAnalytical Methods for Engineers

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsAlgebraic Methods

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 5 -

1.1 Solve the following polynomial division :- - 5 -Solution:- - 5 -Check:- - 6 -

1.2 Solve the following equation using the factor theorem :- - 7 -Solution:- - 7 -Check:- - 7 -

1.3 Use the remainder theorem to find the remainder for :- - 8 -Solution:- - 8 -Check:- - 8 -

1.4 Find the remainder for the following expression :- - 9 -Solution:- - 9 -Check:- - 10 -

1.5 Resolve the following into partial fractions :- - 11 -Solution (a) :- - 11 -Check:- - 11 -Solution (b) :- - 12 -Check:- - 12 -Solution (c) :- - 13 -Check:- - 13 -

TASK 2 - 14 -

2.1 Solve the following equations correct to 2 decimal places :- - 14 -Solution (a) :- - 14 -Check:- - 14 -Solution (b) :- - 15 -Check:- - 15 -Solution (c) :- - 16 -Check:- - 16 -

2.2 The voltage across a capacitor at time t is given by :- - 17 -Graph and Circuit Diagram :- - 17 -Solution (a) :- - 18 -Check:- - 18 -Solution (b) :- - 19 -Check:- - 19 -

2


2.3 Evaluate correct to 4 significant figures. - 20 -Solution (a) :- - 20 -Check:- - 20 -Solution (b) :- - 21 -Check:- - 21 -

2.4 A telegraph wire hangs so that its shape is described by :- - 22 -Solution :- - 22 -Check:- - 23 -

2.5 If , find values for A and B. - 24 -Solution :- - 24 -Check:- - 25 -

2.6 If find values for P and Q. - 26 -Solution :- - 26 -Check:- - 26 -

2.7 Solve the equation correct to 2 d.p’s. - 27 -Solution :- - 27 -Check:- - 29 -

TASK 3 - 30 -

3.1 Determine the 15th term of the series 12, 17, 22, 27,..... - 30 -Solution :- - 30 -Check:- - 30 -

3.2 The sum of the first 10 terms of an arithmetic progression is 200 and the common difference is 4. Find the first term of the series. - 31 -

Solution :- - 31 -Check:- - 31 -

3.3 An Oil & Gas Company drills a hole 10 km deep. - 32 -

Estimate the cost of drilling if the cost is £1000 for drilling the first metre with an increase in cost of £200 per metre for each succeeding metre. - 32 -

Solution :- - 32 -Check:- - 32 -

3.4 Determine the 10th term of the series 2, 6, 18, 54,.... - 33 -Solution :- - 33 -Check:- - 33 -

3.5 Find the sum of the first 12 terms of the series 1, 4, 16, 64,.... - 34 -Solution :- - 34 -Check:- - 34 -

3


3.6 Find the sum to infinity of the series 4, 2, 1, 1/2, 1/4,..... - 35 -Solution :- - 35 -

3.7 Use the Binomial Series to expand - 36 -Solution :- - 36 -Check:- - 36 -

3.8 (a) Expand the following in ascending powers of x as far as the term in using the Binomial Series. - 37 -

Solution (a) :- - 37 -Check:- - 38 -

(b) State the limits for which the expansion is valid. - 38 -Solution (b) :- - 38 -

TASK 4 - 39 -

Please refer to Assignment 3 (Calculus). - 39 -

EVALUATION - 40 -

CONCLUSION - 40 -

BIBLIOGRAPHY - 41 -

Books - 41 -

Catalogues - 41 -

Websites - 41 -

4


Task 1

1.1 Solve the following polynomial division :-

Solution:-

(Subtracting -6 from 7 gives a new

denominator)

(Subtracting -13 from -5 leaves the

remainder)

Check:-

5


6


1.2 Solve the following equation using the factor theorem :-

Solution:-

Hence (x-1) is a factor and x=1 is a root.

Hence (x-2) is a factor and x=2 is a root.

Factors of x begin to predominate at the first term, so move onto Negative Numbers!

Hence (x+1) is a factor and x=-1 is a root.

Therefore:

Check:- Let :

7


1.3 Use the remainder theorem to find the remainder for :-

Solution:-

If is divided by the remainder will be .

Therefore:

a = 1b = -6c = 1p = 3 (After dropping the minus sign)d = -5

Check:-

8


1.4 Find the remainder for the following expression :-

Solution:-

(The difference between -4 from 3 gives a new denominator)

Check:-

If is divided by the remainder will be .

Therefore:9


a = +3b = -4c = +2p = -1

10


1.5 Resolve the following into partial fractions :-

(a)

Solution (a) :-

Let :

Find the coefficients of x on both sides:

Check:-

11


(b)

Solution (b) :-

Let :

Find the coefficients of x on both sides:

Check:-

Let :

12


(c)

Solution (c) :-

Let :

Let :

Therefore:

Check:-

Let :

13


Task 2

2.1 Solve the following equations correct to 2 decimal places :-

(a)

Solution (a) :-

to 2 Decimal Places

Check:-

14


(b)

Solution (b) :-

Transposing gives:

Take Log of both sides:

Using the Laws of Logs:

So:

(Seconds) to 2 Decimal Places

Check:-

15


(c)

Solution (c) :-

Take the Log of both sides:

Multiply out the brackets

Join like terms

So:

to 2 Decimal Places

Check:-

16


2.2 The voltage across a capacitor at time t is given by :-

where C = 10 μF and R = 20 k Ω. Determine :-

(a) The time for the voltage to reach 5 V.(b) The voltage after 1 mS.

Graph and Circuit Diagram :-

17


Solution (a) :-

So:

(Seconds) to 3 Decimal Places

Check:-

18


Solution (b) :-

If and then:

Volts

Check:-

19


2.3 Evaluate correct to 4 significant figures.

(a) cosh 2.47

Solution (a) :-

to 4 significant figures

Check:-

20


(b) sinh 1.385

Solution (b) :-

to 4 significant figures

Check:-

21


2.4 A telegraph wire hangs so that its shape is described by :-

Evaluate correct to 3 significant figures the value of y when x is 10.

Solution :-

to 3 Significant Figures

22


Check:-

23


2.5 If , find values for A and B.

Solution :-

RHS of Equation

Collecting the like terms together gives:

Factorise by looking at the LHS, giving:

Multiply both equations by 2, giving:

Adding equations (3) and (4) together gives:

Therefore

24


Substituting B back into equation (1) gives:

So:

Therefore the equation is:

Check:-

Let :

25


2.6 If find values for P and Q.

Solution :-

So:

Multiply out the brackets:

Check:-

Let :

26


2.7 Solve the equation correct to 2 d.p’s.

Solution :-

Step 1: replace COSH(x) and SINH(x) with definitions in terms of EXP(x).

Step 2: remove the divisor of 2.

Step 3: collect the like terms together.

Step 4: multiply the LHS by .

27


This then looks like a Quadratic Equation, where:

Using the formula for solving Quadratic Equations in terms of y gives:

(this has no real solution so ignore!!!)

28


Replacing y with gives:

to 2 Decimal Places

Check:-

(Note with the full answer of the equation equals 5.319999999, shown in the assignment as 5.32!)

29


Task 3

3.1 Determine the 15th term of the series 12, 17, 22, 27,.....

Solution :-

a = 1st Termd = Common Difference

Check:-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1512 17 22 27 32 37 42 47 52 57 62 67 72 77 82

30


3.2 The sum of the first 10 terms of an arithmetic progression is 200 and the common difference is 4. Find the first term of the series.

Solution :-

Check:-

1 2 3 4 5 6 7 8 9 102 6 10 14 18 22 26 30 34 38

31


3.3 An Oil & Gas Company drills a hole 10 km deep.

Estimate the cost of drilling if the cost is £1000 for drilling the first metre with an increase in cost of £200 per metre for each succeeding metre.

Solution :-

Check:-

32


3.4 Determine the 10th term of the series 2, 6, 18, 54,....

Solution :-

The nth term is determined using

Therefore:

Check:-

1 2 3 4 5 6 7 8 9 102 6 18 54 162 486 1458 4374 13122 39366

33


3.5 Find the sum of the first 12 terms of the series 1, 4, 16, 64,....

Solution :-

When r<1

When r>1

Check:-

1 2 3 4 5 6 7 8 9 10 11 121 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304

34


3.6 Find the sum to infinity of the series 4, 2, 1, 1/2, 1/4,.....

Solution :-

When -1 < r < 1.

35


3.7 Use the Binomial Series to expand

Solution :-

The binomial expansion of is:

Therefore the binomial expansion of is:

Check:-

Let :

36


3.8 (a) Expand the following in ascending powers of x as far as the term in using the Binomial Series.

Solution (a) :-

Therefore expanding in terms of gives:

37


Check:-

Let :

There is a difference of 0.02591369175.

With the additional terms of the 0.02591369175 will gradually decrease and eventually equal 0.

The discrepancy comes from the expansion only being to .

(b) State the limits for which the expansion is valid.

Solution (b) :-

The expansion in (a) is valid provided . i.e.

38


Task 4

Please refer to Assignment 3 (Calculus).

39


Evaluation

During this assignment I found that the notes taken from classes helped considerably.Beginning with the Polynomial Division, I found confusing at first. This is simply because today we very rarely use long division, and the method used to confuse me at school so I have very easily forgotten how to use it! To resolve the problem I decided to recap on the long division method by using the Higher Engineering Mathematics Book by John Bird. I could then move onto inserting letters in the place of numbers using the same methods and all soon became clear.With the Factor Theorem I quickly understood that the key part of the theory is to guess a suitable number to use. Fortunately I was able to determine the required factors quite quickly.The Remainder Theorem question was also straight forward as it simply involved inserting numbers for letters in a predetermined equation.I had a bit of difficulty with 1.5a as for some reason I was getting the denominators on the RHS around the wrong way, this meant after getting values for A,B and C I was getting dissimilar matches between the two sides.Using my notes from class I was able to complete and check 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6, but I struggled a little bit on 2.7.When I came to check 2.7, I found that when replacing x with my answer, I did not get 5.32. I went back over the work a couple of times before reaching the 0.20 answer required to get the 5.32.We had touched on Arithmetic and Geometric Progressions in previous years so this was fairly straight forward. I had learned to do the Binomial Series from the Statistics Assignment, however expanding a reciprocal square root term was new to me. I completed this before the going through this in class and then used my notes from class to double check my workings, which seem to be accurate.

Conclusion

I am very pleased with this assignment and believe it is one of the best I have handed in. My consistency in the presentation has taken a lot of time and effort.Due to the amount of work already completed I decided to leave Task 4 until the Calculus Assignment.I feel I understand the use of checking my answers by simply using a different method. This also helps relieves the pressure when handing the assignment in as the answers are unlikely to be incorrect if they have also been checked.I will continue to check my answers by using multiple methods, as this is a similar approach to what happens on site at work.

40


Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

BooksHigher Engineering Mathematics (John Bird) ISBN: 0-7506-8152-7

CataloguesN/A

WebsitesN/A

41

btec hnc - analytical methods - algebraic methods

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