preparatory problems for 1st years

8/13/2019 Preparatory Problems for 1st Years

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Preparatory Problems for Electronic Engineers Page 1 of 25

Preparatory Problems

for Electronic Engineers

Congratulations on accepting your offer for a place at UCL to read Electronic and

Electrical Engineering. All the courses within the epartment of Electronic and

Electrical Engineering ha!e a common first and second year" which co!ers the

fundamental material you will need for your subse#uent studies and specialisations.

$he first year at UCL builds on the foundations of your A%le!el studies in

&athematics and Physics" and as such it is important to ma'e sure you are familiar

with this material" and to fill any gaps in your 'nowledge prior to arri!ing at UCL.

$o facilitate this" enclosed are a set of problems which you should wor' throughbefore arri!ing at UCL. You are required to hand in your final set of answers in at

the first tutorial. Your tutor will also discuss your solutions in your first tutorial.

(f you find #uite a few of the problems difficult or unfamiliar" do not be discouraged

from wor'ing on the rest.

(n this boo'let there are a series of problems based on the Core A%le!el )yllabus" all

of which should be attempted prior to arri!ing at UCL. (f you do not recognise the

problems as being on material you ha!e co!ered" or if you cannot do them" you should

consult an A%Le!el te*t boo'.

At the end of the boo'let a short section has been added to gi!e you a tester on the

lin' between the mathematics you will learn and electronic engineering.

$he mathematics problems in this boo'let are based on the Core )yllabus for A%le!el

&aths" with many #uestions ta'en from Core Maths for Advanced Level, L. Bostock& S. Chandler, ISBN: 074870!8. $he physics problems in this boo'let are based onthe Core )yllabus for A%le!el Physics and are ta'en from "ract#ce #n "h$s#cs Akr#ll,

Bennett and M#llar, %'l#shed '$ (odder and Sto)hton ISBN 0 *40 78+* !

r )eb )a!ory

r Cyril +enaud

r )ally ay

)eptember 2,1-

UCL ELECTRONIC & ELECTRICAL ENGINEERING


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Table of contents

UCL Electronic Electrical Engineering......................................................................1

&athematics Problems ..................................................................................................-

Answers for &athematics Problems.........................................................................../Circuits Problems...........................................................................................................0

Answers for C Circuit Analysis Problems.............................................................11

Added Problems...........................................................................................................12

Answers for ptional Problems...............................................................................10


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Preparatory Problems for Electronic Engineers Page - of 25

Mathematics Problems(f you ha!e any #ueries please contact r )eb )a!ory email s.sa!ory3ee.ucl.ac.u'

$hese problems co!er the mathematics which most courses will assume you are

familiar with" howe!er it may ha!e been some time since you studied some of these

topics. As mentioned earlier" if you find #uite a few of the problems difficult or

unfamiliar" do not be discouraged from wor'ing on the rest.

1. etermine the !alue of the rational number for which

31/ 4 331/ 6

3=3

p

2. )ol!e the simulaneous e#uations

2x + y=3, 2x2 xy=1 0

-. E*press

5(x3) (x+1)(x12)(x+3)

3(x+1)x12

as a single fraction in its simplest form

4. rite lnx3 + lnxyln y3as a single term. 6ence obtain an e*pression forin terms of xif lnx3 + lnxyln y3 =0

5. )implify2+ 2

2 2

e*pressing your answer in surd form 7i.e. in the form a+b 2 where aand b are numbers to be determined8.

9. $he cubic polynomial x32x

22x+4has a factor (xa)" where ais an

integer.

a. Use the factor theorem to find the !alue of ab. 6ence find e*actly all three roots of x

32x

22x+4

/. :ind" correct to three decimal places" the !alue of ygi!en that2y+1

=35y

;. $he !ertices of the triangle AB are A(3,1)" B(10,8)and C(1,4).:ind an e#uation for the line passing through Aand B" gi!ing your answerin the form px + qy + r=0" where p" qand r are integers. )how bycalculation that C and CBare perpendicular.

0. An e#uilateral triangle of side 25 cm circumscribes a circle 7i.e. the circle is

enclosed by the triangle8. :ind the radius of the circle.

1,. A cur!e in the trac' of a railway line is a circular arc of length 4,,m andradius 12,,m. $hrough what angle does the direction of the trac' turn


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11. $wo discs" of radii 5 cm and 12 cm" are placed" partly o!erlapping" on a table.

(f their centres are 1- cm apart find the perimeter of the =figure%eight> shape

12. :ind the e#uation of the tangent to the cur!e x3 +2x2 +3x +6at the point

where x=1

1-. A piece of wire of total length 12m is cut into two pieces. ne piece of wire is

bent into a rectangle of sides xm and 3xm and the other is bent to form theboundary of a s#uare. )how that the total area" in m

2" enclosed by the

rectangle and the s#uare is gi!en by

A=7x212x+9

:ind the !alue of xif the total area enclosed is a minimum. ?erify that yourstationary !alue is a minimum.

14. A landscape gardener is gi!en the following instructions about laying a

rectangular lawn. $he length xm is to be 2 m longer than the width. $hewidth must be greater than 9.4 m and the area is to be less than 9- m

2. @y

forming an ine#uality in x" find the set of possible !alues of x.

15. An odd function is one for which f(x)=f(x)" and an e!en function isone for which f(x)= f(x). )'etch each function and state whether it ise!en" odd or neither.

a8 cosx b8 tanx c8 ex d8 ln(1+x)

e8 cot x f8 (x1)2 g8 (x1) h8 1

x

19. $he fourth term of an arithmetic series is 2, and the ninth term is 4,.

:ind

a. the common difference

b. the first term

c. the sum of the first 2, terms

1/. :ind"correct to the nearest degree" all the !alues of between 0 and 360

satisfying the e#uation

8 2

cos +2sin=7

1;. :ind

a.

x+ 6

x

2

dx1

4

b.

x cos2xdx


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10. A pump is used to e*tract air from a bottle. $he first operation of the pump

e*tracts 59cm

3of air and subse#uent e*tractions follow a geometric

progression. $he third operation of the pump e*tracts -1.5cm

3of air.

a. etermine the common ratio of the geometric progression and

calculate the total amount of the air that could be e*tracted from thebottle" if the pump were to e*tract air indefinitely

b. After how many operations of the pump does the total amount of air

e*tracted from the bottle first e*ceed 22,cm

3

2,. ifferentiate each of the following functions with respect to x

a81

x

b8 ex c8 x2 cos(2x)

d8x

x2 1

e8 x2 lnx f8 e x g81+x2

lnx

h8 2cos x

21. $he !olume" V" of a sphere of radius r is gi!en by V= 43r3

a. btain an e*pression fordV

dr

b. A balloon when almost fully inflated can be modelled by a sphere.

here the radius of the balloon is 15cm" it is obser!ed that the rate of

increase of the radius is ,.1 cm s

1 . :ind" to two significant figures"

the rate of increase of the !olume at this time.

22. A spea'er uses an amplifier to carry her wor's to members of the audience

metres away. $he power output" Pwatts" if gi!en by the formulaP=0.0004x

2

a. $o increase the distance by a small amount x metre" the output mustbe increased by Pwatt. :ind an appro*imate e*pression for Pinterms of xand x .

b. )how thatP

P2

x

x

c. (f the power output of the amplifier is increased by 2 " by what

percentage appro*imately is the distance her !oice will carry increase

preparatory problems for 1st years

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