1. (a) Givenf(z)=z²+3z+2,
   1. expressf(z)intheformu+jv.
   2. Showthatuandv.satisfytheCauchy-Riemannequations. (5marks)

/

1. Givenu=Coshxcosy,
   1. Showthatuisaharmonicfunction.
   2. determineaharmonicconjugatefunctionvsuchthatf(z)=u+jvisanalytic.

(8marks)

1. FindtheimageofthecircleIzI=2inthew-planeunderthetransfonnation

1

w=z+j

1. (a)j Determinetheeigenvaluesandthecorrespondingeigenvectorsofthematrix

(7marks)

A=[3 2] (10marks)

45

1. Asystemischaracterizedbythevector-matrixdifferentialequation(ft=,

whereA=[3!4]

,r,

1. determinethesystemstate transitionmatrix,(t);
2. showthat(o)=I,whereIia2x2unitmatrix. (lOmarks)
3. (a) Findthehalf-rangeFouriercosineseriesofthefunctionf(t)=rr²-t²,O

(8marks)

(b) Thecharge q(t)ontheplatesofacapacitorisashowninfigurel.

Figurel

Detemiinetheanalyticaldescriptionof q(t)andfinditsFourierseriesrepresentation.

(12marks)

1. t (a) Sketchthe domainofintegration,andevaluatetheintegra1

f2f/2x-x2

Joo Jx2+y2dy (8marks)

1. AregionRofthexy-planeisboundedbythecurvey=x2andthelinesy=0and

x=l.Showthat

(5marks)

R

2521/303 260143/303 2

1. Useadouble integraltodeterminetheareaoftheregionenclosedbythecurve

y=- 4-xandthestraightliney= x-2. (7marks)

1. (a) (i) Giventhatx

0

isanapproximationtotherootoftheequationx3+2x- 1=0,

UsetheNewton-Raphsonmethodthatabetterapproximationisgivenby

-2x!+l /

X"+1- 3x;+2.

(ii) Talcingx0=0.3,detenninetheroot,correcttofourdecimalplaces. (9marks)

(b) Thetable belowrepresentsa polynomialfunctionf(x):

UsetheNewton-Gregoryinterpolationtodeterminef(x),andfind:

(i)

J (ii)

f(-1.3); /

f(3.7). (IImark)

1. (a) Showthatthelineintegral

(1")

lIro,o')

2xsinyda;+(x2cosy-3y²)dyispathindependent,andueapotentialfunctionto

determineitsvalue. (8mark)

(b) VerifyGreen'stheoremintheplanefortheline integralfx²yd.x+(x-y)dy,wherec is theboundaryofthetrianglewithvertices(0,0),(1,1)and(O,J),withcounter clockwise orientation. (12marks)

1. (a)-1- Determinetheareaoftheparaboloid

z=2(.x2+y2)cutoffbytheconeZ=x2a--+-y....,,..²

(10marks)

(b)

Evaluatethesurfaceintegralffyzds,wheresisthepartof theplanez=y+3

thatliesinsidethecylinderx2+y2=l. (10marks)

1. (a) Use thedivergencetheoremtoevaluatefJf. ,forthevectorfield

F-- xi+'Y:f.+_zkgiventhatSisthesp^sherex2+y2+z2=9.

(10marks)

(b)/ Usestokes'theoremtoevaluatethelineintegralf]!.dr,wherethevectorfield

f=3yj+4zl-6xand Cistheboundaryofheparaboloidz=9-x2-y²thatlies above thexy-plane. (10 marks)

Discuss current trends in using formal education for development.

• Relate how assessment of personality type, work behaviors, and job performance can be used for employee development.
• Explain how job experiences can be used for skill development.
• Develop successful mentoring programs.
• Describe how to train managers to coach employees.
• Discuss what companies are doing to melt the glass ceiling.