H2 Mathematics Pure Mathematics Sectio A Comprehesive Checklist of Cocepts ad Skills by Mr Wee We Shih Visit: weshih.wordpress.com Updated: Ja 2010 Syllabus topic 1: Fuctios ad graphs 1.1 Checklist o Fuctios - Determie whether composite/iverse fuctio exists. - Fid composite/iverse fuctio. - Fid restricted domai for composite/iverse fuctio to exist. - Fid domai/rage for a fuctio. - Sketch graphs of fuctios, their iverses ad the lie of reflectio. - Fid the poit of itersectio betwee the fuctio ad its iverse, usig the fact that it occurs o the lie y = x. - Show a uderstadig of piecewise fuctios which are periodic. 1.2 Checklist o Graphig techiques - Sketch basic graphs of parabolas, cubic/quartic fuctios, rectagular hyperbolas, hyperbolas, ellipses, circles ad other stadard fuctios (trigoometric, expoetial ad logarithmic). - Sketch graphs of ratioal fuctios with emphasis o skills like: 1. Fid axial itercepts. 2. Fid equatios of asymptotes. 3. Fid statioary poits. 4. Fid rage of values for which curve does ot lie. 5. Fid coditios for curve to have statioary values. 6. Fid ukow variables give certai properties of curve. 7. Fid the umber of roots for a give equatio, requirig oe to isert a additioal graph ad fidig the umber of itersectios betwee both. 8. Fid a iequality, requirig oe to ivestigate the slopes of asymptotes. - Describe liear trasformatios (of reflectios, traslatios, scaligs) i words. - Sketch curves after beig trasformed by reflectios, traslatios, scaligs or their combiatios. y = f ' x ad appreciate cocepts like cocave up/dow. - Sketch graphs of ( ) - Sketch graphs of the forms y 1 = f ( x), 2 y f ( x ) combiatios of reflectios, traslatios, scaligs. =, y f ( x ) =, y f ( x) = with Page 1 of 5
- Give trasformed graph(s), sketch the origial y = f ( x) ; e.g. give y 2 f ( 2x 1) sketch y = f ( x). 1.3 Checklist o Equatios ad iequalities = + +, - Formulate ad solve a system of liear equatios. - Solve iequalities via algebraic method. - Solve iequalities via graphical method. - Solve a ew iequality that looks similar to the previously solved oe with a suitable substitutio. Syllabus topic 2: Sequeces ad series 2.1 Checklist o Summatio 2 3 r 1. Sum of a, r, r, r, a ad the like. 2. Break up the summatio of a complex expressio ito a simpler oe, 2 r 2 2 3 = 2 3 r. r e.g. ( r ) r= 1 r= 1 r= 1 3. Usig a summatio result to deduce other results. 2.2 Checklist o Biomial expasio 1. Give the expasio i ascedig/descedig powers. 2. State validity rage. 3. Use suitable substitutio to fid a approximate value. 4. Fid coefficiet of x term. 2.3 Checklist o AP/GP 1. Show series is AP/GP. 2. Solve a pure AP/GP problem (e.g. fid term, sum, commo differece, commo ratio, sum to ifiity). 3. Solve a problem where AP/GP are related (e.g. terms of AP are terms of GP). 4. Solve moetary problems (compoud iterest). 5. Solve problems ivolvig patters, e.g. fid first term i the th bracket give that {1}, {3, 5}, {7, 9, 11},... 6. Solve AP/GP problems described i the recurrece form, e.g. U+ 1 = d + U, U+ 1 = r U. Page 2 of 5
2.4 Checklist o Method of differeces 1. Use of partial fractios, trigoometric idetities or appropriate algebraic maipulatios to obtai differece of two similar expressios. 2. Cacellatio of terms. 3. Fid expressio i terms of or N. 4. Fid the sum as or N approaches ifiity (cocept of covergece/divergece). 5. Fid a iequality for a summatio of a similar form, based o the previous oe. 2.5 Checklist o recurreces 1. Fid the limit as approaches ifiity. 2. Prove some results or iequalities, e.g. x > x + 1 or x < x + 1. 3. Relate to graphs whe provig, for istace, x > x + 1 whe x < α. 4. Repeated use of the defiitio of recurrece to obtai a formula for U i terms of. 5. Formulate simple recurrece relatios based o give problems. 2.6 Checklist o Mathematical iductio 1. Prove a result that ivolves summatio, the at times use the result to fid sums. 2. Prove a result that ivolves recurrece. 3. Prove a cojecture, obtaied by observig a patter that comes from guided steps. Syllabus topic 3: Vectors - Use of ratio theorem ad mid-poit theorem to fid positio vectors ad reflectios of poits ad lies. - Carry out scalar ad vector product operatios. - Fid legths of projectios (e.g. vector oto vector, vector oto lie, vector oto plae, vector oto ormal of plae). - Fid shortest distaces (e.g. poit & lie, poit & plae) with or without the eed to fid foot of perpedicular. - Fid areas (via vector product or stadard formula like 1 bh ). 2 - Fid itersectios (e.g. lie & lie, lie & plae, plae & plae, 3 plaes). - Fid agles (e.g. vector & vector, lie & lie, lie & plae, plae & plae). - Fid equatios of lies, plaes, give iformatio from the problem. - Solve vector problems that ivolve give diagrams that describe some real-life situatios. - Give geometrical iterpretatios about lie/plae relatioships. - Give geometrical meaigs of a. p ad a p. Page 3 of 5
Syllabus topic 4: Complex umbers z1 - Fid moduli ad argumets of complex umbers, applyig kowledge of z1z 2, z, z* ad z. - Covert betwee differet forms i.e. cartesia, polar, expoetial. - Represet complex umbers as poits o argad diagram ad prove geometrical properties. - Solve simple or simultaeous complex equatios. - Solve polyomial equatios with real coefficiets. - Solve z = c (c is ay complex umber) type of equatios. - Factorise the equatio based o roots foud. - Sketch loci (e.g. circles, perpedicular bisectors, half-lies) with or without iequalities. - Fid (via trigoometry or usig GC to fid itersectios betwee loci) max/mi moduli/argumets based o loci sketched. - Fid (via trigoometry or usig GC to fid itersectios betwee loci) itersectios of loci sketched. Syllabus topic 5: Calculus 5.1 Checklist o Differetiatio ad its applicatios - Differetiatio techiques o various types of fuctio. - Equatios of tagets ad ormals of curves defied i cartesia form, i parametric form or i implicit form: 1. fid equatios of tagets ad ormals; 2. fid poit where taget/ormal meets curve agai; 3. fid area formed by taget/ormal with the x- or y-axis; 4. fid equatios/poits where taget/ormal is parallel to x-/y-axis. - Rates of chage (via chai rule). - Maxima ad miima: 1. Formulate a expressio (usually area or volume), the fid max/mi. 2. Graph sketchig with statioary poits. - Maclauri's expasio: 1. Fid expasio via stadard series i formulae booklet. 2. Fid expasio via repeated differetiatio. 3. Small agle trigoometric approximatios, which may ivolve the use of sie/cosie rule ad/or trigoometric idetities. 4. Use series to approximate values, to fid defiite itegrals, to fid limits, to fid equatio of taget at the origi. 5. Solve a iequality ivolvig the error betwee the actual fuctio f ad the Maclauri's expasio of f. 2 Page 4 of 5
5.2 Checklist o Itegratio ad its applicatios - Stadard itegratio techiques. - Itegratio by partial fractios. - Itegratio by give substitutios. - Itegratio by parts. - Approximate areas via sum of rectagles, the fid limit as umber of rectagles approaches ifiity. - Fid areas of curves defied parametrically. - Fid areas/volumes of curves defied i cartesia form or i implicit form. - Use the idea of limit to fid the area of a regio betwee the curve ad the positive x- axis. 5.3 Checklist o Differetial equatios - Solve DE via direct itegratio. - Solve DE via variable separable. - Solve DE via substitutio, followed by ay of the above. - Formulate DE based o a give problem descriptio (some stadard oes are listed below): 1. Newto's coolig model; 2. I-out rate flow model; 3. Birth-death model; 4. Moey iterest model. Sometimes, the chai rule may be used. - Fid particular solutios. - Sketch solutio curves. - Commet o appropriateess of model ad/or iterpret solutios (e.g. log-term behaviour). Page 5 of 5