.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including the conditions for rel nd repeted roots. m n = m + n, m n = m n, ( m ) n = mn m The equivlence of n n m nd should be known. Students should be ble to simplify lgebric surds using the results ( ) =, y = y nd ( + y)( y) = y The nottion f() my be used Need to know nd to use b 4c > 0, b 4c = 0 nd b 4c < 0 b b Completing the squre. + b + c = + + c 4 Solution of qudrtic equtions including solving qudrtic equtions in function of the unknown..4 Solve simultneous equtions in two vribles by elimintion nd by substitution, including one liner nd one qudrtic eqution. Solution of qudrtic equtions by fctoristion, use of the formul, use of clcultor nd completing the squre. These could include powers of, trigonometric of, eponentil nd logrithmic of. The qudrtic my involve powers of in one unknown or in both unknowns, e.g. solve y = + 3, y = 4 + 8 or 3y = 6, y + 3 = 50 1 Person Edecel Level 3 Advnced GCE in Mthemtics Specifiction Issue 3 Jnury 018 Person Eduction Limited 018
.5 Solve liner nd qudrtic inequlities in single vrible nd interpret such inequlities grphiclly, including inequlities with brckets nd frctions. e.g. solving + b > c + d, p + q + r 0, p + q + r < + b nd interpreting the third inequlity s the rnge of for which the curve y = p + q + r is below the line with eqution y = + b These would be reducible to liner or qudrtic inequlities e.g. < b becomes < b Epress solutions through correct use of nd nd or, or through set nottion. Represent liner nd qudrtic inequlities such s y > + 1 nd y > + b + c grphiclly..6 Mnipulte polynomils lgebriclly, including epnding brckets nd collecting like terms, fctoristion nd simple lgebric division; use of the fctor theorem. Simplify rtionl epressions, including by fctorising nd cncelling, nd lgebric division (by liner epressions only). So, e.g. < or > b is equivlent to { : < } { : > b } nd { : c < } { : < d } is equivlent to > c nd < d Shding nd use of dotted nd solid line convention is required. Only division by ( + b) or ( b) will be required. Students should know tht if f() = 0 when =, then ( ) is fctor of f(). Students my be required to fctorise cubic epressions such s 3 + 3 4 nd 6 3 + 11 6. Denomintors of rtionl epressions will be liner or qudrtic, e.g. 1 b +, + b p + q + r, 3 3 + Person Edecel Level 3 Advnced GCE in Mthemtics Specifiction Issue 3 Jnury 018 Person Eduction Limited 018 13
.7 Understnd nd use grphs of ; sketch curves defined by simple equtions including polynomils The modulus of liner function. y = nd y = (including their verticl nd horizontl symptotes) Interpret lgebric solution of equtions grphiclly; use intersection points of grphs to solve equtions. Understnd nd use proportionl reltionships nd their grphs. Grph to include simple cubic nd qurtic, e.g. sketch the grph with eqution y = ( 1) Students should be ble to sketch the grphs of y = + b They should be ble to use their grph. For emple, sketch the grph with eqution y = 1 nd use the grph to solve the eqution 1 = or the inequlity 1 > The symptotes will be prllel to the es e.g. the symptotes of the curve with eqution y = + b re the + lines with equtions y = b nd = Direct proportion between two vribles. Epress reltionship between two vribles using proportion symbol or using eqution involving constnt e.g. the circumference of semicircle is directly proportionl to its dimeter so C d or C = kd nd the grph of C ginst d is stright line through the origin with grdient k. 14 Person Edecel Level 3 Advnced GCE in Mthemtics Specifiction Issue 3 Jnury 018 Person Eduction Limited 018
.8 Understnd nd use composite ; inverse nd their grphs..9 Understnd the effect of simple trnsformtions on the grph of y = f(), including sketching ssocited grphs: y = f(), y = f() +, y = f( + ), y = f() nd combintions of these trnsformtions.10 Decompose rtionl into prtil frctions (denomintors not more complicted thn squred liner terms nd with no more thn 3 terms, numertors constnt or liner). The concept of function s one-one or mny-one mpping from R (or subset of R) to R. The nottion f : nd f() will be used. Domin nd rnge of. Students should know tht fg will men do g first, then f nd tht if f 1 eists, then f 1 f() = ff 1 () = They should lso know tht the grph of y = f 1 () is the imge of the grph of y = f() fter reflection in the line y = Students should be ble to find the grphs of y = f() nd y = f( ), given the grph of y = f(). Students should be ble to pply combintion of these trnsformtions to ny of the in the A Level specifiction (qudrtics, cubics, qurtics, reciprocl,,, sin, cos, tn, e nd ) nd sketch the resulting grph. Given the grph of y = f(), students should be ble to sketch the grph of, e.g. y = f(3), or y = f( ) + 1, nd should be ble to sketch (for emple) π y = 3 + sin, y = cos( + ) 4 Prtil frctions to include denomintors such s ( + b)(c + d)(e + f) nd ( + b)(c + d). Applictions to integrtion, differentition nd series epnsions. Person Edecel Level 3 Advnced GCE in Mthemtics Specifiction Issue 3 Jnury 018 Person Eduction Limited 018 15
3 Coordinte geometry in the (,y) plne.11 Use of in modelling, including considertion of limittions nd refinements of the models. 3.1 Understnd nd use the eqution of stright line, including the forms y y 1 = m( 1) nd + by + c = 0; Grdient conditions for two stright lines to be prllel or perpendiculr. Be ble to use stright line models in vriety of contets. 3. Understnd nd use the coordinte geometry of the circle including using the eqution of circle in the form ( ) + (y b) = r Completing the squre to find the centre nd rdius of circle; use of the following properties: the ngle in semicircle is right ngle the perpendiculr from the centre to chord bisects the chord the rdius of circle t given point on its circumference is perpendiculr to the tngent to the circle t tht point. For emple, use of trigonometric for modelling tides, hours of sunlight, etc. Use of eponentil for growth nd decy (see Pper 1, Section 6.7). Use of reciprocl function for inverse proportion (e.g. pressure nd volume). To include the eqution of line through two given points, nd the eqution of line prllel (or perpendiculr) to given line through given point. m = m for prllel lines nd m = for perpendiculr lines 1 m For emple, the line for converting degrees Celsius to degrees Fhrenheit, distnce ginst time for constnt speed, etc. Students should be ble to find the rdius nd the coordintes of the centre of the circle given the eqution of the circle, nd vice vers. Students should lso be fmilir with the eqution + y + f + gy + c = 0 Students should be ble to find the eqution of circumcircle of tringle with given vertices using these properties. Students should be ble to find the eqution of tngent t specified point, using the perpendiculr property of tngent nd rdius. 16 Person Edecel Level 3 Advnced GCE in Mthemtics Specifiction Issue 3 Jnury 018 Person Eduction Limited 018