Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6
How To Use This Booklet You could use this booklet on your own, but it my be better if you hd someone else to test you out on its content this could be friend or fmily member. All your friend hs to do is red out the question on the left-hnd column of the pge, nd ll you hve to do is give the correct nswer (more or less) which is shown on the right. Or, if you prefer, your friend could simply show you the question, nd cover up the nswer t the sme time ll you hve to do is try to give the correct nswer! As you progress throughout the course try to mke point of using this useful booklet gin nd gin - lwys trying to chieve higher score. It is good to be competitive with yourself now nd gin!! Hey, why not print this booklet out nd use it with peer t the strt of Higher Mths study support clsses? Plese lwys remember to sk your Mths techer if you re unsure with your homework, question in n eercise or ny spect of the course. There is study support in Higher Mths t QAHS some fternoons plese check with your techer when these re vilble. And finlly - good luck with your Higher Mths studies!... No pin, no gin!
Wht is the Sine Rule? Revision from Ntionl 5 sin A b sin B c sin C Wht is the Cosine Rule? b c bccos A Wht is the formul for the re of tringle? How do you solve qudrtic eqution? How do you fctorise qudrtic? Are bsin C Mke one side zero, then fctorise the other (or use the qudrtic formul) Look for:. Common fctor. Difference of squres 3. Double brckets Wht is the qudrtic formul? b b 4c How do you find the solution to simple trig eqution, with solutions from 0 to 360 degrees? Find the cute ngle then use n ASTC digrm to find the solutions (usully two) How do you write s power of? ( to the power hlf) How do you write n m s power of? (the nth root of to the power m) m n Wht is 0? How do you write power? n with positive n The Stright Line Wht is the grdient of horizontl line? Wht is the eqution of horizontl line? So the eqution of the -is is? Wht is the grdient of verticl line? m 0 y = c y 0 m is undefined
Queen Anne High School Mths Deprtment Wht is the eqution of verticl line? So the eqution of the y-is is? 0 Wht is the grdient formul? m y y How do you find the eqution of stright line? You need to know the grdient of the line nd point on the line, then you use y b m( ) How do you find the grdient of stright line if you know its eqution? How do you find the size of the ngle between line nd the -is? Rerrnge to the form Use m tn y m c Wht is the rule for prllel lines? m m Wht is the rule for perpendiculr lines? m m How do you find where two lines meet? Use simultneous equtions or use y y Wht does it men if three points re sid to be colliner? How do you show tht three points A, B nd C re colliner? SPECIAL LINES Wht is perpendiculr bisector? Wht is medin of tringle? Wht is n ltitude of tringle? They lie in stright line Show tht mab mbc so the lines re prllel, but B is common point so A, B nd C re colliner A line which bisects (cuts in hlf) given line t right-ngles find midpoint nd use m m A line drwn from one verte to the midpoint of the opposite side find midpoint then grdient A line drwn from one verte to the opposite side, meeting it t right-ngles use m m Wht is the distnce formul? AB ( ) ( y y) or use Pythgors
Differentition How do you differentite? How do you prepre for differentition? Multiply by the power, then decrese the power by one Chnge ny roots into powers must not be on the denomintor (bottom) of ny frction Any pirs of brckets should be epnded Wht nottion or phrses cn we lso use to represent differentition? dy f () (f dshed of ), d (dy by d) Rte of chnge, Grdient function, Derived function How do we find the grdient of the tngent to curve t given vlue of? How do we find the eqution of the tngent to curve? A function is incresing when? And decresing when? And sttionry when? How do you find the sttionry points of function? How do you find where function is incresing or decresing? How do you show tht function is ALWAYS incresing (decresing)? How do you find the solution to n optimistion problem? Wht do you get if you differentite Differentite, then sub in to find the grdient As bove, but we lso need to find the y co-ordinte by substituting into the originl epression, then use y b m( ) f ( ) f ( ) f ( ) 0 0 0 At the sttionry points f ( ) 0 Differentite then solve to find the vlues Substitute bck in to find y vlues Use nture tble to determine the nture Differentite then use nture tble (or solve the inequlity) Differentite then complete the squre to show tht f ( ) 0 ( f ( ) 0 ) for ll vlues of Investigte sttionry points (nd end-points if necessry) Speed
distnce? Wht do you get if you differentite speed? Accelertion Fmilies of grphs Given the grph of y f (), how do you sketch y f ( ) k y f ( ) k y kf() y f () Given the grph of y f (), how do you sketch y y f ( k) f ( k) y f (k) y f ( ) Move grph up k units Move grph down k units Stretch grph up/down by fctor of k Reflect grph in the -is Move grph k units to the left Move grph k units to the right Compress the grph by fctor of k horizontlly Reflect grph in the y-is Given the grph of y f (), how do you sketch y f () (the derived grph) The -coordintes of the sttionry points become zeroes of the grph; then look t the grdient of the curve between these points to decide on shpe (positive bove y-is; negtive below y-is) Functions Wht is the domin of function? Wht is the rnge of function? How do you find where function is undefined? The set of numbers which go INTO the function The set of numbers which come OUT of the function Look for vlues of which mke the denomintor of the frction equl to zero or which led to negtive squre roots
How do you find suitble domin for function? If f ( g( )), wht is the connection between functions f nd g? ( f of g of ) Write down n epression for ll the vlues of for which the function is NOT undefined They re inverses of ech other Trigonometry: grphs nd equtions For trig grph of the form y sin b c or y cos b c, how do we work out the vlue of? b? c? Ect Vlues: The results in the tble opposite must be lerned. An esy method to remember the tble is to lern the two ect vlue tringles. The tringles cn then be used, long with SOH-CAH-TOA to find the vlues in the tble Two Tringles; 3 (30,60,90) or,, (45,45,90) Rdin mesure: You need to be ble to convert from degrees to rdins nd vice-vers. Some common emples re given opposite nd should be memorised: prctice sking these both wys. tells us the mplitude (the difference between the mimum nd minimum vlues) b tells us how often the grph repets in (so the period of the grph is 360 b ) 360 c tells us how fr the grph hs been moved up or down from its usul strting position sin 0 cos tn 0 0 30 45 60 90 3 (we sy 3 3 0 3 - tn 90 is undefined) 80 (pi rdins) 360 0 0 90 30 6 45 4 60 3
Wht does sin cos lwys equl? (sine squred plus cos squred ) Wht does sin cos lwys equl? tn Recurrence Reltions Wht is ment if recurrence reltion is sid to be Convergent? Divergent? Wht is the condition for recurrence reltion to hve limit? How do you find the limit of recurrence reltion? Given three consecutive terms in recurrence reltion, how cn you work out the formul? There is limit There is no limit ( is between - nd ) Use the formul b L (or replce u n nd un by L in the originl epression nd solve) Set up two equtions using pirs of vlues then solve simultneously Qudrtic Functions How do you sketch qudrtic curve (prbol)? Completing the squre: Why do we complete the squre? Wht is the process for completing the. Find the shpe hppy or sd?. Find the roots (if they eist) ie. where the curve cuts the -is (solve y 0 ) 3. Find where the curve cuts the y-is ( 0) 4. Use symmetry to find the turning point (or use differentition) To llow us to mke quick sketch of the prbol, which llows us to find the turning point Identify the -coefficient
squre? Wht form must the epression be in before you cn complete the squre? The discriminnt: Wht is the condition for equl roots? two distinct rel roots? rel roots? non-rel roots? (or no rel roots) Hlve it Squre it Add it on/tke it wy OR: epnd brckest nd equte coefficients Must be... nd not etc, so tke out common fctor if you hve to b b b b 4c 0 4c 0 4c 0 4c 0 How do you show tht line is tngent to curve? Wht does it men to sy tht qudrtic is irreducible? Substitute the line into the curve nd solve the eqution to show tht there re equl roots (or show tht b 4c 0) It cnnot be fctorised Polynomils How do you show tht f ()? How do you fctorise cubic? is fctor of How do you sketch the grph of polynomil? Use synthetic division (with ) to show tht the reminder is zero, or show tht f ( ) 0 First find liner fctor, using synthetic division, then fctorise the qudrtic from the bottom row of the tble.. Find where the curve crosses the -is ( y 0 ) nd the y-is ( 0) dy. Differentite nd solve 0 to find the d sttionry points 3. Use nture tble to determine nture 4. Sketch the grph
Integrtion How do you integrte? How do you prepre for integrtion? When integrting n indefinite integrl (one with no limits), wht must we lwys remember? Why do we integrte? Wht do we hve to remember when the enclosed re is below the -is? Wht do we hve to remember when the re is prtly bove nd prtly below the -is? How do we find the re between two curves or line nd curve? How do we find where the curves meet? Wht do we get if we integrte ccelertion? Wht do we get if we integrte speed? Increse the power by one, then divide by the new power Chnge ny roots into powers must not be on the denomintor (bottom) of ny frction Any pirs of brckets should be epnded +C To find the re under curve, or to recover f () from f () The nswer will be negtive, so we eplin this fct nd chnge the nswer to positive We hve to work out the res seprtely (one bove -is, one below) then dd (curve bove curve below ) d Use Speed Distnce y y nd solve Compound Angle Formule cos(a + B) =? cos(a B) =? sin(a + B) =? sin(a B) =? When sked to find the ect vlue of sin, cos or tn, wht should you look for? If right-ngled tringle is not involved, wht should you do? cosacosb sinasinb cosacosb + sinasinb sinacosb + cosasinb sinacosb cosasinb Right-ngled tringles Try to mke n epression up which involves right-ngled tringles nd ect vlues you know (eg 30, 45, 60 degrees) If you re give sin, cos or tn nd told tht Drw right-ngled tringle, use Pythgors
the ngle is cute ( 0 90, 0 ), how cn you find the other rtios s ect vlues? to find the missing side, then use SOHCAHTOA sina =? cosa =? (three possible nswers) sin Acos A cos A sin cos sin A A A How cn you epnd cos4a, sin4a etc? How cn you epnd cos3a etc? When solving trig eqution, wht twostep process shoud you follow? How do you recognise nd solve stright-forwrd solve? How do you recognise nd solve double-ngle solve? Wht should you lwys check t the end of trig question? Write s (A+A) then epnd using the formule Write s (A+A) then epnd using the formule Is it stright-forwrd solve? If not, then double ngle solve sin, cos or tn ppers once only Solve to find cute ngle, then use ASTC Look for double ngle nd single ngle (eg A nd A) Replce the double-ngle formul with n pproprite single epression, then mke one side zero nd fctorise in order to solve Should the nswer be given in degrees or rdins? The Circle Wht kind of circle hs eqution y r? ( r ) ( y b)? y g fy c 0? How do you find the eqution of circle? Do you need to epnd the brckets nd tidy up your nswer? How cn you show tht n eqution does NOT represent circle? centre (0, 0), rdius r centre (, b), rdius r centre ( g, f ), rdius g f c Find the centre nd rdius, then use ( ) ( y b) r No!! Try to find the rdius you should be left with the squre root of negtive number,
which is impossible, or zero How do you find where line meets circle? How do you show tht line is tngent to circle? How do you show tht line does not meet circle t ll How do you find the eqution of tngent to circle? Wht is common tngent? How do I show tht two circles touch eternlly? Wht is ment by congruent circles? Wht is ment by concentric circles? Rerrnge the line into the form y or (whichever is esier) then substitute this into the circle nd solve As bove you should find equl roots, ie only one point of contct (lterntively, show tht b 4c 0) As bove this time show tht there re no rel roots, ie b 4c 0 Find the grdient of the rdius Use m m to find the grdient of the tngent Then use y b m( ) A line which is tngent to two circles Show tht the distnce between the two centres is equl to the sum of the two rdii Circles tht re the sme size Circles with the sme centre Vectors Wht is the difference between vector nd sclr? Wht is ment by giving vector in component form? How would you write this in i, j, k form? A vector hs mgnitude (size) nd direction, wheres sclr only hs mgnitude Writing the nswer s column vector with brckets, eg b c i b j ck How do you find the mgnitude (length) of vector u = b? c u b c
How do you find vector AB? How do you show tht two vectors re prllel? If point P divides AB in the rtio m:n, how do you find the coordintes of P? Wht re the two forms of the sclr (or dot) product? How do you find the ngle between two vectors? How do you show tht two vectors re perpendiculr? Useful rules:? ( b c)? AB b Show tht one vector is multiple of the other Use the Section Formul: p ( n mb) nd then write out the m n coordintes of P (or use rtios to crete n eqution nd solve) b b cos (for this version, remember tht the vectors must NOT be nose-to-til ) b b b 3b3 Use the dot product nd solve to find b - or use formul cos b Show tht b0 b c Further Clculus Wht do you get if you differentite: sin? cos? Wht do you get if you integrte: sin? cos? Wht is the chin rule for differentition? (How do you differentite f ( g( ))?) cos sin cos c sin c f ( g( )) g( ) (differentite round the brckets, then multiply by the derivtive of wht is inside
the brckets) Wht do you get if you differentite: sin( b)? cos( b)? Wht do you get if you integrte: sin( b)? cos( b)? cos( b) sin( b) cos( b) c sin( b) c Wht you get if you integrte n ( b)? n ( b) ( n ) c The Wve Function How do you epress cos bsin in the form k cos( ) or k sin( )? Epnd the brckets Equte coefficients Solve to find k (squre nd dd to get Solve to find (divide to get tn ) k ) How do you know which qudrnt is in? Look t the signs for k cos nd k sin - if both re positive then is cute, otherwise you need to do n ASTC digrm Given the choice, which version of the wve function should you use? How do you find the mimum or minimum vlues of wve function If it strts with cos, use k cos( ) If it strts with sin, use k sin( ) Use the version which keeps both coefficients positive, if possible Think of the grph: when is cos (or sin) t mimum or minimum, then djust s necessry How do you solve cos bsin c? Put the left-hnd side into wve function form, then solve in the usul wy Wht if the question hs or 3 etc? You still solve the problems in the usul wy with k nd found s before but t the end you will need to divide ny nswers to find
Eponentil nd Logrithmic Functions Wht points does the grph of y lwys pss through? (y equls to the power ) Wht points does the grph of y log lwys pss through? (y equls the log of, bse ) ( 0,) nd (, ) (,0) nd (,) How do you rewrite form? y log in power y How do you solve n eqution where is the power? (eg 4 0) How do you solve log eqution? Log rules: log log y? log log y? n log? log? log? How do you get log to bse e (the nturl log) on your clcultor? How do you get log to bse 0 on your clcultor? If the grph of log y ginst log is stright line, how do you find y in terms of? If the grph of log y ginst is stright line, how do you find y in terms of? Tke logs of both sides then use log rules to work out Epress ech side s single log then cncel the logs Or, get logs to one side nd numbers to the other, then rewrite using power form log y log y nlog 0 ln button log button n y k The vlues of k nd n cn be found from the grph y b The vlues of nd b cn be found from the grph