Higher Maths A1.3 Recurrence Relations - Revision

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1 Higher Maths A Recrrence Relations - Revision This revision pack covers the skills at Unit Assessment exam level or Recrrence Relations so yo can evalate yor learning o this otcome It is important that yo prepare or Unit Assessments bt yo shold also remember that the inal exam is considerably more challenging, ths practice o exam content throghot the corse is essential or sccess The SQA does not crrently allow or the creation o practice assessments that mirror the real assessments so yo shold make sre yor knowledge covers the sb skills listed below in order to achieve sccess in assessments as these revision packs may not cover every possible qestion that cold arise in an assessment Topic Unit Sb skills Revision pack Qestions Recrrence Relations A Determining a recrrence relation rom given inormation sing it to calclate a reqired term Finding interpreting the limit o a seqence, where it exists,,, 6, 7, Heinemann Textbook D, - 0 H When attempting a qestion, this key will give yo additional important inormation Key Note Qestion is at nit assessment level, a similar qestion cold appear in a nit assessment or an exam Qestion is at exam level, a qestion o similar diiclty will only appear in an exam # * The qestion incldes a reasoning element typically makes a qestion more challenging Both the Unit Assessment exam will have reasoning qestions I a star is placed beside one o the above symbols that indicates the qestion involves sb skills rom previosly learned topics I yo strggle with this qestion yo shold go back review that topic, reerence to the topic will be in the marking scheme Qestion shold be completed withot a calclator Qestion shold be completed with a calclator Qestions in this pack will be ordered by sb skill typically will start o easier then get more challenging Some qestions may also cover several sb skills rom this otcome or even inclde sb skills rom previosly learned topics (denoted with a *) Qestions are gathered rom mltiple sorces inclding ones we have created rom past papers Extra challenge qestions are or extension are not essential or either Unit Assessment or exam preparation JGHS H A Revision

2 FORMULAE LIST ircle: The eqation y gx y c 0 g c x represents a circle centre The eqation x a y b r represents a circle centre a, b g, radis r radis Scalar Prodct: ab a b cos, where is the angle between a b or ab ab ab ab where a a a a b b b b Trigonometric ormlae: sin A B cosa B sin Acos B cos Asin B cos Acos B sin Asin B sin A sin Acos A cos A cos A sin A cos sin A A Table o stard derivatives: x x sin ax cos ax acos ax asin ax Table o stard integrals: x xdx sin ax cos ax cos ax a sin a ax JGHS H A Revision

3 Q Qestions Marks A seqence is deined by the recrrence relation n+ = n + 8 with 0 = Evalate A seqence is deined by the recrrence relation n+ = n + 6 with 0 = 0 What is the limit o the seqence? A patient is injected with 6ml o a drg Every 8 hors, % o the drg passes ot o the bloodstream To compensate, a rther ml dose is given every 8 hors (a) Find a recrrence relation or the amont o drg in the bloodstream (b) Use yor answer to calclate the amont o drg remaining ater hors A seqence is deined by the recrrence relation n 08, (a) (b) State why this seqence has a limit Find this limit n 0 A pond is treated weekly with a chemical to ensre that the nmber o bacteria is kept low It is estimated that the chemical kills 68% o all bacteria Between the weekly treatments, it is estimated that 600 million new bacteria appear There are n million bacteria at the start o a particlar week (a) Write down a recrrence relation or n+, the nmber o millions o bacteria at the start o the next week (b) Find the limit o the seqence generated by this recrrence relation explain what the limit means in the context o this qestion 6 A seqence is deined by the recrrence relation n = 0 n- +, = (a) alclate the vale o (b) What is the smallest vale o n or which n > 0? (c) Find the limit o this seqence as n 7 A seqence is deined by the recrrence relation constants with, n a b where a n b are (a) Find algebraically the vales o a (b) Hence ind the vale o (c) Find the vale o b JGHS H A Revision

4 8 Two seqences are deined by the recrrence relations n+ = 0 n + p, 0 = v n+ = 06v n + q, v 0 = I both seqences have the same limit, express p in terms o q # Two seqences are generated by the recrrence relations: n+ = a n + 0 v n+ = a v n + 6 The two seqences approach the same limit as n Determine the vale o a evalate the limit 0 (a) A seqence is deined by n+ = n, with 0 = 6 Write down the vales o (b) A second seqence is given by,, 7,, It is generated by the recrrence relation v n+ = pv n + q, with v = Find the vales o p q (c) Either the seqence in (a) the seqence in (b) has a limit (i) (ii) alclate this limit Why does the other seqence not have a limit? (a) The terms o a seqence satisy n+ = k n- + Find the vale o k which prodces a seqence with a limit o Find the vale o k (b) A seqence satisies the recrrence relation, n m n 0 (i) Express in terms o m (ii) Given that 7, ind the vale o m which prodces a seqence with no limit [END OF REVISION QUESTIONS] [Go to next page or the Marking Scheme] JGHS H A Revision

5 Where sitable, yo shold always ollow throgh an error as yo may still gain partial credit I yo are nsre how to do this ask yor teacher Q Marking Scheme Sbstitte correctly evalate = () + 8 = 6 Evalate = (6) + 8 = Sbstitte into limit ormla L = 6 Evalate limit L = 0 (a) orrect orm o a recrrence relation n+ = n + (or n = n + ) 0 = orrect vales in recrrence relation n+ = 078 n +, 0 = 6 (b) Know to calclate Evidenced in working alclate correctly 7ml (a) orrect statement Limit exists as -<08< (b) Evalate the limit L = 08 = 0 Mst explicitly state vale 08 that it mst be greater than - less than (a) orrect recrrence relation n+ = 0 n (b) Know to calclate Evidenced in working Sbstitte into limit ormla L = Evalate limit L = 88 Limit in context In the long term the amont o bacteria in the pond will settle arond 88 million 6 (a) alclate correctly = 6 (b) orrect vale or n n = 6 (c) Sbstitte into limit ormla L = Evalate limit L = 0 0 JGHS H A Revision

6 7 # (a) Interpret Inormation a b Interpret Inormation 7 a b Solve to ind a (b) Find b a 8 b ( ) ( ) 7 (c) orrect strategy 6 Find Simltaneos eqations is the sggested method o solving the two eqations 8 Limit o v L = p 0 Knowing to eqate limits starting simpliication Writing p in terms o q (mst be lly simpliied) p 08 = q 0 p = q L = q 06 Limit o v L = 0 a Knowing to eqate limits starting simpliication L = 6 a Qadratic eqated to zero 0a 6a + 6 = 0 Solve qadratic a = 06 or a = Select appropriate vale with explanation 0 a = 6 a = 0( a ) = 6( a) as limit exists a, so a=06 JGHS H A Revision

7 0 (a) Evalate 8 (b) Interpret inormation p q Interpret inormation 7 p q (or 7 p q ) Solve (c) Start to ind limit o irst seqence p 0 q 6 Evalate limit 6 0 (ii) 7 Explain why the nd seqence has no limit 7 For a limit to exist a Since the nd seqence has no limit p In (c) I yo calclate a limit or the nd seqence, all marks are not available (a) Sitable strategy Solve (b) Find an expression or Find an expression or (ii) ompare vales or solve start to k ( k) k m m k m k 0 m m m(m ) m m m 7 m 0 6 Solve eqation 6 m m 0 So m or m 7 Select correct vale 7 For a limit to exist a So m [END OF MARKING SHEME] [END OF REVISION QUESTIONS] JGHS H A Revision