Answer: A. Answer: A. k k. Answer: D. 8. Midpt. BC = (3, 6) Answer: C
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1 THE STRAIGHT LINE. (, p) p p p. ( ) AB. D p p 9. A(, ) B(k, l) I. ( ) 9 II III. AB. tn - () = o. Midpt. A = (, ) Midpt. BD = (, ). p p p AB A k k k k. Midpt. B = (, ).. perp 9 k ( ) k k k k k Pegss Higher Mthetics - Unit
2 . tn tn. (, ).. ; perp (, ) QR perp AB ; perp D 9 AB tn ( ) ; (, ). AB ; A. PQ = PR =. PQ ; PR Both stteents correct AB perp. (, k) k ( ); k Pegss Higher Mthetics - Unit
3 .. tn tn perp BD A ;. Q (, ) ; P(, ) PQ Digonls of kite re perp 9.. ; ( ). PQ ; QR ; ; Pegss Higher Mthetics - Unit
4 FUNTIONS nd GRAPHS ( ) f ( g( )) ( ) ( ). g() g( ) ( ) ( ) ( ).. ( 9) ( ) 9 ( ) (, ). ( ). ( ) Min vlue [ sin ] ( ). g ( ) () f ( ) ( ). [( ) ] ( ) ( ) (, ). f ( ) ( ) ( ) ( ). Grph hs been oved to the right nd units up. 9. (, ) (, ) (, ) Pegss Higher Mthetics - Unit
5 DIFFERENTIATION. d d Sub = -. d d (). d d this gives sttionr point.. ( ) d d sub = ( ) d d ( ).. f f ( ) ' ( ) d d sub = tn d d ( ). 9. d d d d () 9 (, ) () Pegss Higher Mthetics - Unit
6 . d d Tble of vlues shows in T.P.. d d. f ( ) f ' for decresing functions.. f ( ) f '( ) f ' () d d.. f f f ( ) ' ' ( ) ( ) d d sub = d d () f ( z) z z. f ' z z f ' (). 9. ( )( ), d d Pegss Higher Mthetics - Unit
7 . d d (). g( ) g ' for incresing function, -. d d ( ) f ( ). f ' f ' () (). d d. f ( ) f '( ) 9 This hs turning point nd iu vlue of -9 is so function is never. d 9 d ( )( ),. () (, ) g( ) g '( ) 9. incresing. 9 d d 9. f ( ) f '( ) f '( ) ( ) Pegss Higher Mthetics - Unit
8 . d d when = d d g( ). g ' (). d d. f ( ) ( f '( ) Answer: B ).. ( ), d d (). d d t S.P. ; (, ) Tble of vlues show M.. for decresing function ( )( ) d. d Pegss Higher Mthetics - Unit
9 Pegss Higher Mthetics - Unit REURRENE RELATIONS.. 9 L. k k. U U U U U U U U. L. L U U 9. q q p q p.
10 MIXED EXERISE - UNIT. line cuts the -is t (, ) line cuts the -is t (, ) idpoint is (, ) Answer: f ( ). f ' f '.. A perp A r rh A' r h (r h). f ( g( )) ( ) (9. AB.. ) B so not isosceles AB B U U U U f ( ) f ( ) so not right ngled 9. PQ perp k RS k k. L. f ( ) f, SP sttionr points so not lws incresing. Tble of vlue shows tht it cn be decresing so III onl correct. f '( ) for decresing or sttionr function. Tble of vlues shows decresing for ll vlues less thn nd sttionr when = or Pegss Higher Mthetics - Unit
11 ... ( ) [( ) ( ) ( ) ], b, c d d d d (). d d Tble of vlues gives + + so point of infleion t (, ). Midpoint B = (, ). 9. A perp.... AB perp ( ) f ( ) f '( ) ( ) - Pegss Higher Mthetics - Unit
12 . ( ) ( ) (, ). ( ) - coord - coord Pegss Higher Mthetics - Unit
13 REMAINDER THEOREM/QUADRATIS. 9 ( ).. coefficient degree 9 9. p p 9 9 p p p p 9 9 p p p p [( ) ] ( ) ( ) ( ) 9 ( )( ). ( )( ) -/. 9 k 9 k k k k k k =.. -. nd 9 q q q = Pegss Higher Mthetics - Unit
14 . b c.. k b c. k k k k Roots re rel nd unequl... k k = -. b c.. Roots re rel nd equl. k k b c k.. k. k = - p p p = -.. p b c.. p p p b p c p.. 9. k k Answer: - 9. b c.. p Pegss Higher Mthetics - Unit
15 ... k. ( b ) c.. Onl rel root is k k k k 9 k -k = - k = reinder.. -. b c.. 9. Roots re non rel. Pegss Higher Mthetics - Unit
16 Pegss Higher Mthetics - Unit INTEGRATION. () f d ) (. ( d ) = when = c. d ) 9 ( 9 9. = d. d. Liits re nd Grph: in T.P.. d. 9. d ) ( ) ( - - -
17 Pegss Higher Mthetics - Unit. d ) (. d. =. s = dt t t t t t t s.. + d, ) )( (. d.. d () ()
18 9. () ().. Fro ( )( + ) liits re nd. d ( )( ) ;. d d ( )( ). d. Liits where Pegss Higher Mthetics - Unit ( ) d - d
19 TRIGONOMETRY... sin( ) ( ) sin, ; sin cos sin sin cos sin sin (cos ) cos ;sin cos, ; 9sin sin cos cos sin sin cos 9cos sin cos (sin cos )... sin, cos sin sin cos sin sin( ) sin( ) sin sin( ) sin cos cos sin. sin. 9. Qudrnts III nd IV ; ; cos A cos A cos A cos( ) cos( ) cos A Pegss Higher Mthetics - Unit
20 Pegss Higher Mthetics - Unit. cos ) cos( sin sin cos cos. ) (cos sin sin cos sin sin sin.,, cos sin ) (cos sin Answer: B. sin sin sin cos ) sin cos cos (sin sin cos cos sin. 9 ) cos 9(sin 9cos 9sin cos cos sin sin cos cos sin sin.,9, ) ( ) cos(.,,, sin sin sin., 9, ) sin( sin cos cos sin. cos sin sin
21 9. cos cos. cos cos( ) cos cos sin sin Pegss Higher Mthetics - Unit
22 THE IRLE. 9.. h ( ) h 9 h A(,-) (,-) B(,-). tn. entre (, ) Rdius is distnce fro (, ) to P or Q Rdius =. (, ) becoes, not chnge. Eqution is ( ) ( ) rdius does 9. entre (, ) to, tn hs. entre is idpoint of AB.. (, ) Rdius is distnce fro (, ) to (, ) r (, ) ( ) ( ). entre, Pegss Higher Mthetics - Unit Rdius = (, -). entre (, ) Rdius = 9. uts is when =. ( )( ) ; P(, ) nd Q(, ) Midpoint is (, ). entre, becoes (, )
23 . ( entre (, ) 9 ) 9. Grdient fro(, ) to (, ) is perp. entre is idpoint of PQ (, ) Rdius =. ( ) ( )( ) ; Midpoint = 9. entre (, ) Rdius = 9. entre, Grdient to (, ) =. For circle rdius ust be greter thn. 9 k k k. entre (, ) becoes,. A(-, ) (-, ) B(-, ). Rdius = ( ) ( ) perp Pegss Higher Mthetics - Unit
24 . (9, ) A(, ) (, ) entre (, ) to (, ) is se pth s (, ) to new centre, Stepping out gives (9, ). Grdient fro centre (, ) to (, ) is. tn Pegss Higher Mthetics - Unit
25 MIXED EXERISE - UNIT sin sin. sin. ( )( ) ( )( )( ). (, ); r. d 9 d.. 9. ( ) ( ) : b reinder. b c.. Non rel roots. Grdient fro centre, perp to (, ). entre is idpoint., Rdius = Eqution is ( ) ( ). (9 ) ( ) Pegss Higher Mthetics - Unit
26 . d.. sin sin sin sin sin ( cos ) sin : cos, ; (); () b c 9.. Roots re rel nd unequl.. cos cos 9 ( )( ) ( ) does not fctorise since b c. d. 9. ; ; ( )( )( ) cos cos( ) cos cos sin sin. Liits where ( ) Pegss Higher Mthetics - Unit
27 Pegss Higher Mthetics - Unit VETORS. 9 ) ( PQ.. ),, ( ),, ( : R R R z. 9. cos 9 q p. 9 b b b b. ),, ( ) 9,, ( : 9 R z. 9 ) (.b. b.
28 . (,, ) ( 9,, ) :.. z K K K :.b cos b cos Pegss Higher Mthetics - Unit
29 FURTHER ALULUS.... ( ( ) ) d d ( ) ( ) sin cos cos sin sin sin.... ( ) f '( ) cos cos () cos sin cos sin sin cos ( cos ) ( cos( ) ( ( )) ( ( )) f '( ) ( ( cos( ) cos( ) f '( ) sin ( sin ) ) ) f ' sin () sin ( cos ) sin( cos) sin( ) sin( ) sin Pegss Higher Mthetics - Unit
30 Pegss Higher Mthetics - Unit. ) )( ( ) ( d d. ) ( ) ( ) ( d d. sin sin. d d cos when ) ( cos d d. sin cos sin cos 9. ) ( ) ( ) ( ) (. ) ( ) (. ) ( ) ( n n b n b n. ) ( ) ( ) ( ) ( d d. ) ( cos cos d d. ) ( ) (
31 . d ( d d ( d ( ( ) ) ). Liits where 9 9 Upper lower = cos( ) cos( ) cos sin (cos sin ) ( ) ( ). f ( ) ( f ' ( ( ) ( 9 ) ) ) when Pegss Higher Mthetics - Unit
32 LOGARITHMS nd EXPONENTIALS.. log log. Answer:. ( ) log f reflected in - is f ( ) log log log. log grph hs been oved to the left so =. (, ) hs becoe (, ) so bse b =... log log log log log ( ) log.. log log log log log log. n n 9. log hs been oved up unit prllel to - is. log log log log log Pegss Higher Mthetics - Unit
33 . 9. log log log log log log.. log log log k log k log k log k log k log k. log hs been oved to the right. log log log. Pegss Higher Mthetics - Unit
34 THE WAVE FUNTION k k ( ) ( ).. sin( ) ( ) 9 Answer: A sin( ) ( ). cos sin If this were to written s... tn cos( ) ( ) cos( ) ( ) 9.. tn k tn ( ) cobined wve for then k ( ) this gives iu vlue of. sin( ) hs. vlue sin( ) hs vlue + =. M. occurs when in. of cos( ) is subtrcted fro ( ). sin cos cos( ) cos( ) ( ) ; ; Pegss Higher Mthetics - Unit
35 . M: ( ) Min: k ( ) ( ). tn ;. k tn. Min: M: ( ) Pegss Higher Mthetics - Unit
36 MIXED EXERISE - UNIT..b ( ). d ( ) d ( ) ( ). ( ) d ( ) ( ) ( ).. log log log ( k) ( k) ( k). Since grph psses through (, ) bse is.. Miniu occurs where cos( ) ( ) log. log log 9 log 9 9. Grph of log hs been oved to the left. =. Bse is cos( ). cos( ). (,, ),, ) z : Pegss Higher Mthetics - Unit
37 . cos( ) cos( ). ( ) ( ) ( ). () 9 p. p p. q. 9 sin( ). sin( ) ( cos ).. b b cos. Miu vlue is + nd occurs when cos( ) ( ) (). 9. log e log log log log e e e e log log ( e ( )( e ) e e e ( ) ) Pegss Higher Mthetics - Unit
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