Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

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Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S

1 Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper ethereisnosuchthingscnnotd oonlnotetbelievethtoucnd oitndourehlfwtherether eisnosuchthingscnnotdoonln otetbelievethtoucndoitnd ourehlfwtherethereisnos uchthingscnnotdoonlnotet Believethtoucndoitndour ehlfwtherethereisnosuchthi ertuiopsdfghjklzcvnmrtui opsdfghjklzcvnmqwertuiop

2 Alger() Solving simple equtions. Solve ech of the following equtions ) f) 6 ) 9 c) p g) h 6 0 h) q d) i) e) m j) 9 9 k) d l) t 6 m) k n) 6 o) n p) r 9 q) e r) w 9 s) j t) 6. Solve ech of the following equtions ) ) p p 9 c) m m d) f f e) u u f) 9 g) h) 6 i) r r j) 0e e k) l) 6 m) 6 n) 6 o) 9 p) 6t t q) m m r) k k s) u 6 u t) 6 h h. Solve the following equtions ) ) p c) 6 d) 6 e) m f) 9 g) h) 0 i) p j) k) 9 l) 6 m) r 9 n) r o) v v 6 p) f q) r) g 6 s) 0 r t) 9 t S Cooper

3 Alger () Removl of rckets. Remove the rckets for ech of the following: ) j) m n p s) ) c) 6m d) e) 6 p f) g) 0 h) i) k) r t l) m) n) o) p) m m q) r) t) c u) pq p v) 6rr s t w) m m n ) e e f ) z) h i j. Remove the rckets nd simplif ech of the following: ) h) o) ) c) d) 6 e) 0 f) g) i) 6 j) k) 6 l) m) n) p) q) r) s) t). Remove the rckets for ech of the following: ) ( + )( + )( + ) ) ( )( + )( + ) c) ( + )( )( + ) d) ( + )( )( + ) e) ( )( + )( + ) f) ( + )( ) g) ( + ) h) ( ) S Cooper

4 Alger () Fctorise ech of the following () 9 () m 6 () p () 0 () r 6 (6) () f g () s t (9) 6 z (0) i j Fctoristion common fctor () m 6n () u v w () 0m n p () 9e f g () 6 0 c (6) () () (9) 9 (0) m n mn 0mn S Cooper

5 Alger () Fctoristion. Fctorise ech of the following:.. 6 c. d. 6 e. 6 f. g. h. i. j. 0 k. l. m. 6 n. 6 o. p. 6 6 q. 0 r.. Fctorise ech of the following: ) ) c) d) 6 e) 6 f) 6 6 g) 9 9 h) i) j). Solve ech of the following equtions c. 0 d. 0 e. 0. () Fctorise () Hence, or otherwise solve the eqution 0. () Fctorise () Hence, or otherwise solve the eqution 0 6. Solve ech of the following equtions: ) = 0 ) + + = 0 c) = 0 S. J. Cooper

6 . () Fctorise () Hence, or otherwise solve the eqution 0. () Fctorise () Hence, or otherwise solve the eqution 0 9. () Fctorise 0 () Hence, or otherwise solve the eqution Solve ech of the following equtions: ) 0 ) c) 0 d) e) 0 f) 0 g) 6 0 h) 0 i) Solve ech of the following equtions: ) ) c) d) e) S. J. Cooper

7 Alger (). Fctorise ech of the following: ) ) c) d) e) f) g) h) i) j) 6 k 9 0u v c 0d. Complete the squre for ech of the following: ) f) ) 6 g) c) 6 h) d) 0 i) 6 e) j). Solve ech of the following equtions completing the squre, leving our nswer s surd: ) 9 0 f) 0 0 ) 0 0 g) 0 c) 0 h) c 0c 0 d) 0 i) 0 e) e 6e 0 j) 0 0. Using the formul, giving nswers to deciml plces where pproprite, solve ech of the following: ) 0 h) 6 0 ) 0 i) r 0r 6 0 c) 9 0 j) u 9u 0 d) 0 e) 6 0 f) 0 g) 0

8 Alger (6) Forming Equtions. I think of numer doule it nd sutrct three. If I then hve five wht ws numer ws I thinking of?. Njid thinks of numer, multiplies it five nd sutrcts his nswer is. Wht numer did Njid strt with?. Susn ws told to think of numer. Then she ws sked to multipl her numer four nd dd si. When sked wht ws her nswer she gve. Wht numer did Susn think off?. The perimeter of the rectngle elow is 6cm work out the vlue of.. Given tht the perimeter of the tringle elow is cm work out the vlue of If the perimeter of the regulr pentgon elow hs the sme perimeter s the rectngle find the vlue of.. John is ers older thn Jnet. However in 0 ers time John will e doule Jnet s ge. If John is ers old. Find.. Given tht the re of the rectngle opposite is 6cm, find.

9 9. Given tht the re of the tringle opposite is m, find p. p 0. Given tht the re of the 0cm find the two possile vlues for. ( ). Given tht the re of the rectngle elow is cm find the one possile vlues for. ( 6). Given tht the volume of the cuoid is 0cm, find the one possile vlues for.

10 Alger () Simultneous Equtions I Eercise Solve ech of the following sets of simultneous equtions. Show our working d c d c 6. i h i h.. 6 n m n m 9. 9 t u t u 0. 9 Eercise Solve ech of the following sets of simultneous equtions. Rememer to show our working d c d c. 0 6 u t u t. f e f e s r s r. 6 6 q p q p. 6 k h k h 9. 6 h g h g v u v u. 6 6 j i j i. 9 q p q p. k k j k 6. 0 e d e d. 9 n m n m. 6 9

11 Eercise. Two numers hve sum of nd difference of 6. Find these two numers.. Find two numers which dd to give nd hve difference of.. Three times one numer dded to twice nother gives. If the difference etween the two numers is find the two numers.. Aln nd Srh hve sved etween them. If Aln hs more thn Srh how much does ech person hve?. A g contins 0 counters red nd lue. If there re five more lue counters thn red how mn red counters re there? 6. The cost of si pens nd four pencils in the print room is.6, wheres the cost of five pens nd two pencils is 9pence. Work out the cost of pen nd pencil.. For recent concert Mohmmed sold 0 tickets rising 9 for school funds. If the tickets cost per child nd per dult, how mn tickets did he sell to dults?. The cost of three pers nd seven pples is.. If the difference etween the two prices is pence find the cost of ech item. 9. For recent holid the cost of two dults nd three children ws quoted for 0. Wheres for two dults nd si children the price ws 60. Wht is the price per child nd per dult? 0. Angel hs 0 spending mone. In ig deprtment store Angel cn four lipsticks nd three different nil polishes for 9.. Or lterntivel she could u one lipstick nd si nil polishes for 9. Wht is the cost of ech item?

12 Alger () Simultneous Equtions II Eercise Solve ech of the following sets of simultneous equtions. Rememer to show our working.. 6. r s rs. 9. p q pq. cd cd. 6 h k 0 hk 6. tu tu 9. g h 6 gh. e f 0 e f 0. 6 Eercise. () Drw the grph of for vlues of etween nd () On the sme set of es drw the stright line with eqution 6 (c) Hence solve the set of simultneous equtions nd 6. () Drw the grph of for vlues of etween nd () On the sme set of es drw the stright line with eqution (c) Hence solve the set of simultneous equtions nd. B drwing suitle grphs solve the set of simultneous equtions given nd

13 Alger (9) Simultneous Equtions III Solve ech of the following sets of simultneous equtions. Rememer to show our working

14 Alger (0) Drwing stright line grph. For ech of the following (i) Cop nd complete the tle (ii) drw the grph for the stright line. () 0 () 0 (c) 0 (d) 0 (e) 0. Drw the grphs for ech of the following: () () (c) (d) 6 (e) 6 (f) 0 (g) (h) (i) (j) (k) (l)

15 . ) Drw on the sme set of es the grphs of nd ) Hence solve the simultneous equtions nd. ) Drw on the sme set of es the grphs of nd ) Hence solve the simultneous equtions nd. ) Drw on the sme set of es the grphs of nd 6 ) Hence solve the simultneous equtions nd 6 6. The grph drwn opposite represents Work out the coordintes of A nd B A B 0. The grph drwn elow represents Work out the coordintes of P nd Q P 0 Q. The grph drwn elow represents Work out the coordintes of E nd F 0 E F

16 Alger () Grdients nd Equtions of lines. The grph elow represents the grph of 6 ) Find the coordintes of the points A nd B A B 0 ) Hence find the grdient of the line.. ) Find the grdient of line which psses through points,6 nd, ) Find the grdient of line which psses through the points, nd,. Write down the grdient nd vlue of the intercept for ech of the following grphs. ) f) ) c) d) e) g) h) 6 i) j). Work out the grdient nd the vlue for the intercept of the grph with eqution. A stright line psses through the point, nd hs grdient equl to. ) On set of es drw this grph. ) Write down n epression for the grdient of the line 6. A stright line psses the points, nd, ) Find the grdient of the line ) Find n epression for the eqution of the line

17 . A stright line psses the points,9 nd, ) Find the grdient of the line ) Find n epression for the eqution of the line. A stright line psses the points, nd 6, ) Find the grdient of the line ) Find n epression for the eqution of the line 9. Find n epression for the eqution of the stright line drwn elow Find the eqution of the line drwn elow. 0

18 Alger () Drwing qudrtics curves. () Cop nd complete the tle elow for the grph of for vlues of from to. 0 () Hence drw the grph of.. () Cop nd complete the tle elow for the grph of for vlues of from to. 0 () Hence drw the grph of.. () Cop nd complete the tle elow for the grph of for vlues of from to. 0 () Hence drw the grph of.. () Cop nd complete the tle elow for the grph of for vlues of from to. 0 () Hence drw the grph of.. () Cop nd complete the tle elow for the grph of for vlues of from to. 0 () Hence drw the grph of.

19 6. () Cop nd complete the tle elow for the grph of 6 for from to. 0 () Drw the grph of 6. (c) Hence stte the vlues t which 0.. () Cop nd complete the tle elow for the grph of for from to. 0 () Drw the grph of.. () Cop nd complete the tle elow for the grph of for from to. 0 () Drw the grph of. (c) Hence stte the coordintes where the curve meets the -is.

20 Alger () Qudrtic equtions. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd. ) Cop nd complete the tle elow for the grph of 6 0 ) Drw the grph of 6 for vlues of etween nd

21 6. ) Cop nd complete the tle elow for the grph of 6 0 ) Drw the grph of 6 for vlues of etween nd c) Use our grph to determine vlues of when d) Stte the minimum vlue of on the grph of 6. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd c) Use our grph to determine vlues of when d) Stte the minimum vlue of on the grph of. ) Cop nd complete the tle elow for the grph of 0 ) Drw the grph of for vlues of etween nd c) On the sme set of es drw the grph of d) Using the grph solve the set of simultneous eqution nd e) Hence show tht the two solutions re the sme for the eqution 0 9. () Drw the grph of for vlues of etween nd (d) On the sme set of es drw the stright line with eqution 6 (e) Hence solve the set of simultneous equtions nd 6 0. () Drw the grph of for vlues of etween nd (d) On the sme set of es drw the stright line with eqution (e) Hence solve the set of simultneous equtions nd

22 Alger () Drwing grphs. ) Cop nd complete the tle elow for the grph of 0 ) Hence drw the grph of in the rnge 0. ) Cop nd complete the tle elow for the grph of ) Hence drw the grph of 0 in the rnge 0. 0 c) Eplin wh it is not possile to find vlue for when 0. ) Cop nd complete the tle elow for the grph of 0 0 ) Hence drw the grph of in the rnge. ) Cop nd complete the tle elow for the grph of ) Hence drw the grph of in the rnge ) Cop nd complete the tle elow for the grph of sin ) Hence drw the grph of sin in the rnge 0 60

23 6. Pir of the following equtions with the grphs drwn elow ) ) c) d) e) f) + = 0 A B C D E F. Drw rough sketch to represent ech of the following grphs ) ) c) d) 0 e) f) 6 g)

24 Alger () Inequlities. Solve ech of the following inequlities ) e) ) 0 f) 9 c) 9 g) d) h) i) j). Solve ech of the following inequlities: ) e) ) r r f) w w 6 c) t t g) f 9 f d) u u h) 0k 9k i) h 9 h j) v 9 6v k) 0z z l) 9 9. Solve the following inequlities: ) ) c) 9 9 d) 0 e) 6 f) g) 0 6 h) 0 i) 90 j) 0. Solve the following inequlities: ) ) c) d) e) f) + > 0 g) + 0 h) < 0 i) > 0 j) +. Solve the inequlit 6. Solve the inequlit 6 6

25 Alger (6) Inequlities II. Write down the inequlit shded for ech of the following: ) ) 0 c) d) 0 e) f)

26 g) h) 0 i) j) 0 k) l)

27 . For ech of the following drw the grph which est ech inequlit. () () (c) (d) (e) (f) (g) 0 (h) 6 (i) (j). Write down n eqution which fits ech of the following shded regions ) ) 0 c) d) 0 e) f)

28 g) h) 0 i) j) 0 k) l)

29 . Descrie the shded region in the digrm drwn elow.. () Write down the eqution of the three lines drwn round the shded region elow () Write down three inequlities which est descrie the shded region.. () On one set of es drw the grphs of (i) (ii) (iii) () Shde in the region defined the set of inequlities,,

30 6. () On one set of es drw the grphs of (i) (ii) (iii) () Shde in the region defined the set of inequlities,,

31 Alger () Rerrnging formule Mke the suject for ech of the following. m c. c. t. t d e. w f g k M c d. t 6. c Mke the suject for ech of the following. t r m.. c f t c d. w g. w u v. V. G T E 9. t 0. A B A 6. g

32 Alger() Sequence. For ech of the following sequences write the net two terms in the sequence nd then find formul for the n th term in the sequence. ),,,, ),,, 9, 6 c),, 0,, 6 d),,, 9, e),,, 9, f) 9,,,, g),,,, h), 0,,, i),,,, j) 6, 9,,,. The third term is 6 nd the common difference is. Write down n epression for the nth term.. The sith term of n rithmetic series is nd the thirteenth term is. i. Find the common difference of the series ii. An epression for the nth term.. In n rithmetic progression, the ninth term is, nd the twent-ninth term is equl to twice the fifth term. Determine the first term nd the common difference of the progression.

33 Alger(9) Sequences. Otin the nth term for ech of the following sequences ),,,, ),, 9, 9, c), 9,, 9, d),, 9,, 9 e) 0,,, 9, 6 f) 0,, 0,, g), 9, 0,, h), 0,, 0, 6 i),,,, j) -, -, -, 0,. ) write down the first si terms for the sequence of digrms elow. ) Hence otin the nth term for this sequence. c) Wht nme is given to this specil sequence of numers?. Otin the nth term for the sequence elow 0, 6,,,. Otin the nth term for the qudrtic sequence elow 0,, 9,,. Three sequences re defined elow Sequence A 9 Sequence B Sequence C Using the nth terms of sequence A nd B otin the nth term of sequence C in the form n + n + c

34 Alger(0) Geometric Progressions (GP). For ech of the following geometric progressions find n epression for the nth term. ),, 9,,,, ), 0, 0, 0, 0, 60, c),, 6, 6, 6, 0, d),, 6, 0,, 9, e), 0, 00, 000, 0000, 00000,. A Geometric progression hs first term nd common rtio. Write down the first si terms for this sequence.. A geometric series hs second term 6 nd third term Work out the first term nd the common difference.. A geometric series hs third term 6 nd fifth term Work out the first term nd the common difference.. A geometric series hs second term nd fifth term 0 Work out the first term nd the common difference. 6. A geometric series hs third term nd forth term 0 Work out the first term nd the common difference.. A geometric sequence is,,,, (i) Write down the vlue of the th term (ii) Write down the epression for the nth term s power of. (iii) Otin the sum of the first eight terms of the sequence.. A geometric progression hs first term nd fourth term 9 ) Find the common rtio ) Find n epression for the nth term of the GP.

35 Alger() Itertive formule. The grphs of = nd = re shown on the grid. () Use the grphs to write down solution of + = 0 () Show tht the eqution Cn e rerrnged to give + = 0 (c) Use the eqution = + = + And n itertive method to find n pproimte solution of + = 0, giving our nswer correct to deciml plces.. () Show tht the eqution + = 0 cn e rerrnged s = () Strting with 0 =, use the itertion formul n+ = n. to find solution of + = 0 correct to deciml plces. Clss Y re tring to find solution to the eqution One group re tring to find the solution using the itertion n n Whilst second group re using the itertion n n ) Strting with 0 = find, using oth itertive formule the solution correct to two deciml plces. ) Which itertion otined the solution the quickest?

36 . () Show tht the eqution 0 hs solution etween nd. () Show tht is rerrngement of the eqution. (c) Sketch crefull the grphs of nd, for 0, on the sme es. (d) Use the itertive formul n, strting with, to find the solution to deciml plces. n (e) Show differentition tht the itertion is not convergent to. n n. The grphs of = nd = re drwn on the sme set of is. As shown opposite. The point of intersection cn e otined solving = () Show tht = cn e rerrnged to () Using this itertion n nd strting with n 0 =., find the point of intersection correct to deciml plces. 6. () Show tht the eqution 6 + = 0 cn e rerrnged to form the itertive formul 6 () Hence strting with 0 =, use the itertive formul to find the solution correct to deciml plces

37 Alger() Algeric Frctions. Simplif cncelltion: (i) (ii) (iii) (iv) 6 (v) (vi) (vii) (viii) (i) c c () (i) (ii). Simplif: (i) (ii) c c c 6 (iii) c c (iv) c c c c (v) (vi) 0 6 6

38 Alger() Are Under grph. The grph elow is of = Use this grph to find the pproimte vlue for the re entrpped etween the nd es nd under the curve = 0 +. The grph elow is of = Use this grph to find the pproimte vlue for the re entrpped etween the nd es nd under the curve = + 6

39 . () Drw the grph of = ( )( ) for vlues of etween nd () Otin n estimte for the re under the curve etween = nd =. () Drw the curve of = + for vlues of etween 0 nd () Hence otin n estimte for the re under the curve etween = 0 nd =. () Drw the curve of = + for vlues of etween nd () Hence otin n estimte for the re under the curve etween = nd = 6. The grph elow shows the velocit-time grph of cr over 0 second period. Work out the distnce trvelled the cr over this period of 0 seconds.. A cr is trvelling with velocit given the formul v = + t ) Drw the velocit-time grph to show the motion of the cr over the first si seconds of motion. ) Use this grph to work out the distnce covered during the first si seconds of motion.. A trin trvels etween two points A nd B with velocit given the formul v = 0 + t Given tht the time tken to trvel from A to B is 0 seconds, find the distnce covered in trvelling from A to B.

40 9. For ech of the velocit time grphs elow find the distnce trvelled during the first 6 seconds () () 0. The velocit-time grph elow is tht of cclist during period of 6 seconds Use the grph to estimte the distnce covered during this period of 6 seconds

41 Alger() Tngent To Curve. () Drw the curve of = 9 for vlues of etween = 0 nd = () Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent. (c) Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent.. () Drw the curve of = + for vlues of etween = nd = () Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent. (c) Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent.. () Drw the curve of = for vlues of etween = nd = () Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent. (c) Drw tngent to the curve t = 0. nd otin n estimte vlue for the grdient of this tngent.. () Drw the curve of = ( )( + ) for vlues of etween = nd = () Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent. (c) Drw tngent to the curve t = 0. nd otin n estimte vlue for the grdient of this tngent.. () Drw the curve of = for vlues of etween = nd = () Drw tngent to the curve t = nd otin n estimte vlue for the grdient of this tngent. (c) Drw tngent to the curve t = 0 nd otin n estimte vlue for the grdient of this tngent.

42 6. The grph elow shows the velocit-time grph of cr over 0 second period. Work out the ccelertion of the cr over this period of 0 seconds.. A cr is trvelling with velocit given the formul v = + t t ) Drw the velocit-time grph to show the motion of the cr over the first si seconds of motion. ) Use this grph to work out n estimte for the ccelertion of the cr t the time when (i) t = (ii) t =. A trin is trvelling with velocit given the formul v = t t + ) Drw the velocit-time grph to show the motion of the cr over the first five seconds of motion. ) Use this grph to work out n estimte for the ccelertion of the cr t the time when (i) t = (ii) t = 9. A cr is trvelling with velocit given the formul v = (t + )( t) ) Drw the velocit-time grph to show the motion of the cr over the first five seconds of motion. ) Use this grph to work out n estimte for the ccelertion of the cr t the time when (i) t = (ii) t =

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions