21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae

Size: px
Start display at page:

Download "21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae"

Transcription

1 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4 Sustitution into Formule 1.5 More Comple Formule 1.6 Chnging the Sujet 1.7 More Chnge of Sujet CIMT nd e-lerning Jmi

2 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 1 Formule 1.1 Using Formule In formule, letters re used to represent numers. For emple, the formul A lw l n e used to find the re of retngle. Here A is the re, l the length nd w the width. In this formul, lw mens l w. w Formule re usull written in this w, without multiplition signs. The perimeter of the retngle would e given the formul P l + w Here gin there re no multiplition signs, nd l mens l nd w mens w. Worked Emple 1 The perimeter of retngle n e found using the formul Find the perimeter if l P l + w 8 nd w 4. The letters l nd w should e repled the numers 8 nd 4. This gives P Worked Emple 4 The finl speed of r is v nd n e lulted using the formul v u + t where u is the initil speed, is the elertion nd t is the time tken. Find v if the elertion is m s, the time tken is 10 seonds nd the initil speed is 4 m s 1. The elertion is m s so. The initil speed is 4 m s 1 so u 4. The time tken is 10 s so t 10. Using the formul v u + t gives CIMT nd e-lerning Jmi 1 v m s 1

3 1.1 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Eerises 1. The re of retngle is found using the formul A lw nd the perimeter using P l + w. Find the re nd perimeter if: () l 4 nd w () l 10 nd w 3 () l 11 nd w (d) l 5 nd w 4. The formul v u + t is used to find the finl speed. Find v if`: () u 6, nd t 5 () u 0, 4 nd t 3 () u 3, 1 nd t 1 (d) u 1, nd t 4 3. Use the formul F m to find F if: () m 10 nd 3 () m 00 nd 4. The perimeter of tringle is found using the formul Find P if: P + + () 10, 1 nd 8 () 3, 4 nd 5 () 6, 4 nd 7 5. The volume of o is given the formul V Find V if: (), 3 nd 10 () 7, 5 nd 3 () 4, 4 nd 9 6. Find the vlue of Q for eh formul using the vlues given. () Q () Q + 4 nd 3 nd 5 () Q + 4 (d) Q 5 3 nd 5 10 nd CIMT nd e-lerning Jmi

4 1.1 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet (e) Q (f) Q 10 nd 4 nd (g) Q + 4 (h) Q nd 3 5 nd 11 (i) Q z (j) Q + z 4, nd z 10, 5 nd z 8 (k) Q z (l) Q + 4 z, 5 nd z 3 8, 3 nd z 4 (m) Q + z (n) Q + z 8, 10 nd z 3 50, nd z 3 7. This formul is used to work out Alele's p: P $0 + Numer of hours worked Rte of p Her rte of p is $0 plus $9 per hour. Work out her p for 40 hours. 8. A retngle hs length of m nd width of m. ( ) The perimeter of retngle is given the formul p +. Clulte the perimeter of retngle when 45. nd Construt nd Use Simple Formule A formul desries how one quntit reltes to one or more other quntities. For emple, formul for the re of retngle desries how to find the re, given the length nd width of the retngle. The perimeter of the retngle would e given the formul P l + w Here gin there re no multiplition signs nd l mens l nd w mens w. Worked Emple 1 () Write down formul for the perimeter of the shpe shown. () Find the perimeter if m, 3 m nd 5 m CIMT nd e-lerning Jmi 3

5 1. MEP Jmi: STRAND G UNIT 1 Formule: Student Tet () The perimeter is found dding together the lengths of ll the sides, so the formul will e P ut s nd re oth dded in twie, this n e simplified to P + + () If m, 3 m nd 5 m, Worked Emple P m When repiring ir onditioning sstems, n emergen engineer hrges si fee of J$3000 plus J$100 per hour. Find formul for lulting the engineer's hrge. Let C hrge nd n numer of hours. The hrge is mde up of fied J$3000 nd J$100 the numer of hours or J$100 n. So the totl hrge in J$ is given C n Eerises 1. Find formul for the perimeter of eh shpe, nd find the perimeter for the speified vlues. () () () 6 m, 4 m 5 m (d) 6 m, 10 m 5 m, 6 m, 10 m CIMT nd e-lerning Jmi 4

6 1. (e) MEP Jmi: STRAND G UNIT 1 Formule: Student Tet (f) 10 m 4 m, 5 m, 9 m (g) (h) 60 m, 160 m, 4 m, 9 m 80 m. Find formul for the re of eh of the shpes elow nd find the re for the vlues given. () () 6 m, 10 m 3 m () (d) m, 8 m 3 m, 4 m, 9 m (e) (f) 4 m, 5 m CIMT nd e-lerning Jmi 5 50 m, 00 m

7 1. MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 3. Three onseutive numers re to e dded together. () If is the smllest numer, wht re the other two numers in terms of? () Write down formul for the totl, T, of the three numers in terms of, using our nswer to (). 4. () Write down formul to find the men, M, of the two numers nd. () Write down formul to find the men, M, of the five numers p, q, r, s, nd t. 5. Tikets for shool onert re sold t $6 for dults nd $4 for hildren. () If p dults nd q hildren u tikets, write formul for the totl vlue, T, of the tiket sles. () Find the totl vlue of the tiket sles if p 50 nd q A retngle is 3 m longer thn it is wide. If is the width, write down formul for: () the perimeter, P; () the re, A, of the retngle. 7. Rhel is one er older thn Brdle. Crl is three ers ounger thn Brdle If Brdle is ers old, write down epressions for: () Rhel's ge; () Crl's ge; () the sum of ll three hildren's ges. 8. A window lener hrges fee of J$800 for visiting house nd J$400 for ever window tht he lens. () Write down formul for finding the totl ost C, in J$, when n windows re lened. () Find C if n A ti driver hrges fee of $5, plus $4 for ever km tht the ti trvels. () Find formul for the ost C of journe tht overs km. () Find C if 3. CIMT nd e-lerning Jmi 6

8 1. MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 10. A grdener uilds pths using pving sls lid out in pttern s shown, with white sls on eh side of row of red sls. () () If n red sls re used, how mn white sls re needed? Another grdener puts white sl t eh end of the pth s shown elow. If n red sls re used, how mn white sls re needed? 11. A pth of width is lid round retngulr lwn s shown. 0 () () Find n epression for the perimeter of the grss. Find n epression for the re of the grss. grss Juie drinks ost J$7 eh. Write down formul for the ost, J$C, of n drinks. pth 13. () Gs osts J$80 per litre. Write down formul for the ost, J$C, of l litres of gs. () Gs osts J$ per litre. Write down formul for the ost, J$C, of l litres of gs. 14. Write down n epression for the TOTAL ost, in dollrs, of 8 metres of fri t dollrs per metre nd reels of thred t dollrs per reel. 15. Mr Jmes works si week of 40 hours t rte of $16 n hour. His overtime rte is $4 per hour MORE thn his si rte. Clulte (i) his TOTAL wge for si week. (ii) his wge for week in whih he worked 47 hours. 1.3 Revision of Negtive Numers Before strting the net setion on formule it is useful to revise how to work with negtive numers. Note When multipling or dividing two numers, if the hve the sme sign the result will e positive, ut if the hve different signs the result will e negtive. CIMT nd e-lerning Jmi 7

9 1.3 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Worked Emple 1 Find () () ( 3) ( 7 ) () ( 4) 3 ( 40) ( 5 ) (d) ( 6) 7 () () ( 3) ( 7) 1 () ( 4) 3 8 ( 40) ( 5) 8 (d) ( 6) 7 4 Note When dding or sutrting it n e helpful to use numer line, rememering to move up when dding nd down when sutrting positive numer. When dding negtive numer, move down nd when sutrting negtive numer, move up. Worked Emple Find () 4 10 () ( ) () 4 5 (d) 6+ 7 (e) 7 ( 4) () () Numer line ( ) () (d) (e) 7 ( 4) Eerises () 6 8 () () 5 + ( ) ( ) ( ) ( ) (d) 6 (e) 8 3 (f) 9 6 (g) ( 4) ( 3 ) (h) 16 ( ) (i) ( 81) ( 3) (j) (k) 8 5 (l) ( 5) 7 (m) 3 ( 8) (n) 1 10 (o) (p) (s) + ( ) ( ) (q) 4 ( 7) (r) 1 ( 4) ( ) ( ) 1 7 (t) 4 + (u) 6 5 CIMT nd e-lerning Jmi 8

10 1.3 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet. () ( ) ( ) ( ) + ( ) 1 () 4 () In London, the temperture t midd ws 5 C. At midnight the temperture hd fllen 8 C. Wht ws the temperture t midnight? 4. The temperture ws reorded inside nd outside house in New York. Inside temperture Outside temperture 16 C 8 C How mn degrees wrmer ws it inside the house thn outside? Chllenge! You open ook. Two pges fe ou. If the produt of the two pge numers is 319, wht re the two pge numers? 1.4 Sustitution into Formule The proess of repling the letters in formul is known s sustitution. Worked Emple 1 The length of metl rod is l m. The length hnges with temperture nd n e found the formul l T where T is the temperture. Find the length of the rod when () T 50 C nd () T 10 C () Using T 50 gives () Using T 10 gives l m l 40 + ( 10) 0. 0 ( ) m CIMT nd e-lerning Jmi 9

11 1.4 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Worked Emple The profit in $ mde slesmn when he sells n ooks is lulted the formul P 4n 50 Find the profit if he mkes () 30 sles () 9 sles () Here n 30 so the formul gives () P So the slesmn's profit fter 30 sles is $70. Here n 9 so the formul gives P He mkes loss of $14 if onl 9 sles re mde. Worked Emple 3 If * is defined * 4 wht is the vlue of * when () 4, (), 4? () 4 * ( ) 4 4( ) () ( ) 4 ( ) * CIMT nd e-lerning Jmi 10

12 1.4 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Eerises 1. The formul elow is used to onvert tempertures in degrees Celsius to degrees Fhrenheit, where F is the temperture in degrees Fhrenheit nd C is the temperture in degrees Celsius. Find F if: F 18. C + 3 () C 10 () C 0 () C 10 (d) C 5 (e) C 0 (f) C 15. The formul 1 s ( u + v) t is used to lulte the distne, s, tht n ojet trvels if it strts with veloit u nd hs veloit v, t seonds lter. Find s if: () u, v 8, t () u 3, v 5, t 10 () u 1., v 38., t 45. (d) u 4, v 8, t (e) u 4, v 8, t 5 (f) u 16., v 8., t The length, l, of spring is given the formul l F where F is the size of the fore pplied to the spring to ompress it. Find l if: () F 5 () F 0 () F 4 (d) F The formul P 10n 400 gives the profit, P in $, mde when n phones re sold in d t shop. Find P if: () n 1 () n 3 () n 4 (d) n 10 How mn phones must e sold to mke profit? 5. Work out the vlue of eh funtion sustituting the vlues given, without using lultor. () V p + q p 8 q 4 () p 10 nd 7 nd CIMT nd e-lerning Jmi 11

13 1.4 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet () z + (d) Q 10 nd 6 15 nd 6 (e) (g) (i) (k) P + (f) Q 4 nd nd z 1 1 V (h) R + 5, 5 nd z 8 4 nd S + (j) R , 4 nd nd 0 T + 5 (l) C + 0 nd nd 5 (m) (o) P (n) A 10 nd 4, 3 nd X (p) z + 10, 17. nd 1. 3 nd 4 (q) P (r) Q + + z 10 nd 6 10, 5 nd z Work out the vlue of eh funtion sustituting the vlues given, using lultor if neessr. () () P () V z + 10, nd 31. nd z 1. R (d) D nd nd (e) Q 3 + (f) V nd nd 4. CIMT nd e-lerning Jmi 1

14 1.4 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 7. If * 4, wht is the vlue of * when (), () 3, 1 8. The formul to onvert tempertures from degrees Fhrenheit ( F ) into degrees Celsius ( C) is 5 C ( F 3) 9 Clulte the temperture in degrees Celsius whih is equivlent to temperture of 4 F. 9. Given tht, 3 nd 0, evlute (i) (ii) Given tht m, p, t, lulte 4 ( m + p) () mp+ t () t Chllenge! There re 10 nk notes ltogether. The onsist of $10, $0 nd $50 notes. If the totl vlue of the notes is $180, find the numer of eh tpe of notes. 1.5 More Comple Formule Some formule suh s nd z + f u v rise in siene or mthemtis, ut when used do not led diretl to vlues of f or z. Here we show how to use the formul to lulte these vlues. Worked Emple 1 Use the formul to find f if u 10 nd v f u v CIMT nd e-lerning Jmi 13

15 1.5 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Sustituting into the formul gives f 10 First dd together the two frtions using 40 s ommon denomintor: f f Now to find f, turn oth frtions upside-down to give Worked Emple Find z using the formul if 36. nd 48.. f 1 40 or f z + Sustituting these vlues into the formul gives z z z 36 Now the squre root n e tken of oth sides to give Eerises 1. Use the formul to find f if: z + 36 or 36 z 6 or f v u () v 3 nd u 4 () v 6 nd u 5 () v 7 nd u 3 (d) v 10 nd u 4 CIMT nd e-lerning Jmi 14

16 1.5 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet. Find z using the formul if: z + () 1. nd 05. () 48. nd 64. () 3 nd Find the vlue of z s frtion or mied numer in eh se elow. () () (e) 1 z + () 1 z + 4 nd 10 3 nd (d) z z nd 7 nd (f) z 4 z nd + (g) 1 z + 4 (h) + 1 z nd 1 (i) z 1 nd 6 4. Find z in eh se elow. () z 9 + () z nd 3 () z (d) z 44 nd nd 3 (e) z + 6 (f) z nd nd 7. 9 CIMT nd e-lerning Jmi 15

17 1.5 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 5. When three resistors re onneted in prllel the totl resistne R is given R X Y Z where X, Y nd Z re the resistnes of eh resistor. Find R if: () X 10, Y 0 nd Z 30 () X 1000, Y 5000 nd Z 000 () X 1500, Y 00 nd Z Use the formul 1 ( t v ) to lulte the vlue of given tht 50, t 5. nd v 06. Give our nswer orret to 1 deiml ple. Show ll neessr working. 7. The formul f uv u + v is used in the stud of light. () Clulte f when u nd v 10.. Give our nswer orret to 3 signifint figures. () B rounding the vlues of u nd v in prt () to signifint figures, hek whether our nswer to prt () is resonle. Show our working. Investigtion Find four integers,,, nd d suh tht + + d. 1.6 Chnging the Sujet Sometimes formul n e rerrnged into more useful formt. For emple, the formul F 18. C + 3 n e used to onvert tempertures in degrees Celsius to degrees Fhrenheit. It n e rerrnged into the form C... to enle tempertures in degrees Fhrenheit to e onverted to degrees Celsius. We s tht the formul hs een rerrnged to mke C the sujet of the formul. CIMT nd e-lerning Jmi 16

18 1.6 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Worked Emple 1 Rerrnge the formul to mke C the sujet of the formul. F 18. C + 3 The im is to remove ll terms from the right hnd side of the eqution eept for the C. First sutrt 3 from oth sides, whih gives Then dividing oth sides 1.8 gives So the formul n e rerrnged s Worked Emple F C F C F 3 C 18. The distne, s, trvelled r in time t from initil speed u to finl speed v is given the formul ( u + v) t s Mke v the sujet of the formul. First multipl oth sides of the formul to give Then divide oth sides t, to give Finll, sutrt u from oth sides to give So the formul eomes s ( u + v) t s t s t u + v u v s v u t CIMT nd e-lerning Jmi 17

19 1.6 Eerises MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 1. Mke the sujet of eh of the following formule. () 4 () + 3 () 4 8 (d) + 4 (e) 5 (f) + (g) (h) + (i) + (j) (k) + + (l) + (m) (n) + (o) 4 3 (p) (s) p d ( ) 3 4 q (q) ( + ) (r) (t) v 5( + ) 4 (u) z + ( ) ( 3) 4. Ohm's lw is used in eletril iruits nd sttes tht V IR Write formule with I nd R s their sujets. 3. Newton's Seond lw sttes tht F m. Write formule with m nd s their sujets. 4. The formul C π r n e used to find the irumferene of irle. Mke r the sujet of this formul. 5. The eqution v u + t is used to find the veloities of ojets. () Mke t the sujet of this formul. () Mke the sujet of this formul. 6. The men of three numers, nd z n e found using the formul m + + z 3 Mke z the sujet of this formul. 7. Mke the sujet of the following formule. () v u + s () s t + 1 t CIMT nd e-lerning Jmi 18

20 1.6 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 8. The volume of tin n is given V π r h where r is the rdius of the se nd h is the height of the n. () Mke r the sujet of the eqution. () Find r orret to deiml ples if V 50 m nd h 10 m A o with squre se hs its volume given nd its surfe re given V h A + 4h h () Mke h the sujet of oth formule. () Find h if A 4 m nd m. () Find h if V 50 m nd 10 m The re of trpezium is given () 1 A ( + ) h Write the formul with s its sujet. () In prtiulr trpezium. Use this to write formul tht does not involve, nd mke the sujet. h 11. () Averge speed, v m s 1, time, t seonds nd distne, d metres, re relted the formul d vt. Mke v the sujet of the formul. () Use the formul to find the verge speed for eh of the following performnes t the Beijing Olmpis in 008. Give our nswers orret to deiml ples. (i) Usin Bolt when he won the men's 100 m re in time of 9.69 seonds (ii) Usin Bolt when he won the men's 00 m re in time of seonds. (iii) The Jmin rel tem for the men's re in time of s. Eplin the differenes in the verge speeds. CIMT nd e-lerning Jmi 19

21 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 1.7 More Chnge of Sujet This setion uses some further pprohes to rerrnging formule. Worked Emple 1 The period, T, of pendulum of length, l, is given the formul Mke l the sujet of the formul. T π l g First divide oth sides π to give T π l g Now the squre root n e esil removed squring oth sides of the eqution, to give T 4π Finll, oth sides n e multiplied g to give l g so the rerrnged formul is Worked Emple Mke the sujet of the formul T g l 4π T g l 4π 6 5 To void leving 5 on the right hnd side of the formul, first dd 5 to oth sides to give Then sutrt from oth sides to give 5 6 Finll, divide 5 to give 6 5 CIMT nd e-lerning Jmi 0

22 1.7 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Worked Emple 3 Mke the sujet of the formul 1 1 q + First sutrt 1 from oth sides so tht the right hnd side ontins onl terms involving. 1 1 q Now omine the two terms on the left hnd side of the formul into single frtion, first mking the ommon denomintor. q 1 1 q 1 1 Now oth frtions n e turned upside-down to give Eerises or 1 q 1 q 1 1. Rerrnge eh of the following formule so tht is the sujet. () 5 3 () 8 6 () (d) 6 5 (e) 8 7 (f) (g) p (h) q 8 + (i) r q 5. For eh formul elow mke the sujet. () q 4 () z () z (d) 3 (e) v 1 4 (f) r 5 π (g) p + 4 (h) r 1 3 (i) 3 + CIMT nd e-lerning Jmi 1

23 1.7 MEP Jmi: STRAND G UNIT 1 Formule: Student Tet 3. Mke u the sujet of eh of the following formule. () (d) u () u + 3 (e) 1 () u p u 5 (f) 1 u u 3 (g) (h) r u v (i) q 7 u p u l 4. The formul T π gives the time for pendulum to omplete one full g swing. () Mke g the sujet of the formul. () Find g if l 05. nd T The formul is used to find the fol length of lens. f u v () Mke v the sujet of the formul. () Find v if f 1 nd u If ll is dropped from height, h, it hits the ground with speed, v, given () v gh Mke h the sujet of this formul. () Find h if g 10 nd v 6. () Mke g the sujet of the formul. (d) Find the vlue of g on plnet when h 10, v A ll is thrown so tht it initill trvels t 45 to the horizontl. If it trvels distne R, then its initil speed, u, is given u gr () Mke R the sujet of the formul. () Find R if u 1 nd g When three resistors with resistnes X, Y nd Z re onneted s shown in the digrm, the totl resistne is R, nd () R X Y Z Mke X the sujet of this eqution. () Find X if R 10, Y 30 nd Z 40. X Y Z 9. The volume of sphere is given the formul V 4 π r. 3 () Rerrnge the formul to give r, in terms of V. () Find the vlue of r when V CIMT nd e-lerning Jmi

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

GM1 Consolidation Worksheet

GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

Algebraic fractions. This unit will help you to work with algebraic fractions and solve equations. rs r s 2. x x.

Algebraic fractions. This unit will help you to work with algebraic fractions and solve equations. rs r s 2. x x. Get strted 25 Algeri frtions This unit will help you to work with lgeri frtions nd solve equtions. AO1 Flueny hek 1 Ftorise 2 2 5 2 25 2 6 5 d 2 2 6 2 Simplify 2 6 3 rs r s 2 d 8 2 y 3 6 y 2 3 Write s

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x). Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

Study Guide and Intervention

Study Guide and Intervention - Stud Guide nd Intervention with the Sme Sign The quotient of two integers with the sme sign is positive. Emple. 7 The dividend nd the divisor hve the sme sign. b. () The dividend nd divisor hve the sme

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

Instructions to students: Use your Text Book and attempt these questions.

Instructions to students: Use your Text Book and attempt these questions. Instrutions to students: Use your Text Book nd ttempt these questions. Due Dte: 16-09-2018 Unit 2 Chpter 8 Test Slrs nd vetors Totl mrks 50 Nme: Clss: Dte: Setion A Selet the est nswer for eh question.

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

Australian curriculum NUMBER AND ALGEBRA

Australian curriculum NUMBER AND ALGEBRA 7A 7B 7C 7D 7E 7F 7G 7H 7I Chpter Wht you will lern Equtions review (Consolidting) Equivlent equtions (Consolidting) Equtions with frtions Equtions with pronumerls on oth sides Equtions with rkets Formuls

More information

Perimeter and Area. Mathletics Instant Workbooks. Copyright

Perimeter and Area. Mathletics Instant Workbooks. Copyright Perimeter nd Are Student Book - Series J- L B Mthletis Instnt Workooks Copyright Student Book - Series J Contents Topis Topi - Plne shpes Topi 2 - Perimeter of regulr shpes Topi 3 - Perimeter of irregulr

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

Algebra Basics. Algebra Basics. Curriculum Ready ACMNA: 133, 175, 176, 177, 179.

Algebra Basics. Algebra Basics. Curriculum Ready ACMNA: 133, 175, 176, 177, 179. Curriulum Redy ACMNA: 33 75 76 77 79 www.mthletis.om Fill in the spes with nything you lredy know out Alger Creer Opportunities: Arhitets eletriins plumers et. use it to do importnt lultions. Give this

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a

Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly.

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

Pythagoras theorem and surds

Pythagoras theorem and surds HPTER Mesurement nd Geometry Pythgors theorem nd surds In IE-EM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of right-ngled tringle. This result is known s Pythgors

More information

Review Topic 14: Relationships between two numerical variables

Review Topic 14: Relationships between two numerical variables Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1 MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]

More information

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245.

Pythagoras Theorem. Pythagoras Theorem. Curriculum Ready ACMMG: 222, 245. Pythgors Theorem Pythgors Theorem Curriulum Redy ACMMG:, 45 www.mthletis.om Fill in these spes with ny other interesting fts you n find out Pythgors. In the world of Mthemtis, Pythgors is legend. He lived

More information

Perimeter, area and volume

Perimeter, area and volume 6 Perimeter, re nd volume Syllus topi M. Perimeter, re nd volume This topi will develop your skills to ompetently solve prolems involving perimeter, re, volume nd pity. Outomes Clulte the re of irles nd

More information

5. Every rational number have either terminating or repeating (recurring) decimal representation.

5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI

More information

Integration. antidifferentiation

Integration. antidifferentiation 9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles. 3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

MATHEMATICS AND STATISTICS 1.6

MATHEMATICS AND STATISTICS 1.6 MTHMTIS N STTISTIS 1.6 pply geometri resoning in solving prolems ternlly ssessed 4 redits S 91031 inding unknown ngles When finding the size of unknown ngles in figure, t lest two steps of resoning will

More information

BEGINNING ALGEBRA (ALGEBRA I)

BEGINNING ALGEBRA (ALGEBRA I) /0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes

More information

Naming the sides of a right-angled triangle

Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information

Plotting Ordered Pairs Using Integers

Plotting Ordered Pairs Using Integers SAMPLE Plotting Ordered Pirs Using Integers Ple two elsti nds on geoord to form oordinte xes shown on the right to help you solve these prolems.. Wht letter of the lphet does eh set of pirs nme?. (, )

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

Chapter 8 Roots and Radicals

Chapter 8 Roots and Radicals Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi

More information

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper

More information

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No

More information

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

More information

1. Twelve less than five times a number is thirty three. What is the number

1. Twelve less than five times a number is thirty three. What is the number Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

UNCORRECTED. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED. Australian curriculum NUMBER AND ALGEBRA 0A 0B 0C 0D 0E 0F 0G 0H Chpter Wht ou will lern Qudrti equtions (Etending) Solving + = 0 nd = d (Etending) Solving + + = 0 (Etending) Applitions of qudrti equtions (Etending) The prol Skething = with diltions

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Math Lesson 4-5 The Law of Cosines

Math Lesson 4-5 The Law of Cosines Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude

More information

Key Stage 3 Mathematics. Level by Level. Pack C: Level 6

Key Stage 3 Mathematics. Level by Level. Pack C: Level 6 p Key Stge Mthemtis Level y Level Pk C: Level 6 Stfford Burndred ISBN 1 89960 7 Pulished y Person Pulishing Limited 1997 Person Pulishing 1995 Revised Ferury 1997 A liene to opy the mteril in this pk is

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Polynomials. Polynomials. Curriculum Ready ACMNA:

Polynomials. Polynomials. Curriculum Ready ACMNA: Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression

More information

Objective: To simplify quotients using the Laws of Exponents. Laws of Exponents. Simplify. Write the answer without negative exponents. 1.

Objective: To simplify quotients using the Laws of Exponents. Laws of Exponents. Simplify. Write the answer without negative exponents. 1. Qotients of Monomils Objetive: To simplif qotients sing the Lws of Eponents. Lws of Eponents m n = m n ( b ) m = m b m ( m ) n = m n n m n m = m n n m = m m m b b = Prtie Problems Simplif. Write the nswer

More information

Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.

Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons. si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)

More information

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter

8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter 8A Length nd perimeter 8E Are of irle 8B Cirumferene of irle 8F Surfe re 8C Are of retngles nd tringles 8G Volume of prisms 8D Are of other qudrilterls 8H Are nd volume onversions SA M PL E Mesurement

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

This conversion box can help you convert units of length. b 3 cm = e 11 cm = b 20 mm = cm. e 156 mm = b 500 cm = e cm = b mm = d 500 mm =

This conversion box can help you convert units of length. b 3 cm = e 11 cm = b 20 mm = cm. e 156 mm = b 500 cm = e cm = b mm = d 500 mm = Units of length,, To onvert fro to, ultiply y 10. This onversion ox n help you onvert units of length. To onvert fro to, divide y 10. 100 100 1 000 10 10 1 000 Convert these lengths to illietres: 0 1 2

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

] dx (3) = [15x] 2 0

] dx (3) = [15x] 2 0 Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve

More information

Introduction to calculus

Introduction to calculus Chpter6 Introdution to lulus Sllus referene: 7., 7.5 Contents: A B C D Limits Finding smptotes using limits Rtes of hnge Clultion of res under urves 6 INTRODUCTION TO CALCULUS (Chpter 6) Clulus is mjor

More information

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1.

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1. Exerise Genertor polynomils of onvolutionl ode, given in binry form, re g, g j g. ) Sketh the enoding iruit. b) Sketh the stte digrm. ) Find the trnsfer funtion T. d) Wht is the minimum free distne of

More information

Exercise sheet 6: Solutions

Exercise sheet 6: Solutions Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd

More information

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using

More information

We use metres to measure length. There are 100 centimetres in a metre. a 6 m = cm b 3 m = cm c 9 m = cm

We use metres to measure length. There are 100 centimetres in a metre. a 6 m = cm b 3 m = cm c 9 m = cm Units of length metres We use metres to mesure length. There re 00 entimetres in metre. 00 m = m Convert these metres to entimetres: 6 m = m 3 m = m 9 m = m 600 300 900 Estimte nd then mesure the length

More information

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

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Sample pages. 9:04 Equations with grouping symbols

Sample pages. 9:04 Equations with grouping symbols Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity

More information

2Linear and UNCORRECTED SAMPLE PAGES. simultaneous equations. Australian curriculum. What you will learn. Chapter 2A 2B 2C 2D 2E 2F 2G 2H 2I

2Linear and UNCORRECTED SAMPLE PAGES. simultaneous equations. Australian curriculum. What you will learn. Chapter 2A 2B 2C 2D 2E 2F 2G 2H 2I A B C D E F G H I J K Chpter Wht you will lern Liner n simultneous equtions Algeri epressions (Consoliting) Simplifying lgeri epressions (Consoliting) Epning lgeri epressions Solving liner equtions Equtions

More information

Momentum and Energy Review

Momentum and Energy Review Momentum n Energy Review Nme: Dte: 1. A 0.0600-kilogrm ll trveling t 60.0 meters per seon hits onrete wll. Wht spee must 0.0100-kilogrm ullet hve in orer to hit the wll with the sme mgnitue of momentum

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

CHENG Chun Chor Litwin The Hong Kong Institute of Education

CHENG Chun Chor Litwin The Hong Kong Institute of Education PE-hing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study- onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

More information

Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions 6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy

More information

1 Find the volume of each solid, correct to one decimal place where necessary. 12 cm 14 m. 25 mm. p c 5 ffiffiffi

1 Find the volume of each solid, correct to one decimal place where necessary. 12 cm 14 m. 25 mm. p c 5 ffiffiffi 1 Find the volume of eh solid, orret to one deiml le where neessry. 8 m 6 m m 14 m 65 m 2 2 m d 7.6 mm 2 m 4 m 4 m 7 m 25 mm Stge 5.3 See Chter 1 See Chter 7 See Chter 9 See Chter 9 See Chter 13 2 Simlify

More information

50 AMC Lectures Problem Book 2 (36) Substitution Method

50 AMC Lectures Problem Book 2 (36) Substitution Method 0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

MATH Final Review

MATH Final Review MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Student Book SERIES. Measurement. Name

Student Book SERIES. Measurement. Name Student Book Nme Series Contents Topi Units of length (pp. 9) metres entimetres metres nd entimetres millimetres perimeter length nd deiml nottion onnet nd lok pply te ompleted Topi Are (pp. 0 5) squre

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

SIMPLE NONLINEAR GRAPHS

SIMPLE NONLINEAR GRAPHS S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h

More information

SOLUTIONS TO ASSIGNMENT NO The given nonrecursive signal processing structure is shown as

SOLUTIONS TO ASSIGNMENT NO The given nonrecursive signal processing structure is shown as SOLUTIONS TO ASSIGNMENT NO.1 3. The given nonreursive signl proessing struture is shown s X 1 1 2 3 4 5 Y 1 2 3 4 5 X 2 There re two ritil pths, one from X 1 to Y nd the other from X 2 to Y. The itertion

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information