Investigtion n 34 moelling questions Contents: A Investigtion questions B Moelling questions Mny of the questions in Chpters 33 n 34 re pte from pst exmintion ppers for IGCSE Mthemtis 0580 by permission of the University of Cmbrige Lol Exmintions Synite. The 0580 ourse is ifferent syllbus from tht followe by stuents of the 0607 ourse, but hs mny fetures in ommon. These questions re ertinly pproprite for prtising mthemtil tehniques n pplitions relevnt to the 0607 urriulum, but o not neessrily represent the style of question tht will be enountere on the 0607 exmintion ppers. Tehers re referre to the speimen ppers of the 0607 syllbus for more representtive group of questions. The University of Cmbrige Lol Exmintions Synite bers no responsibility for the exmple nswers to questions tken from its pst question ppers whih re ontine in this publition. A INVESTIGATION QUESTIONS 1 It is given tht 1 2 +2 2 +3 2 +4 2 +5 2 + :::::: + n 2 n(n + 1)(2n +1) = k where k 2 Z : If n =1, the LHS is 1 2. If n =2, the LHS is 1 2 +2 2. Use n =1 to fin the vlue of k, n hek tht you get the sme vlue of k for n =2 n n =3. b Use the given formul to fin the vlue of 1 2 +2 2 +3 2 +4 2 + :::::: + 100 2. Notie tht 2 2 +4 2 +6 2 =(2 1) 2 +(2 2) 2 +(2 3) 2 =2 2 1 2 +2 2 2 2 +2 2 3 2 =2 2 (1 2 +2 2 +3 2 ) Hene, i fin m if 2 2 +4 2 +6 2 +8 2 + :::::: + 100 2 =2 2 (1 2 +2 2 +3 2 + :::::: + m 2 ) ii fin the vlue of 2 2 +4 2 +6 2 +8 2 + :::::: + 100 2 : Use b n iito fin the vlue of 1 2 +3 2 +5 2 +7 2 + :::::: +99 2 : e Use some of the previous nswers to fin the vlue of: 1 2 2 2 +3 2 4 2 +5 2 6 2 + :::::: +99 2 100 2.
662 Investigtion n moelling questions (Chpter 34) 2 A frmer mkes sheep pen in the shpe of qurilterl from four piees of fening. Eh sie of the qurilterl is 5 metres long n one of the ngles is 60 o. b e Using sle of 1 to 100, mke n urte rwing of the qurilterl. Mrk in its xes of symmetry with broken lines n esribe how they ut eh other. Wht is the speil geometril nme of this shpe? Clulte the re enlose by the sheep pen, giving your nswer in squre metres. By hnging the ngles (but leving the lengths of the sies unhnge), the re enlose by the sheep pen n be vrie. Wht is the gretest possible re tht n be enlose? Justify your nswer. 3 Throughout this question, remember tht 1 is not prime number. Fin prime number whih n be written s the sum of two prime numbers. b Consier the sttement All even numbers greter thn 15 n be written s the sum of two ifferent prime numbers in t lest two ifferent wys. For exmple, 20 = 3 + 17 = 7 + 13: i Show tht the bove sttement is true for 16. ii Fin number between 30 n 50 whih shows tht the sttement is flse. Show tht 16 n be written s the sum of three ifferent prime numbers. Consier the sttement All o numbers greter thn 3 n be written s the sum of two prime numbers. Is this sttement true or flse? Justify your nswer. 4 Apte from June 1989, Pper 4 A firm whih mnuftures golf blls is experimenting with the pkging 12.9 m of its prout. 3 golf blls, eh of rius 2:15 entimetres, re pkge in retngulr box, rosssetion 4.3 m of whih is shown in the igrm longsie. The box is 12:9 entimetres long, 4:3 entimetres wie n 4:3 entimetres high. Given tht the volume of sphere of rius r is 4 3 ¼r3, lulte the mount of spe within the box whih is unfille. b The mrketing eprtment suggests tht n equilterl tringulr box of sie 11:75 entimetres n height 4:3 entimetres might be more ttrtive. The igrms show pln n sie view of the new box. 4.3 m 11.75 m 11.75 m Clulte the mount of spe within this new box whih is unfille. Give your nswer to n b s perentges of the pity of eh ontiner. Design box of your own whih gives smller perentge of unfille spe.
Investigtion n moelling questions (Chpter 34) 663 5 Apte from June 1989, Pper 6 Consier the figures P to T: Q R P 4 6 8 S 3 2 8 T 6 6 4 12 6 b All five figures hve something importnt in ommon. Wht is it? Clulte the re of regulr hexgon (H) of sie 4 entimetres. Using the letters P, Q, R, S, T n H, list the res in orer of size, strting with the smllest. Explin ny onlusions you rrive t. 6 June 1988, Speimen Pper 6 1m 2m 3m The igrm shows 3 squres, the sies of whih re 1 m, 2 m n 3 m respetively. Eh of the smll squres on the igrm hs sie of length 1 m n lternte squres re oloure blk n white. The number of smll squres of eh olour use is shown in the following tble. Copy n omplete the tble. Length of sie of given squre L 1 2 3 4 5 Number of blk squres B 1 2 5 Number of white squres W 0 2 4 Totl number of squres T 1 4 9 b i How mny smll white squres will there be when squre of sie 11 m is rwn? ii Fin the length of the sie of squre when 1681 smll blk n white squres re neee to over it. Write own formul onneting T n L. Write own formul onneting T n B when i B is n even number ii B is n o number.
664 Investigtion n moelling questions (Chpter 34) 7 Apte from June 1989, Pper 6 Copy n omplete the following two sets of lultions. 1 = 1+2 = 1+2+3 = 1+2+3+4 = 1 3 = 1 3 +2 3 = 1 3 +2 3 +3 3 = 1 3 +2 3 +3 3 +4 3 = b How re the two sets of results relte? Fin the vlue of 1 3 +2 3 +3 3 +4 3 +5 3 +6 3 +7 3 +8 3 +9 3. Given tht the sum of the first 25 numbers, 1 + 2 + 3 +:::::: + 25, is 325, fin the vlue of 1 3 +2 3 +3 3 + :::::: +25 3. e 1+2+3+4+5+:::::: + n = n 2 + bn. Fin the vlues of n b, n test your nswers. f Fin the vlue of 1 3 +2 3 +3 3 +4 3 + :::::: + 250 3 : 8 June 1988, Pper 6 row 1 The igrm shows the first eight rows of row 2 ontinuing pttern of blk n white row 3 tringles. row 4 row 5 row 6 row 7 Fin formul for eh of the following: i the number of tringles in the nth row row 8 ii the totl number of tringles in the first n rows iii the totl number of white tringles in the first n rows iv the totl number of blk tringles in the first n rows. b Show lgebrilly tht your nswer to iiis the sum of your nswers to iii n iv. 9 Nov 2002, Pper 4 stnr h A h B h C h r r r r Srh investigtes ylinril plnt pots. The stnr pot hs bse rius r m n height h m. Pot A hs rius 3r n height h. Pot B hs rius r n height 3h. Pot C hs rius 3r n height 3h. i Write own the volumes of pots A, B n C in terms of ¼, r n h. ii Fin in its lowest terms the rtio of the volumes of A : B : C. iii Whih one of the pots A, B or C is mthemtilly similr to the stnr pot? Explin your nswer. iv The surfe re of the stnr pot is S m 2. Write own in terms of S the surfe re of the similr pot. b Srh buys ylinril plnt pot with rius 15 m n height 20 m. She wnts to pint its outsie surfe (bse n urve surfe re). i Clulte the re she wnts to pint. ii Srh buys tin of pint whih will over 30 m 2. How mny plnt pots of this size oul be ompletely pinte on their outsie surfes using this tin of pint?
Investigtion n moelling questions (Chpter 34) 665 10 Nov 2002, Pper 4 Write own the 10th term n the nth term of the following sequenes. i 1, 2, 3, 4, 5,...,..., ii 7, 8, 9, 10, 11,...,..., iii 8, 10, 12, 14, 16,...,... b Consier the sequene 1(8 7), 2(10 8), 3(12 9), 4(14 10),...,.... i Write own the next term n the 10th term of this sequene in the form (b ) where, b n re integers. ii Write own the nth term in the form (b ) n then simplify your nswer. 11 Nov 2000, Pper 4 A teher sks four stuents to write own n expression using eh of the integers 1, 2, 3 n n extly one. Ahme s expression ws (3n +1) 2. Bumni s expression ws (2n +1) 3. Cesr s expression ws (2n) 3+1. Dn s expression ws (3 + 1) 2n. The vlue of eh expression hs been worke out for n =1 n put in the tble below. Copy n omplete this tble, giving the vlues for eh stuent s expression for n =2, 0, 1 n 2. n =2 n =1 n =0 n = 1 n = 2 Ahme 16 Bumni 27 Cesr 16 Dn 16 b Whose expression will lwys give the gretest vlue i if n< 2 ii if n>2? Cesr s expression (2n) 3+1 n be written s n b n Dn s expression (3 + 1) 2n n be written s n. Fin the vlues of, b n. Fin ny expression, using 1, 2, 3 n n extly one, whih will lwys be greter thn 1 for ny vlue of n. 12 Apte from Nov 1997, Pper 4 A tin of soup is 11 entimetres high n hs imeter of 8 entimetres (Digrm 1). Clulte the volume of the tin. 11 m b The tins re pke tightly in boxes of 12, seen from bove in Digrm 2. The height of eh box is 11 entimetres. i Write own the length n the with 8m of the box. Digrm 1 Digrm 2 ii Clulte the perentge of the volume of the box whih is not oupie by the tins. A shopkeeper sells the tins of soup for $0:60 eh. By oing this he mkes profit of 25% on the ost prie. Clulte the ost prie of i one tin of soup ii box of 12 tins. The shopkeeper tries to inrese sles by offering box of 12 tins for $6:49. At this prie: i how muh oes ustomer sve by buying box of 12 tins ii wht perentge profit oes the shopkeeper mke on eh box of 12 tins?
666 Investigtion n moelling questions (Chpter 34) 13 Nov 1997, Pper 4 A Pythgoren triple is set of three whole numbers tht oul be the lengths of the three sies of right-ngle tringle. Show tht f5, 12, 13g is Pythgoren triple. b Two of the numbers in Pythgoren triple re 24 n 25. Fin the thir number. The lrgest number in Pythgoren triple is x n one of the other numbers is x 2. i If the thir number is y, show tht y = p 4x 4: ii If x =50, fin the other two numbers in the triple. iii If x = 101, fin the other two numbers in the triple. iv Fin two other Pythgoren triples in the form fy, x 2, xg, where x < 40. Remember tht ll three numbers must be whole numbers. 14 15 Apte from June 1991, Pper 4 Show tht i 2 1 2 1 2 3 =21 2 +12 3 ii 1 3 4 2 1 3 =13 4 +21 3 iii 2 1 5 1 5 6 =21 5 +15 6 b Write 2 1 2 n 1 2 3 in the form 1+ x y n repet for 1 3 4 n 2 1 3. From your observtions in b, fin nother sttement like those in whih is true. Write own generlistion of wht you hve isovere n prove it lgebrilly. Apte from June 1997, Pper 4 Mri thinks of 3 possible svings shemes for her bby son. Sheme A: sve $10 on his 1st birthy, $20 on his 2n birthy, $30 on his 3r birthy, $40 on his 4th birthy,... Sheme B: sve $1 on his 1st birthy, $2 on his 2n birthy, $4 on his 3r birthy, $8 on his 4th birthy,... Sheme C: sve $1 on his 1st birthy, $4 on his 2n birthy, $9 on his 3r birthy, $16 on his 4th birthy,... She puts these ies in tble. Sheme/Birthy 1st 2n 3r 4th A $10 $20 $30 $40 B $1 $2 $4 $8 C $1 $4 $9 $16 b Write own, for eh of the Shemes A, B n C, the mount to be sve on i his 7th birthy ii his nth birthy. The formule for the totl mount sve up to n inluing his nth birthy re s follows. Sheme A: totl =$5n(n +1) Sheme B: totl = $(2 n 1) Sheme C: n(n + 1)(2n +1) totl =$ 6 i For eh of the shemes A, B n C, fin the totl mount sve up to n inluing his 10th birthy. ii Whih sheme gives the smllest totl mount of svings up to n inluing his 18th birthy? iii Fin the birthy when the sheme you hve selete in biifirst gives the smllest totl mount of svings.
Investigtion n moelling questions (Chpter 34) 667 16 Apte from June 1990, Pper 4 b Work out: i 26 93 n 62 39 ii 36 42 n 63 24 Fin one other pir of multiplitions with the sme property. Explin why every two igit number n be written in the form 10 + b where, b 2 Z +. Wht n be eue from the eqution (10m + n)(10r + s) = (10n + m)(10s + r)? 17 Apte from Nov 1992, Pper 4 n 1 2 3 4 n 2 1 9 n 4 1 81 Copy n omplete the tble of vlues bove. b In the tble below, p =1 2 +2 2 q =1 2 +2 2 +3 2 +4 2 r = 3(2 2 ) + 3(2) 1 s = 3(3 2 ) + 3(3) 1 t =1 4 +2 4 +3 4 u =1 4 +2 4 +3 4 +4 4 Clulte the vlues of p, q, r, s, t n u. n 1 2 3 4... 20 Row X 1 p 14 q... Row Y 5 r s 59... Row Z 1 17 t u... For the first four vlues of n in the tble, onsier the (Row X vlue) (Row Y vlue) n the Row Z vlue. Fin the formul whih onnets Row X n Row Y with Row Z. i The vlue in Row X for n =20 n be foun by putting n =20 into the formul n(n + 1)(2n +1) X =. Fin this vlue of X. 6 ii The vlue in Row Y for n =20 n be foun by putting n =20 into the formul Y =3n 2 +3n 1. Fin this vlue of Y extly. e Use your nswers to n to fin the ext vlue of 1 4 +2 4 +3 4 + :::::: +19 4 +20 4. 18 Apte from Nov 1992, Pper 4 One entrl irle, of rius 3 m n entre O, is ompletely x A surroune by other irles whih touh it n touh eh M B other, s shown in the igrm. These outer irles re ientil to eh other. 3m If the rius of eh outer irle is x m, write own the following lengths in terms of x: i OA ii OB iii AB. O b On one osion there re 6 irles ompletely surrouning the entrl irle. i Clulte ngle AOB ii Wht speil type of tringle is AOB in this se? iii Use your previous nswers to fin x.
668 Investigtion n moelling questions (Chpter 34) On nother osion there re 20 smll irles ompletely surrouning the entrl irle. i Clulte ngle AOB. ii M is the mipoint of AB. Consier the tringle MAO n write own the eqution involving x n trigonometri rtio. iii Solve this eqution to fin x orret to 2 eiml ples. Exten the result to n smll irles n test your result when n =20. 19 Apte from Nov 1996, Pper 4 As the prout of its prime ftors, 1080 = 2 2 2 3 3 3 5. Write 135, 210 n 1120 s the prout of their prime ftors. b Copy this gri. =1 b = = The nine igits 1, 2, 3, 4, 5, 6, 7, 8, 9 re to be ple in your gri in suh wy tht the following four sttements re ll true. = e = f = b e = 135 b e f = 1080 e g h = 210 e f h i = 1120 The igits 1 n 8 hve lrey been ple for you. Use your nswers to to nswer the following questions. g = h = i =8 i Whih is the only igit, other thn 1, tht is ftor of 135, 1080, 210 n 1120? ii Whih is the only letter to pper in ll four sttements bove? iii 7 is ftor of only two of the numbers 135, 1080, 210 n 1120. Whih two? Now omplete your gri. 20 June 1994, Pper 4 Clulte the grient of the stright line joining the points (3, 18) n (3:5, 24:5): b The igrm shows prt of the urve y =2x 2. i P is the point (, ). Write own in terms of. ii Q is the point ( + h, e). Write own e in terms of n h. Q ( h, e) iii Write own the length of PR. Fin n expression for the length of QR in terms of n h, n simplify your nswer. R P (, ) iv Show tht the grient of the line PQ is 4 +2h: v If P is the point (3, 18) n Q is the point (3:5, 24:5), stte the vlue of n the vlue of h, n use these vlues to show tht bivgives the sme nswer s. vi If P is the point (3, 18) n Q is the point (3:1, 19:22), stte the vlue of n the vlue of h, n use bivto fin the grient of the line PQ. vii If P is the point (3, 18) n Q gets loser n loser to P, wht hppens to the vlue of h b to the vlue of the grient of the line PQ? Clik on the ion to obtin 4 more investigtion questions. PRINTABLE QUESTIONS
Investigtion n moelling questions (Chpter 34) 669 B MODELLING QUESTIONS 1 June 1990, Pper 4 A grener hs 357 tulip bulbs to plnt. If she plnte retngle of 15 rows, with 23 bulbs in eh row, how mny bulbs woul be left over? b How mny bulbs woul there be in the lrgest squre tht she oul plnt? i If she plnts x rows, with y bulbs in eh row, write own formul for the number of bulbs left over. ii If 10 <x<20 n y>20, fin the vlue of x n the vlue of y suh tht no bulbs re left over. 2 The epth of wter ( metres) in hrbour is given by the formul = + b sin(t) o where, b n re onstnts, n t is the time in hours fter minight. It is known tht both b n re non-zero n 20 <<35. The following tble gives epths t prtiulr times: t minight noon 1300 1400 5 5 7 8:46 b e f Using the first three piees of informtion in the tble, eue the vlues of, bn. Chek tht the formul is orret by substituting the fourth piee of informtion. Fin the epth of wter t 10 m. Wht is the gretest epth of wter in the hrbour? At wht times of y is the epth of wter gretest? Wht is the lest epth of wter in the hrbour? 3 Apte from Nov 1994, Pper 4 In hemil retion, the mss M grms of hemil is given by the formul M = 160 t where is onstnt integer n t is the time (in minutes) fter the strt. A tble of vlues for t n M is given below. t 0 1 2 3 4 5 6 7 M 160 80 40 20 q 5 r 1:25 i Fin the vlue of. ii Fin the vlues of q n r. iii Sketh the grph of M ginst t. iv Drw n urte grph n to it tngent t t =2. Estimte the rte of hnge in the mss fter 2 minutes. b The other hemil in the sme retion hs mss m grms where m = 160 M: i On the sme grph s in iii, sketh the grph of m ginst t. ii For wht vlue of t o the hemils hve equl mss? iii Stte single trnsformtion whih woul give the grph for m from the grph for M.
670 Investigtion n moelling questions (Chpter 34) 4 The surge moel hs form y = t 2 bt where n b re onstnts n t is the time, t > 0: This moel hs extensive use in the stuy of meil oses where there is n initil rpi inrese to mximum n then slow ey to zero. Use grphis lultor to grph the moel (on the sme set of xes) for: i =10, b =2 ii =15, b =3 b The effet of pin killing injetion t hours fter Time (t hours) 0 2 4 6 12 it hs been given is shown in the following tble: The effet E follows surge moel of the form Effet (E units) 0 25 25 r s E = t 2 bt. i By using two of the points of this tble, fin the vlues of n b. ii Hene, fin the vlues of r n s in the tble. iii Use your lultor to fin the mximum effet of the injetion n when it ours. iv It is known tht surgil opertions n only tke ple when the effetiveness is more thn 15 units. Between wht two times n n opertion tke ple? 5 The logisti moel hs form y = 2bt 2 bt + where t is the time, t > 0. The logisti moel is useful in esribing limite growth problems, i.e., when the y vrible nnot grow beyon prtiulr vlue for some reson. Use tehnology to help grph the logisti moel for =3, b = 1 2 n =2. (Use the winow 1 6 x 6 40, 1 6 y 6 5.) b Wht feture of the grph inites limiting vlue? Wht is the limiting vlue? Bteri is present in rton of milk n fter t hours the bteri (B units) ws reore s follows: t 0 1 2 B(t) 10 12:70 15:03 It is known tht =1. i ii iii iv Use the first two sets of t to fin n b, n hene etermine the logisti moel. Use the moel foun in i to hek the thir t set. Wht is the limiting quntity of bteri for this moel? In the generl moel y = 2bt 2 bt, explin why the limiting quntity hs vlue. + 6 June 1991, Pper 4 A frmer keeps x gots n y ows. Eh got osts $2 y to fee n eh ow osts $4 y to fee. The frmer n only ffor to spen $32 y on niml foo. Show tht x +2y 6 16: b The frmer hs room for no more thn 12 nimls. He wnts to keep t lest 6 gots n t lest 3 ows. Write own three more inequlities. Using sle of 1 m to represent 1 unit on eh xis, represent the four inequlities on grph. One possible ombintion whih stisfies ll the inequlities is 6 gots n 4 ows. Write own ll the other possible ombintions. e If he mkes profit of $50 on eh got n $80 on eh ow, whih ombintion will give him the gretest profit? Clulte the profit in this se.
Investigtion n moelling questions (Chpter 34) 671 7 My 2001, Pper 4 A B C D E F G H b Write own whih one of the sketh grphs bove lbelle A to H shows eh of the following: i spee-time grph for r whih strts from the rest n hs onstnt elertion ii y = x 3 +1 iii y is inversely proportionl to x 2 iv the sum of x n y is onstnt v y = os x for 0 o 6 x 6 90 o vi istne-time grph when the spee is eresing. Write own n eqution for sketh grph D if it psses through the points (1, 1) n (2, 4) n, when extene to the left, hs line symmetry bout the vertil xis. 8 June 1994, Pper 4 In shool grening projet, tehers n stuents rry erth to vegetble plot. A teher n rry 24 kg n stuent n rry 20 kg. Eh person mkes one trip. Altogether t lest 240 kg of erth must be rrie. There re x tehers n y stuents. Show tht 6x +5y > 60: b There must not be more thn 13 people rrying erth, n there must be t lest 4 tehers n t lest 3 stuents. Write own three more inequlities in x n/or y. i Drw x n y xes from 0 to 14, using 1 m to represent 1 unit of x n y. ii On your gri, represent the informtion in prts n b. She the unwnte regions. From your grph, fin i the lest number of people require ii the gretest mount of erth whih n be rrie.
672 Investigtion n moelling questions (Chpter 34) 9 Apte from Nov 1993, Pper 4 Ann throws bll from point A, one metre bove the groun, towrs wll. The bll trvels long the rrowe pth from A to B, given by the eqution y = + bx x 2 where the x-xis represents the horizontl groun n the y-xis represents the wll. The bll psses through the point T(2, 4) n hits the wll 4 m bove O. Fin the vlues of n b. b Show tht the x-oorinte of A stisfies the eqution x 2 2x 3=0. Fin the x-oorinte of A. B wll O y G T (, ) The bll rebouns from the wll t B to the groun t G. The eqution of the pth B to G is y = 2x x 2. i Fin the vlue of. ii Fin the x-oorinte of G orret to 2 eiml ples. e How fr from the wll is the bll when it is i 0:5 m ii 3 m bove the groun? f Fin the gretest height of the bll uring its motion. 1m A x 10 My 2005, Pper 4 A txi ompny hs SUPER txis n MINI txis. One morning group of 45 people nees txis. For this group the txi ompny uses x SUPER txis n y MINI txis. A SUPER txi n rry 5 pssengers n MINI txi n rry 3 pssengers. So 5x +3y > 45: The txi ompny hs 12 txis. Write own nother inequlity in x n y to show this informtion. b The txi ompny lwys uses t lest 4 MINI txis. Write own n inequlity in y to show this. Drw x n y xes from 0 to 15 using 1 m to represent 1 unit on eh xis. Drw three lines on your grph to show the inequlity 5x +3y > 45 n the inequlities from prts n b. She the unwnte regions. e The ost to the txi ompny of using SUPER txi is $20 n the ost of using MINI txi is $10. The txi ompny wnts to fin the hepest wy of proviing SUPER n MINI txis for this group of people. Fin the two wys in whih this n be one. f The txi ompny eies to use 11 txis for this group. i The txi ompny hrges $30 for the use of eh SUPER txi n $16 for the use of eh MINI txi. Fin the two possible totl hrges. ii Fin the lrgest possible profit the ompny n mke, using 11 txis.