APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

Transcription

1 TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex or exponent ² prime, omposite ² ftors, prime ftors, ommon ftors, highest ommon ftor (HCF) ² multiples, lowest ommon multiple (LCM) ² nturl numers N = f0, 1, 2, 3, 4,...g ² integers Z = f..., 2, 1, 0, 1, 2,...g ½ ¾ p ² rtionl numers Q = q j p, q 2 Z, q 6= 0 ² rel numers R = fll numers on the numer lineg THE FUNDAMENTAL THEOREM OF ARITHMETIC Every omposite numer n e written s the prout of prime ftors in extly one wy (ignoring orer). ORDER OF OPERATIONS Brkets Exponents Divisions Multiplitions Aitions Sutrtions R ¾ ¾ s you ome to them s you ome to them SCIENTIFIC NOTATION (STANDARD FORM) 10 k Q Z N where 1 6 <10 n k 2 Z For exmple: ² 2565 is sientifi nottion is 2: ² 3: s eiml is 0: INTERNATIONAL SYSTEM OF UNITS (SI UNITS) You shoul unerstn n e le to use SI units, in prtiulr those for istne, mss, temperture, n time. You shoul e le to: ² perform metri onversions 5:3 kg = 5300 g ² use res n volumes 8 m 5 m =40m 2 3 m 2 m 2 m =12m 3 ² use formule to onvert etween temperture units ² use rtes where we ivie units spee (m s 1 istne (m) ) = time (s) APPROXIMATION AND ESTIMATION Rouning numers: ² if the igit fter the one eing roune is less thn 5, roun own ² if the igit fter the one eing roune is 5 or more, roun up. Signifint figures re ounte from the first non-zero igit from the left. For exmple: 3:413 ¼ 3:41 (to 2 eiml ples) 0: ¼ 0:0346 (to 3 signifint figures) 236:5 ¼ 237 (to the nerest whole numer) A mesurement is urte to 1 2 the sle. of the smllest ivision on An pproximtion is vlue given to numer whih is lose to, ut not equl to, its true vlue. An estimtion is vlue whih is foun y jugement or preition inste of rrying out more urte mesurement. If the ext vlue is V E n the pproximte vlue is V A then: ² error = V A V E VA VE ² perentge error E = 100% V E We usully stte perentge error s the solute perentge error whih is the size of the perentge error, ignoring its sign. LAWS OF ALGEBRA Inex or Exponent Lws ³ n m n = m+n n =, 6= 0 n m = n m n, 6= 0 0 =1, 6= 0 ( m ) n = m n x = 1 n 1 x () n = n n x = x Expnsion n Ftoristion Lws Distriutive lw ( + ) = + FOIL rule ( + )( + ) = Differene of two squres ( + )( ) = 2 2 Perfet squres EQUATIONS AND FORMULAE ( + ) 2 = ( ) 2 = You shoul unerstn the menings of: expression, eqution, inequlity, formul, sustitution, evlute. Liner equtions n e written in the form x+ =0 where x is the vrile n, re onstnts. You shoul e le to solve liner equtions: ² using inverse opertions ² using tehnology. You shoul e le to rerrnge formule to mke nother vrile the sujet. n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition) 6

2 Liner simultneous equtions our when there re two equtions in two unknowns. You shoul e le to solve liner simultneous equtions: ² y sustitution ² y elimintion ² using tehnology Exponentil equtions re equtions where the vrile ppers in n inex or exponent. If we n write the eqution so the se numer is the sme on eh sie, we n then solve y equting inies. If x = k then x = k. If we nnot write the eqution so the se numer is the sme on eh sie, we shoul solve using tehnology. QUADRATIC FACTORISATION ² tke out ny ommon ftors ² look for speil ses: I ifferene of two squres x 2 2 =(x + )(x ) I perfet squres x 2 +2x + 2 =(x + ) 2 x 2 2x + 2 =(x ) 2 ² look for sum n prout type x 2 +(p + q)x + pq =(x + p)(x + q) ² splitting the x-term For x 2 + x + I fin I look for two ftors of whih to I if the ftors re p n q, reple x y px + qx I omplete the ftoristion. QUADRATIC EQUATIONS Qurti equtions hve the form x 2 + x + =0 where 6= 0. The solutions of the eqution re lle its roots. These vlues for x re lso lle the zeros of the qurti expression x 2 + x +. 8 < x = p k if k>0 If x 2 = k then x =0 if k =0 : there re no rel solutions if k<0. You shoul e le to solve qurti equtions: ² using ftoristion n the Null Ftor lw ² using tehnology. SEQUENCES A numer sequene is list of numers whih follow pttern. The numers in sequene re lle its memers or terms. We n efine sequene y: ² listing terms ² using wors ² formul for the generl term or nth term. An rithmeti sequene is sequene in whih eh term iffers from the previous one y the sme fixe numer. fu ng is rithmeti, u n+1 u n = for ll n 2 Z +, where is onstnt lle the ommon ifferene. The generl term of n rithmeti sequene is u n = u 1 +(n 1): An pplition of rithmeti sequenes is simple interest. A sequene is geometri if eh term n e otine from the previous one y multiplying y the sme non-zero onstnt. fu ng is geometri, un+1 = r for ll n 2 Z +, u n where r is onstnt lle the ommon rtio. The generl term of geometri sequene is u n = u 1 r n 1. Applitions of geometri sequenes inlue ompoun interest, n growth n ey prolems. SERIES A series is the ition of the terms of sequene. For the sequene fu n g, the series inluing the first n terms is u 1 + u 2 + u 3 + :::: + u n. The result of this ition is S n = u 1 + u 2 + u 3 + :::: + u n. For n rithmeti series: S n = n 2 (u 1 + u n ) or S n = n 2 (2u 1 +(n 1)) For geometri series: S n = u1(rn 1) r 1 or S n = u1(1 rn ) 1 r SKILL BUILDER - SHORT QUESTIONS 1 Ple the following numers in the pproprite region of the Venn igrm. 2:5, ¼, 3, 8, 0, p Drw numer line to represent the set X = fx j 3 6 x 6 3, x 2 R g Q R Use numer line to lerly represent eh of the following: i x< 1 ii 0 6 x<2 iii x > 2 3 If =2:5, =7 n = 137, lulte 2 +. Give your nswer: i orret to 2 eiml ples ii orret to 3 signifint figures. Write your nswer from iiin sientifi nottion. 4 Goron nees to fertilise the lrge gren lnspe shown in the igrm: Lwn 100 m The ens of this gren lnspe re semi-irulr, n the mile is retngulr. N 25 m 7 n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition)

3 Goron lultes the re of the gren using orret formule, ut pproximtes the vlue of ¼ to one signifint figure, so ¼ ¼ 3. Write own Goron s pproximtion for the re of the gren lnspe. Clulte the perentge error in Goron s pproximtion. Roun your nswer to 2 signifint figures. 5 In the tle elow, inite using Y (yes) or N (no) whih numer sets the following numers re memers of: 6 Evlute N Z Q R ( 2) 3 Y p 2 N 0:65 N 1: Y r 32:76, giving your nswer: 3:95 2:63 orret to 3 eiml ples orret to the nerest whole numer orret to 3 signifint figures in stnr form. 7 The spee of light is pproximtely miles per seon. The istne from Mrs to Erth is pproximtely km. Given tht mile : kilometre =1:1:609, etermine the spee of light in kilometres per minute. Give your nswer to 3 signifint figures. Express your nswer to using sientifi nottion. If light soure on Mrs is ignite, how mny minutes will it e efore it n e seen through telesope on Erth? 8 Consier the rithmeti sequene 120,, 98,,... Determine the vlues of n. Write own the nth term of the sequene. Determine the numer of positive terms in this sequene. 9 Lis invests $ into svings ount whih pys 4:5% p.. interest, ompoune nnully. Clulte the vlue of Lis s investment fter: i 1 yer ii 2 yers. Write own formul whih lultes the vlue of the investment fter n yers. Clulte the vlue of the investment fter 13 yers. Determine the minimum length of time require for the investment to e triple its originl vlue. 10 A mesurement of 5:645 m is roune to 3 signifint figures. i Write own the tul error use y rouning. ii Clulte the perentge error. The speeometer of r res 70 km h 1. It is urte to within 3:2%. i Wht is the mximum possile error? ii Write own the extreme possile vlues for the true spee of the r. 11 The first three terms of sequene re 2, 9, n 16. Write own the next two terms of the sequene. Drw mpping igrm of the first 5 terms. Fin formul for the nth term of the sequene. 12 Write own the first 3 terms of the sequene given y u n = n(n +1). Fin the 15th term. Whih term of this sequene is 600? 13 Complete the following tle y pling tiks in the pproprite oxes. p 4 2 3:75 ¼ 2:33 14 Fin the vlue of 2:34 + N Z Q R orret to the nerest integer orret to 4 eiml ples orret to 3 signifint figures in sientifi nottion. 5:25, giving your nswer: 3:10 7:65 15 The first three terms of n rithmeti sequene re 347, k 166 n 185. Fin the vlue of k. Fin formul for the nth term of the sequene. Whih is the first positive term of the sequene? 16 The sixth term of n rithmeti sequene is 49 n the fifteenth term is 130. Fin the ommon ifferene for this sequene. Fin the first term. How mny terms of this sequene hve vlue less thn 300? 17 A setion of tuing is forme into irle with imeter 4:5 m. Clulte the length of the tuing. Roun your nswer to the nerest whole numer. Clulte the perentge error if the length is roune to the nerest whole numer. 18 The first three terms of geometri sequene re 0:75, 2:25 n 6:75. Fin the ommon rtio. Write own formul for the nth term. Clulte the sum of the first 10 terms. 19 The sum of the first 7 terms of n rithmeti series is 329. The ommon ifferene is 14. Fin the vlue of the first term. Fin n given is the sum of the first n terms of the sequene. 20 One stge of yling re overs 80 km on the flt, n the reminer in the mountins. The ylists n verge 45 km h 1 on the flt n 25 km h 1 in the mountins. Determine the time tken y the ylists to over the flt setion of the ourse. Give your nswer orret to the nerest minute. If the totl time tken for the stge is 3 hours, fin the istne overe in the mountins. n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition) 8

4 21 Consier the rithmeti sequene..., 27, x, 42,... where 27 is the fifth term of the sequene. Clulte the ommon ifferene. Fin the first term of the sequene. Fin the first term of the sequene whih is greter thn The sequene 45, x, 281:25,... is geometri. Fin the ommon rtio r given tht x > 0. Fin the sum of the terms u 5 + u 6 + :::: + u The length of setion of pipe is stte s 4 m. Clui refully mesures the pipe n fins the tul length to e 3:94 m. Write own the size of the error in the stte length. Five suh setions of pipe re joine together. Fin the tul length of the joine pipes. Write own the error for the joine pipes. Clulte the perentge error of the stte length ginst the tul length of the joine pipes. 24 A ruer ll is roppe from height of 5 m. It ounes up to height of 4:5 m on the first oune, then to 4:05 m on the seon oune, n so on. Fin the ommon rtio of the sequene forme y these numers. Clulte the height of the thir oune. How fr hs the ll trvelle vertilly y the time it strikes the floor for the fourth time? 25 List the elements of the following sets: i A = fx j 2 <x<3, x 2 Z g ii B = fprime numers less thn 15g iii C = fx j x 2 =8, x 2 R g Stte whether the following sttements re true or flse: i All rtionl numers re integers. ii 4 p>4+p, p 2 Z iii N = f0, 1, 2, 3,...g footll lus enter the first roun of knokout ompetition. In eh roun, hlf of the prtiipnts re eliminte. How mny lus remin in the seon n in the thir rouns? If there re n rouns, how mny prtiipnts remin in the nth roun? Clulte the numer of rouns neee to etermine winner. 27 The ost of 15 ooks n 8 pens is 209 NZD. The ost of 7 ooks n 3 pens is 96:25 NZD. Write 2 equtions involving n p whih represent this informtion. Solve the equtions simultneously to fin n p. Fin the ost of 10 ooks n 6 pens. 28 The seon term of geometri sequene is 14:5 n the fifth term is 1:8125. Determine the ommon rtio. Fin the vlue of the first term. Fin the sum of the first 5 terms. 29 The popultion of smll town inreses y n verge of 9% per nnum. In 2005, the popultion ws Clulte the size of the popultion in In whih yer will the popultion reh 2500? Fin the rte of inrese tht woul result in the popultion rehing 3200 in Consier the sequene 2k 9, k, k 6,... Determine the two possile vlues of k for whih this sequene is geometri. For wht vlue of k is the sequene rithmeti? 31 Solve, using tehnology: i 12s +17r = 277 5s +11r = 135 ii u = = 202 u 1 The ost of hiring txi inlues flt fee of $ plus $p per kilometre. A 12 kilometre txi rie osts $20, n 22 kilometre journey osts $34. Fin the vlues of n p. 32 Fin the term tht the sequenes u n = 178 4n n u n =7n +57 hve in ommon. A firm s revenue funtion is R = 25n n its ost funtion is C = :5n, where n is the numer of goos proue n sol. Fin the vlue of n suh tht the ost equls the revenue. The sum of the first n nturl numers is equl to n(n +1). 2 For wht vlues of n oes the sum exee 435? 33 The nth term of geometri sequene is given y u n = n 1, 3 where n is positive integer. Determine the vlue of the 7th term of the sequene. For wht vlues of n is u n 6 0:1? Determine the sum of the first 8 terms of the sequene. 34 The perimeter of retngle is 80 m. The with is x m. Write own the vlue of the length, in terms of x. Show tht the re of the retngle is given y the funtion A =40x x 2 m 2 : If the re of the retngle is 375 m 2, fin its imensions. 35 The height of sky roket ove the groun t seons fter firing is given y s = ut 5t 2 m, where u represents the initil spee of the sky roket. If the initil spee is 70 ms 1, fin: the mount of time the roket is in the ir the time the roket is ove 30 m. 36 Kren purhses new moile phone. The monthly ost $C of the phone when use for t minutes per month is C = rt +, where is fixe monthly fee, n r is the ll ost for eh minute of usge. In the first month, Kren pi $19:15 for 23 minutes usge. In the seon month, she pi $15:95 for 15 minutes usge. Fin the vlues of n r. 37 Consier the qurti eqution (x 2 + x + ) =0. Given the solutions of the eqution re x =3 n x = 5, fin the vlues of n. If the y-interept of the grph of f(x) =(x 2 + x + ) is 30, etermine the vlue of. 9 n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition)

5 SKILL BUILDER - LONG QUESTIONS 1 The nth term of n rithmeti sequene is given y u n =4+11n. Fin: i the first 2 terms of the sequene ii the sum of the first 10 terms. The nth term of geometri sequene is given y u n = 4(2:2) n 1. Fin: i the fifth term of the sequene ii the sum of the first 10 terms. Fin the first term for whih the vlue of the geometri sequene is greter thn the vlue of the rithmeti sequene. Wht is the ifferene etween the terms of the two sequenes for this vlue of n? 2 Drw numer line to represent x 2 R, n mrk the integers from 5 to 5. Represent the following on the numer line using pproprite nottion. i x > 4 ii 1 6 x<3 iii 3 6 x 6 0 iv x< 4 Let X = fx j 4 6 x 6 4, x 2 Z g n Y = fy j 4 6 y 6 4, y 2 Z g. On set of oorinte xes, plot the points whih represent the set: i W = f(x, y) j y = 1, x 2 X, y 2 Y g x ii V = f(x, y) j x > 0, y > 0, y x =1, x 2 X, y 2 Y g If x, y 2 Q n x>y, write own n exmple whih mkes x 2 <y 2 true. 3 The fifth term of n rithmeti sequene is 51, n the sum of the first 5 terms is 185. If u 1 is the first term of the sequene, n is the ommon ifferene, write own two equtions in u 1 n whih stisfy the informtion provie. Solve the equtions from to etermine the vlues of u 1 n. Fin the first term in this sequene to exee The sum of the first k terms in this sequene is i Deue tht k stisfies the eqution 7k 2 +39k 7470 = 0. ii Hene, fin k. 4 Whih of the following sttements re flse? Justify your nswers. i f 2, 1, 0, 1, 2g ½fx j x<2, x 2 R g ii f0, 1, 2, 3, 4g ½fx j x 6 5, x 2 Z g iii fx j x 2 + x =2; x 2 Z g = f 1, 1, 2g Let U = fx j 14 6 x<30, x 2 N g, A = fmultiples of 7g, B = fftors of 56g, C = feven numers > 20g. i List the memers of eh of the sets A, B n C ontine in U. ii Represent these sets on 3-irle Venn igrm. p n q re ifferent integers. Whih of the following sttements re flse? Give n exmple to support your eision. i p + q = q + p ii p q = q p iii pq = qp 6:057 5 Evlute 25:32 2:4 p, giving your nswer 5:14 orret to: i five signifint figures ii the nerest tenth iii 1 signifint figure. Three setions of fening re erete. Eh setion hs stte length of 3:60 m, mesure to the nerest tenth of metre. The tul length of eh setion is 3:63 m. i Fin the tul length overe y the three setions of fening. ii Clulte the perentge error etween the tul length n the stte length of the three setions of fening. The three setions of fening in form one sie of squre enlosure. The enlosure will hve onrete floor mm thik. Conrete osts E47:50 per ui metre. i Write own the minimum n mximum possile vlues for the volume of onrete neee for the floor. ii Clulte the ifferene in ost etween the mximum possile volume of onrete n the plnne volume se on the stte length. iii Express the ost ifferene in ii s perentge of the plnne ost. 6 Write own the first three terms of the following sequenes: i u n = (n 1) ii u n+1 = u n +7, u 1 =4. e Whih term o the sequenes in hve in ommon, n wht is the vlue of tht term? Whih of the sequenes in hs 151 s one of its terms? The sum of the first n terms of the sequenes in is the sme. Fin n. When will the sum of the first sequene in exee the sum of the seon sequene y 228? 7 A pool hs two retngulr setions, the swimming pool tht is 180 m eep, n the wing pool tht is 50 m eep. The imensions of these regions re shown in the igrm elow. Show tht the totl re of the surfe of the pool n e foun y the expression 2x 2 +4x 16. Given tht the totl surfe re of the pool is 224 m 2, fin the vlue of x. Clulte the volume of wter (in litres) tht the pool ontins. Over perio of time, 2400 litres of wter evportes from the wing pool. i ii swimming pool 180 m eep (x+8)m wing pool (x-2)m 50 m eep x m Fin the volume of wter in litres whih the wing pool now ontins. Fin the new epth of the wing pool. 8 The first 3 terms of sequene re 56, 28 n 14. i Show tht the sequene is geometri. ii Fin the 8th term of the sequene. iii Fin the sum of the first 8 terms. n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition) 10

6 The thir term of nother geometri sequene is 24:5 n the 5th term is 12:005. All of the terms of this sequene re positive. i Fin the first term n the ommon rtio. ii Write own the generl formul for term of this sequene. The first n terms of the sequene in re lrger thn the orresponing terms for the sequene u n =20 (0:8) n 1. Fin n. For the sequene u n =20 (0:8) n 1, fin the sum of the first: i 30 terms ii 50 terms iii 100 terms. Answer to 3 signifint figures in eh se. Comment on your results. 9 Mish tkes out lon to purhse genertor for his usiness. He orrows $ t 12:5% per nnum, ompoun interest. At the en of eh yer, Mish is require to py $k. i If Mih oes repy $k eh yer, explin why the mount owing on the lon t the en of the first yer is $( :125 k). ii Write own n expression, in terms of k, for the mount owing on the lon t the en of the seon yer. iii At the en of the seon yer, the mount owing on the lon will e $17 131:25. Fin k. Mish pi totl of $ for the genertor. Its vlue epreite fter purhse so tht, t the en of the first yer, it ws only worth $ The vlue fter eh susequent yer elines t the sme rte. i Fin the perentge erese in the vlue of the genertor in the first yer. ii Clulte the vlue of the genertor t the en of the seon yer. iii Write own the formul for the vlue fter n yers. iv Sketh the grph of the vlue of the genertor for the first 6 yers. 10 Consier the rithmeti sequene efine y A n =19n 13. Write own: i the first term ii the ommon ifferene. G n is geometri sequene with 3r term 1920 n 10th term 15. i Clulte the ommon rtio for the sequene G n. ii Show tht G 1 = By onsiering the eqution A n = G n, fin the term whih A n n G n hve in ommon. Let S An represent the sum of the first n terms of A n, n S Gn represent the sum of the first n terms of G n. i Show tht S An = 19 2 n2 7 2 n. ii Write own n expression for S Gn in terms of n. iii Determine the minimum numer of terms require for the sum of the first n terms of A n to exee the sum of the first n terms of G n. 11 Thli orrows $5000 from her nk. Her repyments eh month re $250 plus the interest for tht month. Interest is hrge t 1:2% eh month. Determine the pyments she must mke t the en of the: i first month ii seon month iii thir month. e The monthly pyments form n rithmeti sequene. Fin the first term n ommon ifferene for the sequene. Hene write own the formul to fin the vlue of the nth pyment for this lon. Clulte the size of the pyment t the en of the tenth month. Determine the numer of pyments require for Thli to py off the lon. Clulte the totl mount Thli will py for this lon. 12 A riotive mteril loses 10% of its weight per yer. The weight of the mteril t the strt of the first yer ws 200 g. i Write own the weight of the mteril t the strt of the seon n thir yers. ii The weights t the strt of eh yer form geometri sequene. Write own the ommon rtio of the sequene. iii Fin the weight of the mteril t the strt of the 6th yer. iv Sketh the grph of the nnul weight of the mteril for the first 6 yers. v At the strt of whih yer will the weight of the mteril fll elow 20 g? TOPIC 3: SETS The weight of seon smple of riotive mteril erese from 120 gto49:152 g fter 6 yers. Fin the nnul perentge rte of erese. A set is olletion of numers or ojets. Every numer or ojet in set is lle n element or memer of the set. The empty set fgor? ontins no elements. The numer of elements in set A is n(a). A set whih hs finite numer of elements is lle finite set. An infinite set hs infinitely mny elements. The universl set U is the set of ll elements uner onsiertion. P is suset of Q if every element of P is lso n element of Q. P is proper suset of Q if P is suset of Q ut is not equl to Q. The intersetion of P n Q onsists of ll elements whih re in oth P n Q. The union of P n Q onsists of ll elements whih re in P or Q or oth. P n Q re isjoint or mutully exlusive if they hve no elements in ommon. The omplement of P, enote P 0, is the set of ll elements of U whih re not in P. VENN DIAGRAMS SETS, LOGIC AND PROBABILITY P µ Q P ½ Q P \ Q P [ Q P \ P 0 =? P [ P 0 = U A Venn igrm onsists of universl set U represente y retngle. Sets within the universl set re usully represente y irles. 11 n Mthemtil Stuies SL Exm Preprtion & Prtie Guie (2 eition)