A LEVEL TOPIC REVIEW. factor and remainder theorems

Transcription

1 A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of ( mrks) +. ( mrks). Use lgebric division to find the quotient nd reminder (if ny) when : ) is divided by ( + ). ( mrks) + is divided by ( + ). ( mrks) + is divided by ( ). ( mrks). In the following equtions one solution is given. Fctorise ech epression nd hence solve the eqution: ) 6 + 6=; = is one solution. ( mrks) = ; = is one solution. ( mrks). If ( + ) is fctor of + + 6, find the vlue of. ( mrks). Find the reminder when : ) + is divided by ( ). ( mrks) + + is divided by ( ). ( mrks) 6. f( ) 7 = +. Show tht ( ) is not fctor of f( ), nd find the reminder when f( ) is divided by ( + ). ( mrks) 7. The epression f( ) = + + b + is divisible by ( + ) but leves reminder of when divided by ( + ) ) Find the vlues of the constnts nd b. (6 mrks) Solve the eqution f( ) =. ( mrks) 8. When is divided by ( ) the reminder is three times the reminder when divided by ( ). Find the vlue of. (6 mrks)

2 A LEVEL TOPIC REVIEW : ANSWERS unit C fctor nd reminder theorems. ) f () = + = f( ) = = f = + = quotient : ( + )( ) =. ) quotient : + + reminder : reminder : 7 quotient : + + reminder :. ) ( )( + 6) = ( )( ) = =, = (+ )( ) = =, = 7. ) f( ) = 8+ b+ = f( ) = + b+ = =, b= 7 ( )( 6 ) + + = ( + )( )( ) = =, =, = A 8. f() = = + f() = = + + = ( + ) =. f( ) = = =. ) f () = + = f = + + = M 6. f () = 6 + = 9 f = + + = M 7

3 A LEVEL TOPIC REVIEW unit C coordinte geometry of the circle. Write down the centres nd rdii of the following circles. ) + y = ( ) y 9 + = ( ) ( y ) = (6 mrks). Find the equtions of the following circles with the following properties. ) centre (, ), rdius 6. ( mrks) centre (, ), rdius. ( mrks) centre (, ), pssing through the point (, ). ( mrks) d) the points (, 8) nd (, ) re opposite ends of the dimeter. ( mrks). ) Show tht the point P (, ) lies on the circle ( ) ( y ) 7 + =. ( mrk) Find the eqution of the tngent to the circle t the P. giving your nswer in the form + by + c =, where, b nd c re integers. ( mrks). The circle ( ) + ( y ) = hs centre C nd rdius r. ) Write down the coordintes of C nd the vlue of r. ( mrks) A tngent to the circle psses through the point P (, 7) nd touches the circle t T Drw sketch of the circle showing clerly the positions of P, T nd C. ( mrks) Hence clculte the length PT. ( mrks). The points A (, ), B (7, 9) nd C (, ) lie on circle. ) Find the grdients of AC nd BC. ( mrks) Eplin wht your nswer to ) tells you bout the line AB. ( mrk) Find the eqution of the circle pssing through A, B nd C. ( mrks) 6. ) Find the coordintes of the two points A nd B where the line y = intersects with the circle ( ) + y =. ( mrks) Sketch the circle nd the line showing clerly the position of A nd B. ( mrks) M is the mid point of AB. Write down the eqution of the line pssing through M nd the centre of the circle. ( mrks) 7. A circle hs eqution ( ) + ( y+ ) = 6. Find the distnce between the point P (6, 9) nd the nerest point on the circle to P. ( mrks)

4 A LEVEL TOPIC REVIEW : ANSWERS unit C coordinte geometry of the circle. ) (, ); (, ); 7 (-, );.. ) ( ) + ( y ) = 6 ( ) ( ) + y+ = r = ( ) + ( ) = ( ) ( y ) + = d) centre (.,.) B r = (.) + (8.) =. (.) + ( y.) =.. ) ( ) + ( ) = 7 B grdient of rdius = = y = ( ) y+ 7=. ) (, ); G P C CP = + = PT = = T. ) AC : = BC 9 : = 7 C is right ngle, nd so AB must be dimeter ( ngle in semicircle ) centre (, 7) r = (7 ) + (9 7) = ( ) + ( y 7) = 6. ) ( ) + ( ) = 8 8= ( )(+ ) = (, ); (.,.8) y G A M (.8,.) B centre (, ) B. grdient = =.8 y = ( ) y= + 7. rdius = centre C (, ) CP = (6 ) + (9 ) = distnce = = 9 B

5 A LEVEL TOPIC REVIEW unit C geometric series. A geometric series hs first term of nd common rtio of. ) Write down the first four terms of the series nd the n th term. ( mrks) Clculte the sum of the first terms. ( mrks). The th term of geometric sequence is nd the 9 th term is 8. All the terms re positive. ) Find the common rtio. ( mrks) Find the first term. ( mrks) Find the sum of the first terms. ( mrks). The first three terms of geometric progression re 6, 6 nd. Find two possible vlues of nd the corresponding common rtios of the sequence. (8 mrks) is geometric series ) Find the vlue of. ( mrks) Find the number of terms of the series. ( mrk) Find the sum of the series. ( mrks). Evlute r r= ( mrks) 6. Find the sum of the following infinite geometric series. ) ( mrks) , leving your nswer in the simplest surd form. ( mrks) 7. A mn invests in svings ccount on Jnury st every yer, strting in. The ccount pys % interest on the st December ech yer. ) How much money does he hve in his ccount (i) On st December. (ii) On st December. ( mrks) ( mrks) Write down geometric series, the sum of which gives the mount of money in his ccount on st December. Find the sum of this series. ( mrks) After how mny yers will the ccount first eceed? ( mrks) A LEVEL TOPIC REVIEW : ANSWERS

6 unit C geometric series. ), 6,, n ( ) = 7. ) r =. 8 r = 8 8 r = = 8 r = = = ( ) = 76 6 r = = 6 6 (6 ) = ( )( 6) = + + = ( + 6)( ) = = 6 r = = 8 6 = r = = 9. ) = 9 = = 968 terms ( ).. =... = 8 =, r = = ) (i). = (ii) +. = ( ).. ( n ).. = 7. > (. n ) > n log. > log.9 n >... n = (or tril nd improvement M). = r =, 6. ) = =, r =

7 A LEVEL TOPIC REVIEW unit C binomil theorem. Use the binomil theorem to epnd: ) ( + ) ( 7) ( mrks) ( mrks) ( mrks). ) Epnd ( +. ( mrks) Hence write down the epnsion of (. ( mrk) Hence simplify ( + ) ( ), giving your nswer in the form b. ( mrks). ) Epnd ( ) 9 in scending powers of up to nd including the term in. ( mrks) 9 Use your epnsion to find n pproimtion to.98, correct to d.p. ( mrks). Show tht the first three terms of the epnsion of ( ) ( ) re + +. (7 mrks). When ( + ) n is epnded, the first three terms re + +. Find the vlues of nd n. 8 (8 mrks) 6. Find the coefficient of the term in 8 in the epnsion of 6 (7 + ). ( mrks) 7. ) Epnd ( ) +. Hence write down the epnsion of ( ) ( mrks). ( mrk) Hence solve the eqution ( + ) + ( ) = 6. ( mrks)

8 A LEVEL TOPIC REVIEW : ANSWERS unit C binomil theorem. ) M M M ) + b+ 6 b + b + b. ). B b+ 6 b b + b 8b+ 8b =. B ( ) = ( ) 8 + = ( ) ( ) = + + M n = nn ( ) = 8 nn ( ) = n 8 6( n ) = n n = 6 = ( ) 6. ( ) 7. ) 9 8 = 9 M = 6 6 B + = + 8 = ( )( ) + = = ±

9 A LEVEL TOPIC REVIEW unit C trigonometry. Convert the following ngles, which re given in rdins, to degrees: ) π π π d) π ( mrks). Epress the following ngles in rdins, giving ech nswer in terms of π : ) 9 6 d) ( mrks). A sector AOB is formed from circle, centre O, rdius cm where ngle AOB = 6 π. ) Clculte the length of the rc. ( mrks) If the chord AB is drwn, clculte the re of the segment formed. ( mrks). ) Given tht B is obtuse, find the missing lengths nd ngles of this tringle. (6 mrks) Clculte the re of the tringle. ( mrks). Solve the following equtions for vlues within the given rnge: A 6 cm ) sin =., 8 8 ( mrks) cos =, π ( mrks) tn( + ) =, d) ( ) cos + π =, π π B cm ( mrks) ( mrks) C 6. Solve the following equtions for 6, giving your nswers correct to the nerest degree: ) sin sin = ( mrks) c os sin = ( mrks) + t n = tn ( mrks) 7. Prove the following identities: ) tn + tn sin cos ( mrks) (sin + cos ) = sin cos ( mrks)

10 A LEVEL TOPIC REVIEW : ANSWERS unit C trigonometry. ) 8 B B 6 B d) 6 B 6. ) sin (sin ) = sin = =, 8, 6 sin = =, ( ) cos cos =. ) π B. ) π B 6 π B d) π. ) π =.9 cm 6 sector = π = π 6 tringle = sin π = 6 segment = π = 6.7 cm sin B sin = 6 B =.8 C = 8.8 = 7. c = cos 7. c =. cm 6 sin7. =.87 cm 7. ) cos cos = (cos + )(cos ) = cos = =, cos = =, 6 tn tn + = tn + tn = (tn + )(tn ) = tn = = 7, 97 tn = =, sin cos + cos sin sin + cos sin cos sin cos sin + sin cos + cos sin cos. ) = = 8 = = π = π π = π 7 + =,,,8 d) = 7,, + π = π, π, π = π, = π, = π 6

11 A LEVEL TOPIC REVIEW unit C eponentils nd logrithms. Sketch, on the sme set of es, the grphs of : ) y = y = y = d) ( ) y = ( mrks). Evlute: ) log log log d) log e) log log ( mrks). Epress s single logrithm: ) log + log log log log + log log 6 d) log 6 + log + (7 mrks). Epress in terms of log, logb nd lo g c : ) log b c log b c ( mrks). Solve the following equtions: ) = 6 = + 6= d) 7 = ( mrks) ( ) 6. Solve the following equtions: log log ) + = ( n n) log 9 = ( mrks) 7. ) Eplin why < log7 6 <. ( mrks) Find the vlue of log7 6, giving your nswer to three deciml plces. ( mrks) n r 8. Find log ( ), giving your nswer in terms of nd n. ( mrks) r=

12 A LEVEL TOPIC REVIEW : ANSWERS unit C eponentils nd logrithms. y G y = y = G y = G. ) G = = = = = d). = = log = e) log log = (, ) = y =. ) = = = log = log =. + = ( ) 6 ( )( 8) = =, = d) ( )log7= log log7 log= log7 log7 = = 7.6 log 7 log 6. ) log = = = n 9n= ( n+ )( n ) = n=, n=. ) log ( ) = log6 log = log. log = log. 6 d) log (6 ) = log. ) log + log b log c log + log b log c 7. ) 7 < 6< 7 B 7 = 6 log 8. log 7 = log 6 =. n r= r nn ( + )log

13 A LEVEL TOPIC REVIEW unit C differentition. Use differentition to find the vlues of for which the function f( ) = is n incresing function. ( mrks). Use differentition to find the coordintes (s frctions, not decimls!) nd ntures of the turning points of the following curves. ) y = y = (7 mrks). f( ) =. ) Find f( ). ( mrk) Find the coordintes of the sttionry points, nd determine their ntures. (6 mrks) Find the rnge of vlues for which the function is decresing. ( mrks) d) Sketch the curve y = f( ) mrking clerly the coordintes of ny turning points nd intercepts with the es. ( mrks). The height, h metres, of bll bove ground level is given by the formul h= + 9t t, where t is the time elpse in seconds. ) Find the height of the bll when t =. ( mrk) Find the time t which the bll hits the ground. ( mrks) Find the time t which the bll is t its gretest height nd find this height. ( mrks). A seled cylindricl cn of height h cm nd rdius r cm hs totl surfce re of π cm nd volume of V cm. ) Write down n epression for the surfce re nd show tht r h =. ( mrks) r Obtin n epression for V in terms of r nd hence find the vlue of r which will mimise the volume. Find this volume, nd verify tht your nswer is indeed mimum nd not minimum. (7 mrks)

14 A LEVEL TOPIC REVIEW : ANSWERS unit C differentition. ) + 9> ( )( ) > <, >. ) y = 9 + = = 9 9 = y = = = y = = 7 d y d ( ) 9 6 =, d y = minimum 9 7 d, d y = mimum 9 7 d ( ) = ( 9) = = y = = y = = 6 = y = + = 6 d y 9 d = ( ) d y, = inconclusive d grdient negtive either side of (, ) point of infleion d y = = 7 minimum d d y = = 7 mimum d. ) 6 ( ) = (, ) nd (, ) d y 6 6 d = ( ) d y, = 6 minimum d ( ) d y, = 6 mimum d ( ) < <, > d) y coordintes B shpe G. ) metres B. ) + 9t t = (t+ )( t ) = t = seconds 9 t = t =.9 seconds, h = 6. metres π r πrh r + = π + rh= r h = r r V = π r = πr πr r π πr = r = cm V = π 8π = 6π cm dv 6 dr (, ) (, ) (, ) = π r neg so mimum

15 A LEVEL TOPIC REVIEW unit C integrtion. Evlute: ) + d d + d (9 mrks). ) Sketch on the sme digrm the grphs of y = + nd + y = for. ( mrks) Find the coordintes of the point of intersection of the two grphs. ( mrks) Use integrtion to clculte the re enclosed by + y =, y = + nd the y-is. ( mrks). Use the trpezium rule with the number of trpezi indicted to find pproimtions to the following integrls. ) 7 d, 6 trpezi d, trpezi (6 mrks). Find the res enclosed by the following lines nd curves. In ech cse drw sketch to show the re concerned. ) y = +6 nd the -is. ( mrks) y = nd y =. (7 mrks). ) Sketch the curve y = ( )( ). ( mrks) Find the eqution of the tngent to the curve t the point where =. ( mrks) Show this tngent meets the curve gin t (, ) nd drw the tngent on your sketch. ( mrks) d) Find the re enclosed between the tngent, the curve nd the -is. ( mrks)

16 A LEVEL TOPIC REVIEW : ANSWERS unit C integrtion. ) + ( ) ( 6 ) + + =8 = + d ( ) ( ) = 8. ) G y = + (, ) (, ) + = + = ( + )( ) = (, ) + d d d y ( ) + y = (., ) =. ) or. ( ) or.77. ) ( )( ) = =, = + 6 ( 9 8) ( ) 8 6 G + + = G = = =, = d d (, 6) y y d (, ) (, ) (, ) ( ) 6 =

17 6. ) y G (, ) (, ) y = + = y= = 8 dy = 6+ d dy d = 6 = ( ) y = 8 y= + = y= + = y G (, ) (, ) d) bh + d 9 = 8 + ( ) ( ) + + = = 6 6