Math Sequences and Series RETest Worksheet. Short Answer

Transcription

1 Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws t the beginning of 000 Assuming rte of growth of 37% per yer since 000, wht will the popultion be t the beginning of 07? [ mrks]

2 Nme: Mth 0- Formul Sheet Sequences nd Series t n = t + ( n )d S n = n È t ÎÍ + ( n )d S n = n Ê Á t + t n ˆ t n = t r n Ê t r n ˆ Á S n =, r r S n = rt n t r, r S = t r, r Trigonometry Exponent Lws m n = m + n m n = m n, 0 Ê m ˆ n Á = mn ( b) m = m b m Ê ˆ m Á b = m b, b 0 m n = n, 0 n n = m n n = m or Ê n Á ˆ m sin A = sin B = sin C b c sin A = b sin B = c sin C c = + b b cos C cos C = + b c b Qudrtic Functions y = Ê Áx pˆ + q y = x + bx + c Qudrtic Equtions Given x + bx + c = 0, 0 then Rdicls Ê k ˆÊ k m Á n b Á k m = m k k n b n Frctions b + c d b c d = c bd b c d ˆ = mn b = d + bc bd = b d c k b x = b ± b 4c

3 Nme: Multiple Choice The common difference in the rithmetic sequence 4 5, 3 0, 9 5, 3 0, 4 5, is 6 A B C 4 D 5 Determine the next three terms on the following sequence? 39, 7, 5, F 3, 8, 9 G 3, 0, H 3, 9, J 3, 9, 3 Find the first five terms for the following rithmetic sequence t n = 43 3n A 30, 7, 4, 7, 30 C 43, 30, 7, 4, 9 B 30, 9, 8, 7, 6 D 30, 7, 4, 9, 4 Determine the number of terms in the sequence 48, 35,,, 343 F 3 G 09 H 08 J 0 5 Complete the following rithmetic sequence 83,,,8, A 58; 33; 7 B 57; 33; 43 C 58; 33; 6 D 57; 3; 43 6 Which term of the rithmetic sequence 6, 6, 8, hs vlue of 40? F t 0 G t 9 H t J t 40 7 Determine the generl term of the sequence 3,, 7,, 7, A t n = 5n + 8 B t n = 5n + C t n = 5n + 8 D t n = 5n + 8 Wht is the 30 th term of the sequence 3, 36, 4, 48, 54,? F 004 G 06 H 06 J The sum of n rithmetic series where t =, d =, nd n = 49 is A 485 B 4753 C 97 D Determine the sum of the rithmetic series ë + 07 F 856 G 805 H 885 J 907 The common rtio for the geometric sequence 7, 7 3, 7 9, 7 7, is A 3 B 3 C 3 D 3 3

4 Nme: In the formul for the generl term of geometric sequence t n = 4 5 n Ê ˆ, the common rtio is Á F 9 G H J The sum of n infinite geometric series is 6 nd its common rtio is 3 Wht is the first term of the series? A B 46 C D In n rithmetic sequence, t 8 = 79 nd t 45 = 30 Wht is the vlue of t? F 43 G 49 H 37 J 30 5 Which of the following is geometric sequence? A 9, 35, 05 B 9, 35, 05 C 9,, 5 D 9,, 5 6 Stte the common rtio of the geometric sequence 88, 43, 648, F G 3 3 H 3 J 88 7 Complete the following geometric sequence,,,6, A 8; 8; 6 B 4; 8; 3 C 8; 8; 6 D 4; 8; 3 8 Determine the sum (to 3 deciml plces) of the first 7 terms of the geometric series: ë F 0944 G 0944 H 0976 J Determine the sum of the infinite geometric series: A S = 53 B S = 3 C S = 4 D S = 64 0 A geometric series hs r = 3 5 nd S = 35 Determine t F t = G t = 5 H t = 5 J t = 3 4

5 Sequences nd Series RETest Worksheet Answer Section SHORT ANSWER OTHER Mth 0- Formuls MULTIPLE CHOICE A J 3 D 4 H 5 A 6 G 7 A 8 F 9 D 0 J B H 3 B 4 H 5 B 6 H 7 B 8 F 9 B 0 G

6 Trigonometry RE-test Worksheet Mth 0- Nme: Given sinθ = 07547, where 0 θ < 360, determine the mesure of θ, to the nerest degree Given cos θ = 09986, where 0 θ < 360, determine the mesure of θ, to the nerest degree Clculte the length of AC in BAC to deciml plce (The digrm is NOT drwn to scle) 3 Given cos θ = 0340, where 0 θ < 360, determine the mesure of θ, to the nerest degree 4 Given tnθ = 0405, where 0 θ < 360, determine the mesure of θ, to the nerest degree 5 Given tnθ = 460, where 0 θ < 360, determine the mesure of θ, to the nerest degree 6 Given the ngle 79 is in stndrd postions Determine the reference ngle 3 Clculte the mesure of A in CBA to the nerest tenth of degree (The digrm is NOT drwn to scle) 7 Given the ngle 66 is in stndrd postions Determine the reference ngle 8 Given tht n ngle hs reference ngle of 54, determine the ngle in stndrd position if the ngle is in qudrnt one 9 Given tht n ngle hs reference ngle of 68, determine the ngle in stndrd position if the ngle is in qudrnt three 0 Determine the exct vlue of sinθ if the terminl rm of n ngle in stndrd position psses through the point Ê Á8, 8ˆ 4 Clculte the length of AB in CAB to deciml plce (The digrm is NOT drwn to scle) Determine the exct vlue of cos θ if the terminl rm of n ngle in stndrd position psses through the point Ê Á 9, 4ˆ

7 Nme: 5 Given tht B is obtuse, clculte the mesurement of B in BAC to deciml plce (The digrm is NOT drwn to scle) 6 Given C = 47, b = 6, c = 5 in ABC, clculte two possible mesurements of A to deciml plce [ mrks] 7 Given C = 8, b = 5, c = 8 in ABC, clculte two possible mesurements of A to deciml plce [ mrks] 8 Srh nd Simone re wlking in wlk--thon down stright street tht leds to the finish line At the sme time, they both notice tethered hot-ir blloon directly over the finish line Srh sees tht the ngle from the ground to the blloon s 0, nd Simone (who is 053 km closer to the finish line thn Srh) sees the ngle from the ground to the blloon s 56 Determine the height of the blloon, to the nerest hundredth of kilometre 9 Two irplnes leve the Hy River irport in the Northwest Territories t the sme time One irplne trvels t 370 km/h The other irplne trvels t 50 km/h About h lter, they re 930 km prt Determine the ngle between their pths, to the nerest degree

8 Trigonometry RE-test Worksheet Answer Section θ = 49, θ = 3 θ = 77, θ = 83 3 θ = 0, θ = 50 4 θ = 7, θ =35 5 θ = 4, θ = sinθ = cos θ = 9 97 b = 94 3 A = 6 4 c = 35 5 ngle B = A = 87, or 43 7 A = 903, or The height of the blloon is 06 km 9 60

9 Mth 0- Nme: Qudrtic Functions: Retest Worksheet Short Answer Given the eqution y = x + 4x + 3, determine the following: 3 The grph of qudrtic function is shown below Determine its eqution: ) y-intercept: b) x-intercept(s): c) vertex: d) xis of symmetry: e) domin: e) rnge: Given the eqution y = x 6x + 8, determine the following: ) y-intercept: 4 The grph of qudrtic function is shown below Determine its eqution: b) x-intercept(s): c) vertex: d) xis of symmetry: e) domin: e) rnge:

10 Nme: 5 Chnge the eqution y = 6x 7x + to the form y = Ê Áx pˆ + q by completing the squre 7 Chnge the eqution y = 4x 6x + 9 to the form y = Ê Áx pˆ + q by completing the squre 6 Chnge the eqution y = x 4x + 6 to the form y = Ê Áx pˆ + q by completing the squre 8 Chnge the eqution y = 3x + 30x + 79 to the form y = Ê Áx pˆ + q by completing the squre

11 Nme: 9 Given the vertex is Ê Á, 3ˆ nd point on the grph is Ê Á4,4ˆ Determine the eqution of the prbol in the form y = Ê Áx pˆ + q A store sells energy brs for $5 At this price, the store sold n verge of 0 brs per month lst yer The mnger hs been told tht for every 5 decrese in price, he cn expect the store to sell eight more brs monthly ) Write qudrtic function you cn use to model this sitution? b) Determine the mximum revenue the mnger cn expect the store to chieve c) Wht price will give tht mximum? 0 Given the vertex is Ê Á, 6ˆ nd point on the grph is Ê Á, ˆ Determine the eqution of the prbol in the form y = Ê Áx pˆ + q A dinner thetre hs 600 seson ticket holders The owners of the thetre hve decided to rise the price of seson ticket from the current price of $400 According to recent survey of seson ticket holders, for every $50 increse in the price, 30 seson ticket holders will not renew their sets ) Write qudrtic function you cn use to model this sitution? b) Determine the mximum revenue the concert could chieve c) Wht should the owners chrge for ech seson ticket in order to mximize their revenue? 3

12 Nme: 3 A hospitl sells rffle tickets to rise funds for new medicl equipment Lst yer, 000 tickets were sold for $0 ech The fund-rising coordintor estimtes tht for every $ decrese in price, 00 more tickets will be sold ) Wht decrese in price will mximize the revenue? 5 A rectngulr dog pen is to be fenced with 8 m of fencing Determine the mximum re nd the width of this rectngle b) Wht is the price of ticket tht will mximize the revenue? 6 A rectngulr lot is bordered on one side by building nd the other 3 sides by 600 m of fencing Determine the re of the lrgest lot possible c) Wht is the mximum revenue? Show nd explin your work 4 Three rectngulr res re being enclosed long the side of building, s shown There is 64 m of fencing mteril Assume tht ll the mteril is used 7 A science museum wnts to build n outdoor ptio The ptio will be bordered on one side by wll of the museum nd the other 3 sides by 40 m of fencing Determine the re of the lrgest ptio possible 8 Two numbers hve difference of 6 nd their product is minimum Determine the numbers ) Write the function tht represents the totl re in terms of the distnce from the wll 9 Two numbers hve difference of 8 The sum of their squres is minimum Determine the numbers b) Determine the mximum re 0 The sum of two numbers is Their product is mximum Determine the numbers c) Determine the length nd width of the overll enclosure 4

13 Qudrtic Functions: Retest Worksheet Answer Section SHORT ANSWER ) 3 b), 3 c) Ê Á, ˆ d) x = e) y ) 8 b), 4 c) Ê Á3, ˆ d) x = 3 e) y 3 y = x 4 y = 05x + x 5 y = 6( x 6) 4 6 y = ( x ) y = 4( x ) y = 3( x + 5) y = 3( x ) 3 0 y = x + ( ) + 6 ) R = (5 005x)(0 + 8x) b) $360 c) $50 (x is 5) ) R = ( x)(600 30x) b) $ c) $700 (x is 6)

14 3 Determine n eqution to represent the sitution For ech $ decrese in price, 00 more tickets will be sold Let x represent the number of $ decreses in the price of ticket When the price decreses by $ x times: the price, in dollrs, of ticket is 0 x the number of tickets sold is x the revenue, in dollrs, is (0 x)( x) Let the revenue be R dollrs An eqution is: R = (0 x)( x) Use grphing clcultor Grph: R = (0 x)( x) Use the CALC function to determine the coordintes of the vertex ) From the grph, the mximum revenue occurs when the number of $ decreses is 5 So, the decrese in price tht will mximize the revenue is $5 b) The price of ticket tht will mximize the revenue is: $0 $5 = $5 c) Substitute x = 5 in R = (0 x)( x) to determine the mximum revenue R = (0 5)( (5)) R = The mximum revenue is $ ) A = 3( 64 4d)d or A = 9d d b) 768 m² c) 8 m 3 m 5 A = 49 m ; w = 7 m m 7 00 m 8 3 nd nd nd 6

15 Mth 0- Nme: Clss: Qudrtic Equtions RE-Test Worksheet Fctor this polynomil expression: 6( x 4) 9Ê Á4y + 3ˆ [ mrks] Fctor this polynomil expression: 4( x + 4) + 7( x + 4) + 4 [ mrks] 3 Determine the discriminnt of x + 8x + 8 = 0 [ mrk]

16 Mth 0- Nme: Absolute Vlue nd Reciprocl Functions RE-Test workheet Evlute ( 3) Write n eqution for the bsolute vlue function Show your work Drw the grph of y = x 3 3 Given f(x) = x + 5x + 4, sketch grph of the reciprocl function y = nd identify the f(x) verticl symptotes, if they exist 5 Determine the eqution of the following grph Show your work

17 Nme: 6 Solve this eqution: 7 8 x 3 = 9 Given the qudrtic function y = f( x) below, sketch the grph of y = Use the sme coordinte f( x) plne shown 7 Solve this eqution: x x 65 = 0 8 Write this bsolute vlue function in piecewise nottion y = (x + 3) 9 0 The cross-section of the sloping roof of house is represented on coordinte grid so tht the points representing the bottom of the roof lie on the x-xis The eqution of the function describing the cross-section is h( x) = 3 x + 4, where h is the height of the roof, in metres, nd x is the horizontl distnce from the centre of the roof, in metres Wht is the width of the bottom of the roof?

18 Absolute Vlue nd Reciprocl Functions RE-Test workheet Answer Section 55 3 The grph of y = f(x) opens up nd hs x-intercepts nd 4 So, the grph of the reciprocl function hs verticl symptotes x = nd x = 4 Plot points where the lines y = nd y = intersect the grph of y = f(x) These points re common to both grphs Using these points nd the symptotes, drw smooth curves tht pproch the symptotes but never touch themthe grph of the reciprocl function hs Shpe 3

19 4 Choose two points on the line to determine the slope of the liner function: ( 4, 5) nd (, ) m = y y x x m = (5) ( 4) m = y-intercept: 3 An eqution for the bsolute vlue function is: y = x 3 5 y = x 8 6 x = 4 nd x = The solutions re: x = 5, x = 7, x = 5, nd x = 3 Ï 8 y = Ô (x + 3) 9, if x 6 or x 0 Ì ÓÔ (x + 3) + 9, if 6 < x < Therefore, the distnce between the two points is m The width of the bottom of the roof is m

20 Nme: 4 Describe the nture of the roots of x x + = 0 (Do not solve) [ mrk] 5 Solve x + x 3 = 0 to the nerest hundredth [ mrks] 6 Determine the exct solution(s) for (x 5) = 6 [ mrks] 7 Solve 6x 6x = 0 by fctoring [ mrks] 8 Solve x + 7x 0 = 0 by fctoring [ mrks] 9 Algebriclly solve 4x + x + 3 = 0 [ mrks]

21 Nme: 0 Determine the exct roots of x x 9 = 0 [ mrks] Determine the exct roots of 9x + x 9 = 0 [ mrks] Problem A penny is dropped from the top of the High Level Bridge It s height, h meters, bove the river t seconds fter it is relesed is modeled by the qudrtic function: h( t) = 0 48t To the nerest tenth of second, how long hs the penny fllen for when it is 9 m bove the river? [ mrk] Mrc s rectngulr grden mesures 9 m by m He wnts to double the re of his grden by dding equl lengths to both dimensions Determine this length to the nerest tenth of metre Show your work [ mrks] 3

22 Qudrtic Equtions RE-Test Worksheet Answer Section SHORT ANSWER Ê Á 4x + y 7 ˆ Ê Á 4x y 5 ˆ 4(x + 8)(x + 8) Discriminnt = 0 nd therefore equl rel roots 5 x =Ö 054 or x = 5 ± 6 7 x = 0 or 8 x = 5 or 9 x = or x = ± x = ± PROBLEM 4 s The length to be dded is pproximtely 43 m

23 Mth 0- Nme: Qudrtic Systems of Equtions RE-Test worksheet ( point) Solve by grphing: y = 3 4 ( x 4) 3x + y 8 = 0 3 ( points) Algebriclly solve: y =8x 9 4x 4x y +35 = 0 ( point) Solve by grphing: x + 6x + 4y 7 = 0 x + y = 0

24 4 ( points) Algebriclly solve:: x + 8x y + = 0 3x + y + 6 = 0 5 ( point) Algebriclly solve: A line with slope = nd y-intercept of + intersects prbol with vertex (, ) nd point Ê Á 3, 6 ˆ

25 Problems: (Show ll work) Grphicl solutions require equtions, sketch, nd window settings (6 points) After footbll is kicked, it reches mximum height of 3 m nd it hits the ground 6 m from where it ws kicked After soccer bll is kicked, it reches mximum height of 9 m nd it hits the ground 40 m from where it ws kicked The pths of both blls re prbols ) Consider both blls to be kicked from the sme strting point nd plce this t the origin of coordinte grid Drw digrm to model the given informtion for the two kicked blls ( point) b) Determine qudrtic eqution in the form of: y = (x p) + q to model the footbll height compred to the horizontl distnce it trvelled nd qudrtic eqution in the form of: y = (x p) + q to model the soccer bll height compred to the horizontl distnce it trvelled ( point) c) Solve the system of two equtions If you re using grphicl pproch, mke sure you include ll relevnt informtion - including the window settings nd equtions you used to solve the problem for full mrks Round your nswer to the nerest centimeter ( points) d) Wht is occuring t the intersection point, in the context of this problem? ( point) 3

26 (5 points) The perimeter of the right tringle is 50 m The re of the tringle is 30y squre metres ) Write simplified expression for the tringle s perimeter in terms of x nd y ( point) b) Write simplified expression for the tringle s re in terms of x nd y ( point) c) Write system of equtions nd explin how it reltes to this problem ( point) d) Solve the system for x nd y Wht re the dimensions of the tringle? ( points) 4

27 Other (0 points) Mth 0- Formul Sheet Sequences nd Series t n = t + ( n )d S n = n È t ÎÍ + ( n )d S n = n Ê Á t + t n ˆ t n = t r n Ê t r n ˆ Á S n =, r r S n = rt n t r, r S = t r, r Trigonometry Exponent Lws m n = m + n m n = m n, 0 Ê m ˆ n Á = mn ( b) m = m b m Ê ˆ m Á b = m b, b 0 m n = n, 0 n n = m n n = m or Ê n Á ˆ m sin A = sin B = sin C b c sin A = b sin B = c sin C c = + b b cos C cos C = + b c b Qudrtic Functions y = Ê Áx pˆ + q y = x + bx + c Qudrtic Equtions Given x + bx + c = 0, 0 then Rdicls Ê k ˆÊ k m Á n b Á k m = m k k n b n Frctions ˆ = mn b b + c d + bc = d bd b c d = c bd b c d = b d c k b x = b ± b 4c 5

28 Qudrtic Systems of Equtions RE-Test worksheet Answer Section SHORT ANSWER P(,) P(4, ) P(3, 5) P(,3) 3 Solution: Ê Á4,3ˆ 4 P( 7,5) P( 4, 4) 5 y = x+ y = ( x ) Solution: Ê Á, 3ˆ

29 PROBLEM b footbll: y = 5 x( x 6) soccerbll: y = x( x 40) c (0, 900) d The intersection point is where the two blls re t the sme height given the sme horizontl distnce trvelled ) 8x + y + 9 b) 5 x x c) y = 8x + 38 y = 4 x + 0 x d) x = 0 nd y = bse =, height = 60, hypotenuse = 9 OTHER Mth 0- Formuls