Higher. Specimen NAB Assessment

Transcription

1 hsn uknet Higher Mathematics UNIT Specimen NAB Assessment HSN50 This dcument was prduced speciall fr the HSNuknet website, and we require that an cpies r derivative wrks attribute the wrk t Higher Still Ntes Fr mre details abut the cpright n these ntes, please see

2 Higher Mathematics Unit Specimen NAB Assessment UNIT Specimen NAB Assessment utcme A line passes thrugh the pints A (, ) and B( 6,) Find the equatin f this line A line makes an angle f 0 with the psitive directin f the -ais, as shwn in the diagram 0 Find the gradient f this line (a) Write dwn the gradient f a line parallel t = + (b) Write dwn the gradient f a line perpendicular t = + utcme The diagram belw shws part f the graph f = f ( ) = f ( ) (a) Sketch the graph f = f ( ) (b) n a separate diagram, sketch the graph f = f ( + ) hsnuknet Page HSN50

3 Higher Mathematics Unit Specimen NAB Assessment 5 (a) The diagram belw shws the curve = sin and a related curve = sin 60 Write dwn the equatin f the related curve (b) The diagram belw shws the curve = cs and a related curve 60 = cs Write dwn the equatin f the related curve 6 The curve = a is shwn in the diagram belw = a (, ) Given that the curve passes thrugh the pint (, ), write dwn the value f a 7 The diagram belw shws the graph f the functin f ( ) = and its inverse functin = (, ) (,) Write dwn the frmula fr the inverse functin hsnuknet Page HSN50

4 Higher Mathematics Unit Specimen NAB Assessment 8 (a) Tw functins f and g are defined b f ( ) = and g ( ) = ( ) Find an epressin fr f g ( ) (b) Functins h and k are defined n suitable dmains b h( ) = 5 and k ( ) = tan Find an epressin fr k ( h ( )) utcme 9 Given that 5 = fr 0, find d d 0 The curve with equatin = is shwn belw = ( 5, 6) Find the gradient f the tangent t the curve at the pint ( 5,6 ) A curve has equatin = + Find the statinar pints n the curve and, using differentiatin, determine their nature 8 utcme A pnd is treated weekl with a chemical t ensure that the number f bacteria is kept lw It is estimated that the chemical kills 68% f all bacteria Between the weekl treatments, it is estimated that 600 millin new bacteria appear There are u n millin bacteria at the start f a particular week (a) Write dwn a recurrence relatin fr u n +, the number f millins f bacteria at the start f the net week (b) Find the limit f the sequence generated b this recurrence relatin and eplain what the limit means in the cntet f this questin hsnuknet Page HSN50

5 Higher Mathematics Unit Specimen NAB Assessment Marking Instructins Pass Marks utcme utcme utcme utcme utcme Straight Lines ( ) m =! = 6 ( + 6 )! 5 = 6 = = 0 =! m = tan 0 = 0 8 (t dp)! Use gradient frmula Calculate gradient Equatin f line Calculate gradient (a)! State gradient (b)! State gradient utcme Functins and Graphs (a) Sketch shwing images = f ( ) f given pints (b) ( ) = f + (, ) Sketch shwing images f given pints hsnuknet Page HSN50

6 Higher Mathematics Unit Specimen NAB Assessment 5 (a) = sin Identif equatin (b) = cs Identif equatin 6 Since = when = : a = a = State the value f a 7 f ( ) = lg State frmula fr inverse 8 (a) f g ( ) = f ( ) ( ) (b) k h( ) = k 5 ( ) ( ) ( ) = = tan5 utcme Differentiatin 5 9 = = d = + 9 d 0 Gradient f tangent is given b d d d 5 d = At = 5, m = 5 5 = 5 Epressin fr cmpsite functin Epressin fr cmpsite functin Simplif first term Simplif secnd term Differentiate first term Differentiate secnd term Knw t differentiate Differentiate Knw t evaluate derivative Calculate gradient hsnuknet Page 5 HSN50

7 Higher Mathematics Unit Specimen NAB Assessment d = 8 + d d Statinar pints eist where 0 d = 8 + = 0 ( 6)( ) = 0 = r = 6 T find -crdinates: =, = ( 6) ( 6) + ( 6) At 6 = At =, = ( ) ( ) + ( ) = 7 Statinar pints are at ( ) 6 d d sketch ( ),7 and ( 6, ),7 is a maimum turning pint! ( 6, ) is a minimum turning pint! Knw t differentiate Differentiate Set derivative equal t 0 Find -crdinates f statinar pints Find -crdinates f statinar pints Methd t determine nature Nature f ne statinar pint Nature f secnd statinar pint utcme Sequences (a) u n+ = 0 un State recurrence relatin (b) A limit l eists since < 0 < Knw hw t calculate 600 limit l = 0 Calculate limit = 88 5 (t dp) Interpret limit In the lng term, the number f bacteria will settle arund 88 millin! 8 hsnuknet Page 6 HSN50

8 Practice Assessment () Unit - Mathematics (H) utcme Marks A line passes thrugh the pints (,-7) and (6,) Find the equatin f this line A line makes an angle f 50 with the psitive directin f the -ais, as shwn in the diagram, where the scales n the aes are equal 50 Find the gradient f the line () () (a) Write dwn the gradient f an line parallel t = + () (b) Write dwn the gradient f a line perpendicular t =!! () utcme See wrksheet Diagrams and n the wrksheet shw part f the graph f = f () (a) n Diagram, draw the graph f =! f () () (b) n Diagram, draw the graph f = f ( + ) () 5 (a) The diagrams belw shw part f the graph f = sin and the graph f a related functin Write dwn the equatin f the related functin = sin () (b) The diagram shws part f the graph f = sin (drawn as a brken line) and the graph f a related functin Write dwn the equatin f the related graph ()

9 6 See wrksheet The graph f = is shwn in Diagram n the wrksheet Write dwn the equatin f the graph f the epnential functin f the frm = a which passes thrugh the pint (,9) as shwn n the wrksheet () 7 See wrksheet Diagram n the wrksheet shws part f the graph f the functin its inverse functin = 6 and Write dwn the equatin f the inverse functin () 8 (a) Tw functins f and g are given b f ( ) =! and g ( ) =! btain an epressin fr f ( g() ) () (b) Functins h and k, defined n suitable dmains, are given b h( ) = and k ) cs k h() () ( = Find ( ) utcme + 9 Given =, find d d () 0 The diagram shws a sketch f the curve with equatin =! with a tangent drawn at the pint (5,) =! Find the gradient f this tangent (5,) () Find the crdinates f the statinar pints f the curve with equatin =! + Using differentiatin determine their nature (8) utcme In a small cln 0% f the eisting insects are eaten b predatrs each da, hwever during the night 00 insects are hatched There are U n insects at the start f a particular da (a) (b) Write dwn a recurrence relatin fr U n+, the number f insects at the start f the net da Find the limit f the sequence generated b this recurrence relatin and eplain what the limit means in the cntet f this questin () ()

10 Unit - Practice Assessment () Wrksheet fr Questins, 6 and 7 Questin Name : Class : (8,0) (8,0) (,-6) (,-6) Diagram Diagram Questin 6 The equatin f the graph passing thrugh (,9) is 0 8 (,9) = 6 (,) = - Diagram Questin 7 The equatin f the graph passing thrugh (,0) is = 6 (0,) (,0) Diagram

11 Practice Assessment () Unit - Mathematics (H) utcme Marks A line passes thrugh the pints (-,) and (-,) Find the equatin f this line A line makes an angle f 75 with the psitive directin f the -ais, as shwn in the diagram, where the scales n the aes are equal 75 Find the gradient f the line () () (a) Write dwn the gradient f an line parallel t =!! () (b) Write dwn the gradient f a line perpendicular t =! () utcme See wrksheet Diagrams and n the wrksheet shw part f the graph f = f () (a) n Diagram, draw the graph f =! f () () (b) n Diagram, draw the graph f = f (! ) () 5 (a) The diagrams belw shw part f the graph f = sin and the graph f a related functin Write dwn the equatin f the related functin = sin () (b) The diagram shws part f the graph f = sin (drawn as a brken line) and the graph f a related functin Write dwn the equatin f the related graph ()

12 6 See wrksheet The graph f = is shwn in Diagram n the wrksheet Write dwn the equatin f the graph f the epnential functin f the frm = a which passes thrugh the pint (,6) as shwn n the wrksheet () 7 See wrksheet Diagram n the wrksheet shws part f the graph f the functin its inverse functin = 7 and Write dwn the equatin f the inverse functin () 8 (a) Tw functins f and g are given b f ( ) = and ( ) = + g btain an epressin fr f ( g() ) () (b) Functins h and k, defined n suitable dmains, are given b h( ) = sin and ) k h() k ( = Find ( ) () utcme + 9 Given =, find d d () 0 The diagram shws a sketch f the curve with equatin =! 0 + with a tangent drawn at the pint (7,) =! 0 + Find the gradient f this tangent (7,) () Find the crdinates f the statinar pints f the curve with equatin = +! Using differentiatin determine their nature (8) utcme In a small rabbit cln ne eighth f the eisting rabbits are eaten b predatrs each summer, hwever during the winter rabbits are brn There are U n rabbits at the start f a particular summer (a) Write dwn a recurrence relatin fr U n+, the number f rabbits at the start f the net summer () (b) Find the limit f the sequence generated b this recurrence relatin and eplain what the limit means in the cntet f this questin ()

13 Unit - Practice Assessment () Wrksheet fr Questins, 6 and 7 Questin Name : Class : (0,5) (0,5) - - Diagram Diagram Questin 6 The equatin f the graph passing thrugh (,6) is = (,6) (,9) = Diagram Questin 7 The equatin f the graph passing thrugh (,0) is = 7 (0,) (,0) Diagram

14 Practice Assessment () Unit - Mathematics (H) utcme Marks A line passes thrugh the pints (,-) and (,6) Find the equatin f this line A line makes an angle f 0 with the psitive directin f the -ais, as shwn in the diagram, where the scales n the 0 aes are equal Find the gradient f the line () () (a) Write dwn the gradient f an line parallel t =! + 9 () (b) Write dwn the gradient f a line perpendicular t =!! () utcme See wrksheet Diagrams and n the wrksheet shw part f the graph f = f () (a) n Diagram, draw the graph f =! f () + () (b) n Diagram, draw the graph f = f (! ) () 5 (a) The diagrams belw shw part f the graph f = sin and the graph f a related functin Write dwn the equatin f the related functin = sin careful! () (b) The diagram shws part f the graph f = sin (drawn as a brken line) and the graph f a related functin Write dwn the equatin f the related graph ()

15 6 See wrksheet The graph f = 5 is shwn in Diagram n the wrksheet Write dwn the equatin f the graph f the epnential functin f the frm = a which passes thrugh the pint (,) as shwn n the wrksheet () 7 See wrksheet Diagram n the wrksheet shws part f the graph f the functin its inverse functin = and Write dwn the equatin f the inverse functin () 8 (a) Tw functins f and g are given b f ( ) = + and g ( ) = + btain an epressin fr f ( g() ) () (b) Functins k and h, defined n suitable dmains, are given b k( ) = cs and ( ) = ( +! ) k h() () h Find ( ) utcme Given =, find d d () 0 The diagram shws a sketch f the curve with equatin =! + 5 with a tangent drawn at the pint (,) =! + 5 Find the gradient f this tangent (,) () Find the crdinates f the statinar pints f the curve with equatin =!! Using differentiatin determine their nature (8) utcme Fr an established ant hill 6% f the wrker ants fail t return at the end f each da Hwever, during the night 50 wrker ants are hatched There are U n wrker ants at the start f a particular da (a) Write dwn a recurrence relatin fr U n+, the number f wrker ants at the start f the net da () (b) Find the limit f the sequence generated b this recurrence relatin and eplain what the limit means in the cntet f this questin ()

16 Unit - Practice Assessment () Wrksheet fr Questins, 6 and 7 Questin Name : Class : (-,) (-,) (,-) (,-) Diagram Diagram Questin 6 The equatin f the graph passing thrugh (,) is = (,5) = 5 (,) Diagram Questin 7 The equatin f the graph passing thrugh (,0) is = (0,) (,0) Diagram

17 Unit - Practice Assessments Answers Practice Assessment utcme : m =, =! (! = (! 6) r + 7 = (! ) ) m =! 9 (a) m = (b) m = utcme : (a) diagram (reflectin in -ais) (b) diagram (translated units left) 5 (a) = sin! (b) = sin 6 = 7 = lg 6 8 (a) f ( g( )) = (! )! " 9! 6 (b) k( h( )) = cs d! utcme : 9 =! + d 0 m =!, 5 ), ma, (, ), min ( utcme : (a) U = 0! 8U 00 (b) L = 000, + eplanatin n+ n + Practice Assessment utcme : m = = + 0 (! = ( + ) r! = ( +, ) ) m =! 7 (a) m =! (b) m =! utcme : (a) diagram (reflectin in -ais) (b) diagram (translated units right) 5 (a) = sin + (b) = sin 6 = 6 7 = lg 7 8 (a) f ( g( )) = ( + )! + + (b) k h( )) = sin ( utcme : 9 d d =! 6! 0 m =!,6 ), ma, (,! ), min ( 7 utcme : (a) U = U (b) L = 9, + eplanatin n+ + 8 n Practice Assessment utcme : m =! 5, =! ( + =! 5(! ) r! 6 =! 5(! ) ) m = 0! 8 (a) m =! (b) m = utcme : (a) diagram (reflectin in -ais then up ) (b) diagram (translated units right) 5 (a) = sin + (b) sin = 6 = 7 = lg 8 (a) f ( g( )) = ( + ) + +! (b) k( h( )) = cs( +! ) d! utcme : 9 =! 5 + d 0 m =! (!,0), ma, (,! ), min utcme : (a) U = 0! 9U 50 (b) L = 9000, + eplanatin n+ n +