Higher. Specimen NAB Assessment

Transcription

1 hsn.uk.net Higher Mathematics UNIT Specimen NAB Assessment HSN0 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For more details about the copright on these notes, please see

2 Unit Specimen NAB Assessment UNIT Specimen NAB Assessment utcome. A line passes through the points A (, ) and B( 6,). Find the equation of this line.. A line makes an angle of 0 with the positive direction of the -ais, as shown in the diagram. 0 Find the gradient of this line.. (a) Write down the gradient of a line parallel to = +. (b) Write down the gradient of a line perpendicular to = +. utcome. The diagram below shows part of the graph of = f ( ). = f ( ) (a) Sketch the graph of = f ( ). (b) n a separate diagram, sketch the graph of = f ( + ). Page HSN0

3 Unit Specimen NAB Assessment. (a) The diagram below shows the curve = sin and a related curve. = sin 60 Write down the equation of the related curve. (b) The diagram below shows the curve = cos and a related curve. 60 = cos Write down the equation of the related curve. 6. The curve = a is shown in the diagram below. = a (, ) Given that the curve passes through the point (, ), write down the value of a. 7. The diagram below shows the graph of the function f ( ) = and its inverse function. = (, ) (,) Write down the formula for the inverse function. Page HSN0

4 Unit Specimen NAB Assessment 8. (a) Two functions f and g are defined b f ( ) = and g ( ) =. ( ) Find an epression for f g ( ). (b) Functions h and k are defined on suitable domains b h( ) = and k ( ) = tan. Find an epression for k ( h ( )). utcome 9. Given that = for 0, find d d. 0. The curve with equation = + 6 is shown below. = + 6 (, 6) Find the gradient of the tangent to the curve at the point (,6 ).. A curve has equation = +. Find the stationar points on the curve and, using differentiation, determine their nature. 8 utcome. A pond is treated weekl with a chemical to ensure that the number of bacteria is kept low. It is estimated that the chemical kills 68% of all bacteria. Between the weekl treatments, it is estimated that 600 million new bacteria appear. There are u n million bacteria at the start of a particular week. (a) Write down a recurrence relation for u n +, the number of millions of bacteria at the start of the net week. (b) Find the limit of the sequence generated b this recurrence relation and eplain what the limit means in the contet of this question. Page HSN0

5 Unit Specimen NAB Assessment Marking Instructions Pass Marks utcome utcome utcome utcome utcome Straight Lines ( ). m = = 6 ( + 6 ) = 6 = = 0 =. m = tan 0 = 0 8 (to d.p.) Use gradient formula Calculate gradient Equation of line Calculate gradient. (a) State gradient (b) State gradient utcome Functions and Graphs. (a) Sketch showing images = f ( ) of given points (b) ( ) = f + (, ) Sketch showing images of given points Page HSN0

6 Unit Specimen NAB Assessment. (a) = sin Identif equation (b) = cos Identif equation 6. Since = when = : a = a = State the value of a 7. f ( ) = log State formula for inverse 8. (a) f g ( ) = f ( ) ( ) (b) k h( ) = k ( ) ( ) ( ) = = tan utcome Differentiation 9. = = d = + 9 d 0. Gradient of tangent is given b d d d d = At =, m = = Epression for composite function Epression for composite function Simplif first term Simplif second term Differentiate first term Differentiate second term Know to differentiate Differentiate Know to evaluate derivative Calculate gradient Page HSN0

7 Unit Specimen NAB Assessment. d = 8 + d d Stationar points eist where 0 d = 8 + = 0 ( 6)( ) = 0 = or = 6 To find -coordinates: =, = ( 6) ( 6) + ( 6) At 6 = At =, = ( ) ( ) + ( ) = 7 Stationar points are at ( ) 6 d d sketch ( ),7 and ( 6, ),7 is a maimum turning point ( 6, ) is a minimum turning point Know to differentiate Differentiate Set derivative equal to 0 Find -coordinates of stationar points Find -coordinates of stationar points Method to determine nature Nature of one stationar point Nature of second stationar point utcome Sequences. (a) u. n+ = 0 un State recurrence relation (b) A limit l eists since < 0. < Know how to calculate 600 limit l = 0. Calculate limit = 88. (to d.p.) Interpret limit In the long term, the number of bacteria will settle around 88 million 8 Page 6 HSN0