Mathematics. Notes. Higher. Higher Still. HSN21510 Unit 1 Level C Assessment

Transcription

1 Higher Mathematics HSN Unit Level C Assessment These notes were created speciall for the wesite, and we require that an copies or derivative works attriute the work to us. For more details aout the copright on these notes, please see

2 Unit Level C Assessment. Find the equation of the line which passes through the points A(, ) and B( 6,).. A line makes an angle of with the positive direction of the -ais as shown in the diagram. Find the gradient of this line.. a) Write down the gradient of a line parallel to = +. ) Write down the gradient of a line perpendicular to = +.. The diagram shows part of the graph of = f ( ). = f ( ) On separate diagrams, sketch the graphs of: a) = f ( ) ) = f ( + ). Write down the equation of each of these trigonometric graphs: a) ) 6 6 Page HSN

3 6. The diagram elow shows part of the graph of = and also of Unit Level C Assessment = a. = a = If the graph of = a passes through the point (,), calculate the value of a and hence write down the equation of the graph = a. 7. The diagram elow shows part of the graph of a logarithmic function. = log a. (7,) a) Write down the equation of the function. ) Sketch the graph of = log showing where is cuts the -ais and also one other point on the graph. 8. a) Two functions, f and g are given f ( ) = and g ( ) =. Find an epression for f ( g( )). ) Functions h and k are defined on suitale domains as h ( ) = and k ( ) = sin. Find kh ( ( )). 9. Given =, find d d. Page HSN

4 Unit Level C Assessment. The diagram shows a sketch of the curve with equation = ( )( ), with a tangent drawn a the point (,6). = ( )( ). (,6) Find the gradient of the line at (,6).. Find the co-ordinates of the stationar points on the curve with equation = + and use differentiation to determine their nature. 8. A pond is treated monthl with a chemical to ensure that the numer of acteria is kept low. The chemical compan that makes it claim it will kill 68% of all acteria. Between the monthl treatments, it is estimated that 6 million new acteria are evolved. a) If there are U n million acteria at the start, write down a recurrence relation for U n +, which is numer of millions of acteria at the end of the first month. ) Find the limit of this recurrence relation and eplain what it means in the contet of the question. Total Marks: 6 [END OF QUESTIONS] Page HSN

5 Unit Level C Assessment Pass Marks Outcome Outcome Outcome Outcome Outcome The Straight Line m = ( ) = 6 = = m = tanθ = tan =.8 Marking Scheme m = A( 6,) = m( a) = ( + 6) = = Co-ordinates are sued into m = m is calculated Equation of straight line is found m is calculated a m = m is stated m = m is stated Outcome Functions and Graphs a Graph is sketched and = f ( ) points are shown Page HSN

6 Unit Level C Assessment. (-,) Graph is sketched and points are shown = f ( + ) a = sin Graph is identified 6 = cos Graph is identified = a Since = and = : a = a = therefore = Equation is stated 7 a = log7 Equation of function is identified Graph is sketched and = log root and one other point. is shown (,) 8 a f ( g( )) = f ( ) = ( ) kh ( ( )) = k( ) = sin Outcome Differentiation 9 = = ( ) = d 6 = + d Page Composite functions are solved Composite functions are solved () Brackets are correctl epanded () function is differentiated correctl HSN

7 = ( )( ) = + 6 = + 6 d Equation of gradient = d d d = At =, m = () = d Stationar points occur when d = = + d = 8 + d = 8 + = ( 6)( ) = 6 = To find points: at = 6, = (6) (6) + (6) = at =, = () () + () = 7 Stationar points occur at (,7 ) and (6, ) Unit Level C Assessment Brackets are epanded Equation of gradient d = d is stated or implied d d is calculated value of is sued in to find gradient m is calculated Know to differentiate Differentiation is carried out correctl Differential set equal to zero Both values of are found Both -co-ordinates of points are found Nature of stationar points are determined () Nature of stationar points are stated 6 d d + + sketch (,7 ) is a maimum turning point (6, ) is a minimum turning point 8 Page 6 HSN

8 Unit Level C Assessment Outcome Recurrence Relations a Un+ =.Un + 6 Recurrence relation is set up correctl a =. = 6 a and are correctl Limit eists since a identified and sued L = into L = a a 6 L is calculated =. Limit is interpreted correctl = 88. In the long term, the numer of acteria will settle at roughl 88 million (88 ) Page 7 HSN