# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#7

Question      i.       The diagram shows part of the curve  and part of the straight line  meeting at the point , where  and  are positive constants. Find the values of  and .    ii.             The function f is defined for the domain  by Express  in a similar way. Solution i.   It is evident from the diagram that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#11

Question The function  is such that , for .      i. Find the coordinates and the nature of the stationary point on the curve . The function g is such that , , where  is a constant.    ii. State the smallest value of  for which  has an inverse. For this value of k,   iii. Find an expression for .   iv. Sketch, on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#10

Question Functions and  are defined by  for  for  ,      i. Obtain expressions, in terms of , for  and , stating the value of  for which  is not defined.    ii. Sketch the graphs of  and  on the same diagram, making clear the relationship between the two graphs.   iii. Given that the equation , where  is a constant, has no solutions, find the set of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#10

Question It is given that a curve has equation , where .     i.       Find the set of values of  for which the gradient of the curve is less than 5.    ii.       Find the values of  at the two stationary points on the curve and determine the nature of each stationary point. Solution i.   Gradient (slope) of the curve is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#6

Question The functions  and  are defined for  by Solve the following equations for .      i.       , giving your answer in terms of .    ii.       , giving your answers correct to 2 decimal places. Solution i.   We have the functions; We write as; We are given that; Therefore; ii.   We have the functions; We write as; We are given that; Therefore; We utilize the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#2

Question The function  is such that  for . Find      i.     in the form  where ,  and  are constants,    ii.       the domain of . Solution i.   We have the functions;  for We write it as; To find the inverse of a given function  we need to write it in terms of  rather than in terms of . Interchanging ‘x’ with ‘y’; ii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#10

Question The function  is defined by  for .      i.       Express  in the form , and hence state the coordinates of the vertex of the graph of  . The function  is defined by , for .    ii.       State the range of .   iii.       Find an expression for  and state the domain of . Solution i.   We have the function; We have the expression; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#8

Question The function  is defined for the domain , where  and  are constants.      i.       Express  in the form , where  and  are constants.    ii.       State the range of  in terms of .   iii.       State the smallest value of  for which  is one-one.   iv.       For the value of  found in part (iii), find an expression for  and state the domain of  , giving your answers […]