# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#8

Question      i.      Express  in the form , where a, b and c are constants.    ii. Functions f and g are both defined for . It is given that  and     . Find .   iii.       Find  and give the domain of .   Solution      i.   We have the expression; We use method of “completing square” […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#10

Question A function f is defined by  for .     i.       Find the range of f.    ii.       Sketch the graph of   iii.       Solve the equation , giving answers in terms of . The function  is defined by  for , where k is a constant.   iv.       State the largest value of k for which g has an inverse.    v.       For this value of k, find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#8

Question The functions f and g are defined by  for  for .      i.       Find and simplify an expression for  and state the range of .      ii.       Find an expression for  and find the domain of . Solution      i.   We are given;  for  for . We are required to find expression for ;   We are also […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#10

Question The function f is such that  for . The function g is such that  for      , where a, b and q are constants. The function fg is such that  for . i.       Find the values of a and b. ii.       Find the greatest possible value of q. It is now given that . iii.       Find the range of fg. iv.       Find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined by  for .      i.       Find the set of values of x for which f(x) ≤ 3.    ii.       Given that the line y=mx+c is a tangent to the curve y = f(x), show that The function g is defined by  for x ≥ k, where k is a constant.   iii.       Express   in the form , where a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#1

Question Functions  and  are defined by , ,, Solve the equation  Solution   First we find left hand side of required equation . We are given; It can be written as; Therefore; Now we find right hand side of required equation . We are given; These can be written as; Therefore; Now for  substitute  ; Finally, equating left […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#11

Question The function  is defined by  for .         i.       State the range of .    ii.       Find the coordinates of the points at which the curve  intersects the coordinate                  axes.    iii.       Sketch the graph of .   iv.       Obtain an expression for , stating both the domain and range of . Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#8

Question The function  is such that  for , where a and b are constants.      i.       For the case where  and , find the possible values of  a and b.    ii.       For the case where  and , find an expression for  and give the domain of  . Solution i.   We are given that function  is represented as; We are given that; First we substitute […]