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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#6

Hits: 613 Question The functions f and g are defined by   for   for      i.       Find an expression for .    ii.       Solve the equation . Solution i.   We are given the function; 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 . As […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#6

Hits: 996   Question a.   The function f, defined by   for , is such that  and .                             i.       Find the values of the constants a and b.                           ii.       Evaluate . b.   The function g is defined by  for for . The range of g is given by  . Find the values of the constants c and d. Solution a.   i.   We […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#2

Hits: 654   Question A function f is defined by  for .      i.       Find an expression for  and find the point of intersection of the graphs of  and  .    ii.       Sketch, on the same diagram, the graphs of  and , making clear the  relationship between the graphs. Solution i.   We have; We write it as; To find the inverse of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#9

Hits: 552   Question Functions f and g are defined for x > 3 by;      i.       Find and simplify an expression for gg(x).    ii.       Find an expression for  and state the domain of .   iii.       Solve the equation .  Solution      i.   We are given that; It can be written as; Therefore for ;    ii. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#2

Hits: 520   Question A function f is defined by  for . It is given that f is an increasing  function. Find the largest possible value of the constant a. Solution We are given that function f is increasing function. To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/12) | Q#10

Hits: 904   Question The function f is defined by , for .     i.       Solve the equation , giving your answer correct to 1 decimal place.    ii.       Find an expression for  and find the domain of .   iii.       Sketch, on the same diagram, the graphs of y=f(x) and . Solution      i.   We are given the function as; We […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#8

Hits: 585   Question The functions f and g are defined for  by i.       Show that  and obtain an unsimplified expression for . ii.       Find an expression for  and determine the domain of . iii.       Solve the equation . Solution      i.   We are given the functions; We can rewrite these as; First we find . Next we find .    […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/11) | Q#9

Hits: 575   Question The function f is defined by  for , .      i.        Find an expression for . The function g is defined by  for , where a is a constant.    ii.       Find the value of a for which .   iii.       Find the possible values of a given that the equation  has two equal roots. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/11) | Q#5

Hits: 837   Question The equation of a curve is .      i.       Sketch the graph of  for , stating the coordinates of the point of  intersection with the y-axis. Points P and Q lie on the curve and have x-coordinates of  and  respectively.    ii.       Find the length of PQ correct to 1 decimal place. The line through […]

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

Hits: 523   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 […]

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

Hits: 947 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 […]

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

Hits: 602 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 ;   […]

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

Hits: 714 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 […]

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

Hits: 1105 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 […]

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

Hits: 479 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, […]

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

Hits: 510 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 […]

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

Hits: 376 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; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#8

Hits: 501 Question The function  is defined by  for , where a is a constant. The function  is  defined  for .      i.       Find the largest value of a for which the composite function can be formed. For the case where ,    ii.       solve the equation ,   iii.       find the set of values of  which satisfy the inequality . Solution i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 562 Question The function f is defined, for , by , where a and b are constants.      i.       In the case where a = 6 and b = −8, find the range of f.    ii.       In the case where a = 5, the roots of the equation f(x)=0 0 are k and −2k, where k is a constant. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#1

Hits: 300   Question Functions f and g are defined by  ,  , Solve the equation . Solution First we find . We have; 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’; Now we find . We have; We write it […]