# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#12

Question A diameter of a circle C1 has end-points at (−3, −5) and (7, 3). a)   Find an equation of the circle C1. The circle C1 is translated by  to give circle C2, as shown in the diagram. b)  Find an equation of the circle C2.  The two circles intersect at points R and S. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#9

Question a)Express in the form , where  and  are constants. The function f is defined by  for . b)Find an expression for and state the domain of . The function is defined by  for . c)For the case where k = −1, solve the equation . d)State the largest value of possible for the composition […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#2

Question The graph of  is transformed to the graph of . Describe fully the two single transformations which have been combined to give the  resulting transformation. Solution We are given function; The following function represents the vertical translation (along y-axis) of graph by units upwards; Therefore, for the given case, following will represent vertical translation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#2

Question The function g is defined by  for . By first completing the square, find an  expression for and state the domain of . Solution We are given that; We use method of “completing square” to obtain the desired form. We complete the square for the  terms which involve . We have the algebraic formula; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#9

Question Functions f and g are defined by;  for   for Where  is a constant.     i.      Find the value of for which the line is a tangent to the curve .   ii.     In the case where , find the set of values of for which .  iii.     In the case where , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#6

Question a)   Given that x > 0, find the two smallest values of x, in radians, for which . Show  all necessary working. b)  The function is defined for . i.     Express f(x) in the form , where a and b are constants. ii.      Find the range of f. Solution a)     We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#7

Question The functions f and g are defined by  for  for . i.       Find the range of f and the range of g.    ii.       Find an expression for fg(x), giving your answer in the form  , where a, b and c are  integers.  iii.      Find an expression for , giving your answer in the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#4

Question The function f is defined by  for . The function g is defined by for   where  and are constants.     i.       Find the greatest value of and the least value of which will permit the formation of the composite function gf. It is now given that the conditions for the formation of gf are satisfied. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#7

Question The functions f and g are defined by , , i.Obtain expressions for and , stating the value of x for which is not defined. ii.Solve the equation Solution i. We are given;  for  for We have; We write them as; To find the inverse of a given function we need to write it […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#9

Question The functions f is defined by  for  ,     i.      State the range of .   ii.     Sketch the graph of . The functions g is defined by for , where is a constant.  iii.     State the largest value of for which g has an inverse.  iv.     For this value of , find an […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#5

Question The function f is defined by  for .     i.      Express in the form of where and are constants.   ii.     State the greatest value of .    The function g is defined by  for .  iii.     Find the value of for which . Solution i.   We have the expression;   We use […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#8

Question     i.      Express in the form of . The function f is defined by  for , where is constant.   ii.     State the largest value of for which is a decreasing function. The value of is now given to be 1.  iii.     Find an expression for and state the domain of .  iv.     The […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#11

Question     i.      Express in the form of where a, b and c are constants. The function f is defined by  for .    ii.       State the largest value of the constant k for which f is a one-one function.  iii.     For this value of k find an expression for and state the domain of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#9

Question The function f is defined by  for .     i.      Express in the form of where a and b are constants.   ii.     State the range of .  The function g is defined by  for .  iii.       State the largest value of k for which g has an inverse.   iv.     Given that g has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#4

Question The functions f and g are defined by  for  ,  for      i.      Solve the equation .   ii.     Sketch the graph of . Solution i.   We are given;  for  for We are required to solve equation ; To solve this equation for , we can substitute . Hence, Since given interval is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#11

Question The one-one function f is defined by  for , where  is a constant. a.                    i.     State the greatest possible value of .              ii.     It is given that takes this greatest possible value. State the range of f and find an expression for .   b.    The function g is defined by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#10

Question The functions f and g are defined by  for  for . i.   a.   State the range of the function f. b.   State the range of the function g. c.   State the range of the function fg.    ii.       Explain why the function gf cannot be formed.   iii.       Find the set of values […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#10

Question The one-one function f is defined by  for , where c is a  constant.   i.       State the smallest possible value of c. In parts (ii) and (iii) the value of c is 4.    ii.       Find an expression for  and state the domain of .   iii.       Solve the equation , giving your […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#7

Question The function is defined by  for .      i.       Express   in the form , where a and b are constants.    ii.       State the coordinates of the stationary point on the curve y = f(x). The function is defined by  for .   iii.       State the smallest value of k for which g […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#4

Question The function f, is such that   for . It is given that  and .                             i.       Find the values of the constants a and b.                           ii.       Find the set of values of k for which the equation f(x) = k has no  solution. Solution i.   We are given the function as; We […]