# 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

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.   We are […]

# 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

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. Find the […]

# 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

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 as; Therefore; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#9

Question      i.       Express  in the form , where a, b and c are constants. The function  is defined for , where  is a constant.    ii.       State the smallest value of  for which  is one-one.   iii.       For the case where , find n expression for and state the domain of . Solution i.   We have the expression; We use method of “completing square” to obtain […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/13) | Q#6

Question The diagram shows the graph of , where  is defined by  for 0 < x ≤ 2.      i.       Find an expression for  and state the domain of .    ii.       The function  is defined by  for x ≥ 1. Find an expression for , giving your  answer in the form ax + b, where a and b are constants to be […]

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

Question Express  in the form , where a, b and c are constants. Solution  We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘2’ from the terms which involve ; Next we complete the square for the terms which involve . We have the algebraic formula; For the given case we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined by  for .      i.       Find the set of values of p for which the equation  has no real roots. The function  is defined by   for .    ii.       Express  in the form , where a, b and c are constants.   iii.       Find the range of . The function  is defined by   for , 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 | May-Jun | (P1-9709/11) | Q#9

Question The equation of a curve is , where  is a positive constant.      i.       Show that the origin is a stationary point on the curve and find the coordinates of the other  stationary point in terms of .    ii.          Find the nature of each of the stationary points. Another curve has equation .   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#8

Question The function  is defined for .      i.       Solve the equation , giving your answer correct to 2 decimal places.    ii.       Sketch the graph of .   iii.       Explain why  has an inverse.   iv.       Obtain an expression for . Solution      i.   We are given; We are required to solve; Therefore; Using calculator;    ii.   Ware required to sketch […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#5

Question A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius r cm.      i.       Show that the area of the sector, A cm2, is given by .    ii.       Express A in the form , where a and b are constants.   iii.       Given that r can vary, state […]