# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#11

Question The function  is defined for .      i.       Find the exact value of  for which .    ii.       State the range of .   iii.       Sketch the graph of .   iv.       Find an expression for . Solution i.   We are given that; We can write it as; We are given that , therefore; Using calculator; ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#6

Question The equation of a curve is , where  and  are constants.      i.       In the case where the curve has no stationary point, show that .    ii.       In the case where  and , find the set of values of  for which  is a decreasing               function of . Solution i.   We are given; Gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#10

Question      i.       Express  in the form .  The function  is defined for , where  and  are positive constants, by The range of  is given by , where  and  are constants.    ii.       State the smallest possible value of . For the case where  and ,   iii.       find  and ,   iv.       find an expression for . Solution i.   We have the expression; We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#5

Question Find the set of values of  for which the line  meets the curve                     at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#8

Question      i.       Express  in the form , where a, b and c are constants, and use your  answer to state the minimum value of .    ii.       Find the set of values of  for which the equation  has no real roots. Solution      i.   We have the expression; We use method of “completing square” to obtain the desired form. We take out […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#5

Question A functions  is such that   for   .      i.       Find an expression for  and use your result to explain why  has an inverse.    ii.       Find an expression for , and state the domain and range of . Solution i.   We have; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#10

Question Functions and  are defined by  for  for        i.       Solve the equation .    ii.       Find the range of .   iii.       Find the set of values of x for which .   iv.       Find the value of the constant  for which the equation  has two equal roots. Function  is defined by  for , and it is given that  has an inverse.    v.       State […]