# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#9

Question The functions  and  are defined by   for     for       i.       Express  in terms of  and solve the equation .    ii.       On the same diagram sketch the graphs of  and , showing the coordinates of their point of intersection and the relationship between the graphs.   iii.       Find the set of values of  which satisfy . Solution i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#5

Question i.    Sketch, on the same diagram, the graphs of  and  for . ii.       Verify that  is a root of the equation , and state the other root of this equation for which .  iii.       Hence state the set of values of , for , for which Solution i.   We are required to sketch  and  for . First we sketch  for . We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#11

Question Functions  and  are defined by         i.       Express  in the form , where ,  and  are constants.    ii.       State the range of .   iii.       State the domain of .   iv.       Sketch on the same diagram the graphs of ,  and , making clear the relationship between the graphs..    v.       Find an expression for . Solution i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#10

Question Functions  and  are defined by ,        i.       Evaluate .  ii.       Sketch in a single diagram the graphs of  and , making clear the relationship between the graphs.   iii.      Obtain an expression for  and use your answer to explain why  has an inverse.   iv.       Express each of  and , in terms of . Solution i.   We have functions; We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#9

Question The function f is such that , for , where  is a constant.      i.       In the case where ,         a.   Find the range of ,        b.   Find the exact solutions of the equation .    ii.       In the case where ,        a.   Sketch the graph of  ,        b.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/11) | Q#3

Question i.       Sketch the curve ii.      The region enclosed by the curve, the x-axis and the y-axis is rotated through 360◦ about the x-axis. Find the volume obtained, giving your answer in terms of . Solution      i.   Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If  (‘a’ is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#7

Question i.   A straight line passes through the point (2, 0) and has gradient . Write down the equation of the line. ii.  Find the two values of  for which the line is a tangent to the curve . For each value of , find the coordinates of the point where the line touches the curve.   iii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#2

Question Find the set of values of  for which the line  intersects 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/11) | Q#7

Question A curve is such that  and the point  lies on the curve. i.       Find the equation of the curve.    ii.       Find the set of values of x for which the gradient of the curve is less than . Solution     i.   We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#2

Question The functions  and  are defined for  by where  and  are constants. Given that  and , find      i.       the values of  and ,    ii.       an expression for . Solution i.   We are given that; We need to find  and . First we find ; We have; For ; Now we evaluate ; We are given that; Therefore; Now we find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#4

Question A function  is defined for  and is such that . The range of the function is given by .      i.       State the value of  for which  has a stationary value.    ii.       Find an expression for  in terms of . Solution i.   We have; The expression for  represents derivative of . A stationary point  on the curve  is the point where gradient of the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#6

Question The function f is defined by , ,      i.     Show that .    ii.     Hence, or otherwise, obtain an expression for . Solution i.   We have function; We can write these as; For , we can write; ii.   We have functions;  , We can write these as; To find the inverse of a given function  we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/11) | Q#11

Question Functions  and  are defined for  by      i.       Find and simplify expressions for  and .    ii.       Hence find the value of a for which .   iii.       Find the value of  for which .   iv.       Find and simplify an expression for . The function  is defined by  ,    v.       Find an expression for . Solution i.   We have functions; We can write these […]