# 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 | Oct-Nov | (P1-9709/13) | Q#8

Question A curve  has a stationary point at . It is given that, where k is a constant. i.       Show that  and hence find the x-coordinate of the other stationary point, Q.    ii.       Find  and determine the nature of each of the stationary points P and Q.   iii.       Find . Solution i.   A stationary point  on the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#8

Question The equation of a curve is . Find     i.       an expression for  and the coordinates of the stationary point on the curve,    ii.       the volume obtained when the region bounded by the curve and the x-axis is rotated through  about the x-axis. Solution i.   Gradient (slope) of the curve is the derivative of equation of the curve. Hence […]

# 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#10

Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.     i.       Find the coordinates of B and C.    ii.       Find the equation of the normal to the curve at C.   iii.       Find the volume obtained when the shaded region is rotated through  about the y-axis. […]

# 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#7

Question The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48m.     i.       Find an expression for  in terms of .    ii.       Given that the area of the garden is A m2, show that .   iii.       Given that  can vary, find the maximum area of the garden, showing that this is a maximum […]

# PPast 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#2

Question A curve has equation . Show that the gradient of the curve is never negative. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: For the given case; Rule for differentiation of  is: Therefore; Rule for differentiation of  is: Rule for differentiation of  is: Hence; […]

# 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#9

Question A curve is such that  and P(9, 5) is a point on the curve.     i.       Find the equation of the curve.    ii.       Find the coordinates of the stationary point on the curve.   iii.       Find an expression for  and determine the nature of the stationary point.   iv.       The normal to the curve at P makes an angle of  with the positive x-axis. Find […]

# 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#4

Question a)   Differentiate  with respect to . b)   Find   and hence find the value of a . Solution a)   We are given that; Rule for differentiation of  is: Rule for differentiation of  is: b)     We are given that; Rule for integration of  is: Now we find the value of;

# 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#3

Question The line  , where a and b are positive constants, meets the x-axis at P and the y-axis at Q. Given that   and that the gradient of the line PQ is  , find the values of a and b. Solution We are given that points P and Q are x and y intercepts, respectively. First we find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#11

Question The diagram shows part of the curve . The curve has a maximum point at M and meets the x-axis at O and A.     i.       Find the coordinates of A and M.    ii.       Find the volume obtained when the shaded region is rotated through 360o about the x-axis, giving your answer in terms of . Solution     i. […]

# 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 | May-Jun | (P1-9709/11) | Q#6

Question The variables x, y and s can take only positive values and are such that  and     i.       Show that .    ii.     Find the stationary value of  and determine its nature. Solution     i.   We are given that; We are also given that; We can find out y from this equation; Substituting this value of y in;   ii.   […]

# 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#2

Question The volume of a spherical balloon is increasing at a constant rate of 50 cm3 per second. Find the rate of increase of the radius when the radius is 10 cm. [Volume of a sphere ] Solution      i.   We are given that; We are required to find rate of change of radius; Therefore; For rate of change of […]