# 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 | Oct-Nov | (P1-9709/11) | Q#5

Question The diagram represents a metal plate , consisting of a sector  of a circle with centre  and radius , together with a triangle  which is right-angled at C. Angle  radians and  is perpendicular to . i.       Find an expression in terms of  and  for the perimeter of the plate. ii.    For the case where  and  , find […]

# 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#9

Question A line has equation  and a curve has equation , where  is a constant.     i.       For the case where , the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB.    ii.       Find the two values of  for which the line is a tangent to the curve. […]

# 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#8

Question Relative to an origin , the point A has position vector  and the point B has position vector  , where  is a constant.      i.       Find .    ii.       Hence show that there are no real values of  for which  and  are perpendicular to each other.   iii.       Find the values of  for which angle AOB =. Solution      i.   A point  has […]

# 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 […]

# 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#3

Question      i.       Sketch, on a single diagram, the graphs of  and  for .    ii.       Write down the number of roots of the equation  in the interval .   iii.       Deduce the number of roots of the equation  in the interval . Solution     i.        ii.   If two lines (or a line and a curve) intersect each other at a […]

# 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 | 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 | Oct-Nov | (P1-9709/11) | Q#1

Question Find the term independent of  in the expansion of  . Solution Expression for the Binomial expansion of  is: In the given case: Hence; The term independent of  in the expansion of is .

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#6

Question a)   The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find  the seventh term. b)   A geometric progression has first term 1 and common ratio r. A second geometric progression  has first term 4 and common ratio . The two progressions have the same sum to […]