# 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#10

Question The function  is defined by  for .      i.       Find  and  and hence verify that the function  has a minimum value at . The points  and  lie on the curve , as shown in the diagram.      i.       Find the distance AB.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are given; Rule for differentiation […]

# 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#9

Question A curve passes through the point A(4,6) and is such that  . A point P is moving along  the curve in such a way that the x-coordinate of P is increasing at a constant rate of 3 units per  minute.      i.       Find the rate at which the y-coordinate of P is increasing when P is at A.    […]

# 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#1

Question A line has equation  and a curve has equation , where c is a constant. Find the set of possible values of c for which the line does not intersect the curve. 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.  […]

# 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#10

Question The diagram shows part of the curve . The point P(2,1) lies on the curve and the  normal to the curve at P intersects the x-axis at A and the y-axis at B.      i.       Show that B is the mid-point of AP. The shaded region is bounded by the curve, the y-axis and the line y = 1. […]

# 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#6

Question Points A, B and C have coordinates A(-3,7), B(5,1) and C(-1,k), where k is a constant.      i.       Given that AB=BC, calculate the possible values of k.  The perpendicular bisector of AB intersects the x-axis at D.    ii.       Calculate the coordinates of D. Solution i.   Expression to find distance between two given points  and is: We are […]

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

Question The diagram shows part of the curve  and a point P(6,5) lying on the curve. The line  PQ intersects the x-axis at Q(8,0).      i.       Show that PQ is a normal to the curve.   ii.       Find, showing all necessary working, the exact volume of revolution obtained when the shaded  region is rotated through 360o about the x-axis. [In part (ii) you […]

# 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#6

Question A curve has equation  and a line has equation , where  is a constant.      i.       Show that the x-coordinates of the points of intersection of the line and the curve are given by  the equation .    ii.       For the case where the line intersects the curve at two points, it is given that the x-coordinate  of one of the points of […]

# 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#10

Question Points A(2,9) and B(3,0) lie on the curve y=9+6x−3×2, as shown in the diagram. The tangent at A  intersects the x-axis at C. Showing all necessary working,     i.       find the equation of the tangent AC and hence find the x-coordinate of C,    ii.       find the area of the shaded region ABC. Solution      i.   To find the equation of […]

# 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#7

Question The point A has coordinates  and the point B has coordinates , where p is a  constant.      i.       For the case where the distance AB is 13 units, find the possible values of p.    ii.       For the case in which the line with equation 2x+3y=9 is perpendicular to AB, find the value of p. Solution      i. […]

# 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#10

Question The equation of a curve is  .      i.       Find, showing all necessary working, the volume obtained when the region bounded by the               curve, the x-axis and the lines x=1 and x=2 is rotated through 360O about the x-axis.    ii.     Given that the line  is a normal to the curve, find […]

# 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#7

Question The point C lies on the perpendicular bisector of the line joining the points A(4,6) and B(10,2). C  also lies on the line parallel to AB through (3,11).     i.       Find the equation of the perpendicular bisector of AB.    ii.       Calculate the coordinates of C. Solution i.   We are required to find the equation of the perpendicular bisector of […]

# 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#10

Question The diagram shows part of the curve  . The curve intersects the y-axis at A . The  normal to the curve at A intersects the line  at the point B.      i.       Find the coordinates of B.    ii.       Show, with all necessary working, that the areas of the regions marked P and Q are equal. Solution […]

# 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#6

Question The line with gradient -2 passing through the point P(3t,2t) intersects the x-axis at A and the y-axis  at B.      i.       Find the area of triangle AOB in terms of t. The line through P perpendicular to AB intersects the x-axis at C.    ii.       Show that the mid-point of PC lies on the line y = x. Solution […]

# 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#2

Question   The diagram shows the curve  and the points  and . The point Q lies on the  curve and PQ is parallel to the y-axis.   i. Express the area, A, of triangle XPQ in terms of .   The point P moves along the x-axis at a constant rate of 0.02 units per second and Q moves along  […]