# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#3

Question The parametric equations of a curve are Find the equation of the tangent to the curve when t = 0 giving your answer in the form ax + by + c =  0, where a, b and c are integers. Solution We are required to find the equation of tangent to the curve at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#6

Question The diagram shows the curve with parametric equations for .     i.       Show that .    ii.       Find the equation of the tangent to the curve at the point where the curve crosses the positive y-axis. Give the answer in the form y = mx +c. Solution      i.   We are required to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#3

Question The variables x and y satisfy the equation y = Kxm, where K and m are constants. The graph of ln y  against ln x is a straight line passing through the points (0.22, 3.96) and (1.32, 2.43), as shown in  the diagram. Find the values of K and m, correct to 2 significant […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#6

Question The diagram shows part of the curve and its point of intersection P with the x-axis.      i.       Find the exact x-coordinate of P.    ii.       Show that the equation of the curve can be written and use integration to find the exact area of the shaded region enclosed by the curve and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#5

Question      i.       By sketching a suitable pair of graphs, show that the equation has two real root.    ii.       Use the iterative formula to find one of the real roots correct to  3 decimal places. Give the result of each iteration to 5 decimal places.   iii.       Hence find the coordinates of each of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#6

Question The diagram shows part of the curve and its point of intersection P with the x-axis.      i.       Find the exact x-coordinate of P.    ii.       Show that the equation of the curve can be written and use integration to find the exact area of the shaded region enclosed by the curve and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#5

Question      i.       By sketching a suitable pair of graphs, show that the equation has two real root.      ii.       Use the iterative formula to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places.   iii.       Hence find the coordinates of each […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#7

Question The equation of a curve is y3+4xy=16      i.       Show that .    ii.       Show that the curve has no stationary points.   iii.       Find the coordinates of the point on the curve where the tangent is parallel to the y-axis. Solution      i.   We are required to find . Hence; To find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#3

Question The equation of a curve is Find the equation of the tangent to the curve at the point  . Give the answer in the form y = mx + c, where the values of m and c are correct to 3 significant figures. Solution We are required to find the equation of tangent to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#2

Question The variables x and y satisfy the equation where A and p are constants. The graph of against x is a straight line passing through the  points (2,1.60) and (5, 2.92) as shown in the diagram. Find the values of A and p correct to 2  significant figures. Solution We are given; Taking natural […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#2

Question      i.       Solve the equation .    ii.       Hence, using logarithms, solve the equation , giving the answer correct to  3 significant figures. Solution i.   SOLVING EQUATION: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#1

Question      i.       Solve the equation .    ii.       Hence, using logarithms, solve the equation , giving the answer correct to 3  significant figures. Solution     SOLVING EQUATION: ALGEBRAICALLY i.   Let, . We can write it as; We have to consider two separate cases; When When We have the equation; It […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#1

Question      i.       Solve the equation .    ii.       Hence, using logarithms, solve the equation , giving the answer  correct to 3 significant figures. Solution i.   SOLVING EQUATION: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have […]

# 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 […]