Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Feb-Mar | (P2-9709/22) | Q#7

Question The equation of a curve is 2×3+y3=24      i.       Express in terms of x and y, and show that the gradient of the curve is never positive.    ii.       Find the coordinates of the two points on the curve at which the gradient is −2. Solution      i.   Gradient (slope) of the curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Feb-Mar | (P2-9709/22) | Q#6

  Question The diagram shows the part of the curve  for , and the stationary point M.        i.       Find the equation of the tangent to the curve at the origin.    ii.       Find the coordinates of M, giving each coordinate correct to 3 decimal places. Solution      i.   We are given that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#6

Question A curve has parametric equations      i.       Find an expression for in terms of t.    ii.       Find the exact value of at the stationary point.   iii.       Find the gradient of the curve at the point where it crosses the x-axis. Solution      i.   We are required to find  for the parametric […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#7

Question The diagram shows the curve with parametric equations for .     i.       Show that can be expressed in the form    ii.       Find the equation of the normal to the curve at the point where the curve crosses the positive y-axis. Give your answer in the form y = mx +c, where the constants […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#5

Question The diagram shows the curve and its stationary point M. The x-coordinate of M is m.       i.       Find an expression for and hence show that .    ii.       Use an iterative formula based on the equation in part (i) to find the value of m correct to 4  significant figures. Give the result […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#3

Question A curve has equation y = 2 sin 2x − 5 cos 2x +6 and is defined for 0 ≤ x ≤ π. Find the x-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. Solution We are required to find the x-coordinates of stationary points of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#7

Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q.     i.       Show that .    ii.       Find the coordinates of M.   iii.       Find the gradient of the curve at P and at Q. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#5

Question The equation of a curve is  . At the point on the curve with x-coordinate p, the gradient of  the curve is .      i.       Show that .      ii.       Show by calculation that 3.3 < p < 3.5.     iii.       Use an iterative formula based on the equation in part (i) […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#7

Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q.     i.       Show that .    ii.       Find the coordinates of M.   iii.       Find the gradient of the curve at P and at Q. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#5

Question The equation of a curve is  . At the point on the curve with x-coordinate p, the gradient of  the curve is .      i.       Show that .      ii.       Show by calculation that 3.3 < p < 3.5.     iii.       Use an iterative formula based on the equation in part (i) […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#6

Question The equation of a curve is  . At the point on the curve with positive x-coordinate p, the gradient of the curve is .        i.       Show that .      ii.       Show by calculation that 2 < p < 3.     iii.       Use an iterative formula based on the equation in part […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#5

Question A curve is defined by the parametric equations for .       i.       Show that .    ii.       Find the coordinates of the stationary point.   iii.       Find the gradientof the curve at point . Solution      i.   We are required to show that  for the parametric equations given below; If a curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#1

Question Find the gradient of the curve at the point for which x = 0. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is: We are given that; Therefore; Rule for differentiation of  is: Rule for differentiation of natural exponential function is; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#11

  Question A curve has equation , where k is a non-zero constant.      i.       Find the x-coordinates of the stationary points in terms of k, and determine the nature of each  stationary point, justifying your answers.    ii.   The diagram shows part of the curve for the case when k = 1. Showing all necessary working, find  the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#10

  Question A curve is such that  , where a is a positive constant. The point A(a2, 3) lies on the  curve. Find, in terms of a,      i.       the equation of the tangent to the curve at A, simplifying your answer,    ii.       the equation of the curve.  It is now given that B(16,8) also lies on the curve.   iii.       Find […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#4

  Question The function f is such that  for , where n is an integer. It is given that  f is an increasing function. Find the least possible value of n. Solution We are given function; We are also given that it is an increasing function. To test whether a function  is increasing or decreasing at a particular point , […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#7

Question The equation of a curve is .     i.       Obtain an expression for    ii.       Explain why the curve has no stationary points.  At the point P on the curve, x = 2.   iii.       Show that the normal to the curve at P passes through the origin.   iv.       A point moves along the curve in such a way that its x-coordinate is decreasing […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#11

Question The point P(3,5) lies on the curve  .      i.       Find the x-coordinate of the point where the normal to the curve at P intersects the x-axis.    ii.       Find the x-coordinate of each of the stationary points on the curve and determine the nature of  each stationary point, justifying your answers. Solution      i.   The point where […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#10

Question A curve has equation  and it is given that . The point A is the only point  on the curve at which the gradient is −1.      i.       Find the x-coordinate of A.    ii.       Given that the curve also passes through the point (4,10), find the y-coordinate of A, giving  your answer as a fraction. Solution      i.   […]