Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#5

  Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#3

Question Find the equation of the normal to the curve at the point (3, 1). Solution  We are given equation of the curve as; We are required to find the equation of normal to the curve at the point (3, 1). To find the equation of the line either we need coordinates of the two […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#3

Question A curve has equation Find the exact gradient of the curve at the point for which y = 4. Solution We are required to find the gradient of the curve at a point where y=4. Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#3

Question A curve has equation Find the exact gradient of the curve at the point for which y = 4. Solution We are required to find the gradient of the curve at a point where y=4. Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#6

  Question The diagram shows the curve with parametric equations  ,    for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1.     i.       Show that the value of t at the point P satisfies the equation .    ii.       Use the iterative formula  with t1 =0.7 to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#3

  Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for gradient […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/22) | Q#6

  Question The diagram shows the curve with parametric equations  ,    for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1.     i.       Show that the value of t at the point P satisfies the equation .    ii.       Use the iterative formula  with t1 =0.7 to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/22) | Q#3

  Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#5

Question The curve with equation and the point P on the curve has y-coordinate 10. i.               Show that the x-coordinate of P satisfies the equation ii.               Use the iterative formula to find the x-coordinate of P correct to 4 significant figures. Give the  result  of each iteration to 6 significant figures. iii.               Find the gradient […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#6

Question a.   Show that b.   Find the exact value of showing all necessary working. Solution a.     We are required to show; Rule for integration of  is: Division Rule; Power Rule;   b.     We are required to find; We know that Therefore, we can rearrange; Hence; Similarly, we know that; Therefore, we can rearrange; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#5

Question The curve with equation crosses the x-axis at only one point. This point has coordinates .   i.               Show that satisfies the equation ii.               Show by calculation that 0.75 < p < 0.85.  iii.               Use an iterative formula based on the equation in part (i) to find the value of p  correct to 5 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#4

  Question Find the gradient of the curve at the point (0, 2). Solution We are required to find gradient of the curve ta the point (0 , 2). Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that particular point. Gradient (slope) of the curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/23) | Q#5

Question A curve has parametric equations      i.       Find the equation of the tangent to the curve at the origin.    ii.       Find the coordinates of the stationary point, giving each coordinate correct to 2  decimal places. Solution i.   We are required to find the equation of tangent to the curve at the origin. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/21) | Q#5

Question A curve has parametric equations      i.       Find the equation of the tangent to the curve at the origin.    ii.       Find the coordinates of the stationary point, giving each coordinate correct to 2  decimal places. Solution i.   We are required to find the equation of tangent to the curve at the origin. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/23) | Q#2

Question A curve has equation      i.       Find the x-coordinate of the stationary point.    ii.       Determine whether the stationary point is a maximum or minimum point. Solution i.   First we are required to find the x-coordinate of the stationary point of the curve. A stationary point on the curve is the point where […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/22) | Q#2

Question A curve has equation      i.       Find the x-coordinate of the stationary point.    ii.       Determine whether the stationary point is a maximum or minimum point. Solution i.   First we are required to find the x-coordinate of the stationary point of the curve. A stationary point on the curve is the point where […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/21) | Q#4

Question The diagram shows the curve . The curve crosses the x-axis at the point P and has a maximum point M.      i.       Find the gradient of the curve at the point P.    ii.       Show that the x-coordinate of the point M satisfies the equation   iii.       Use an iterative formula based on […]

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 2017 | Oct-Nov | (P2-9709/23) | Q#6

Question The parametric equations of a curve are  ,        i.       Find  in terms of t and hence find the coordinates of the stationary point, giving each  coordinate correct to 2 decimal places.    ii.       Find the gradient of the normal to the curve at the point where the curve crosses the x-axis. Solution […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/22) | Q#5

Question The diagram shows the curve y=4e−2x and a straight line. The curve crosses the y-axis at the point  P. The straight line crosses the y-axis at the point (0, 9) and its gradient is equal to the gradient of  the curve at P. The straight line meets the curve at two points, one of […]