# 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#7

Question A curve is defined by the parametric equations for .       i.       Find the exact gradient of the curve at the point for which .    ii.       Find the value of at the point where the gradient of the curve is 2, giving the value correct to 3 significant figures. Solution      i. […]

# 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#7

Question The parametric equations of a curve are for . At the point P on the curve, the gradient of the curve is 2.      i.       Show that the value of the parameter at P satisfies the equation .    ii.       By first expressing in the form , where R > 0 and   , find […]

# 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#3

Question The diagram shows the curve with equation for values of x such that . Find the x-coordinate of the stationary point M, giving your answer correct to 3  significant figures. Solution We are required to find the x-coordinate of stationary point M. A stationary point on the curve is the point where gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve with equation  for . At  the points P and Q on the curve, the gradient of the curve is 3.      i.    Find an expression for .    ii.     By first expressing  in the form , where R > 0 and ,  find the x-coordinates of […]

# 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#7

Question The diagram shows the curve with equation  for . At  the points P and Q on the curve, the gradient of the curve is 3.      i.     Find an expression for .    ii.     By first expressing  in the form , where R > 0 and ,  find the x-coordinates of […]

# 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#5

Question A curve has equation Find the equation of the tangent to the curve at the point where it crosses the y-axis. Solution We are given that curve with equation  and we are required to find the equation of  the tangent to the curve at point where it crosses the y-axis i.e.  . To find […]

# 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#5

Question A curve has equation Find the equation of the tangent to the curve at the point where it crosses the y-axis. Solution We are given that curve with equation  and we are required to find the equation of  the tangent to the curve at point where it crosses the y-axis i.e.  . To […]

# 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#5

Question The parametric equations of a curve are for .     i.       Find the gradient of the curve at the point for which  radian.    ii.       Find the value of at the point on the curve where the tangent is parallel to the y-axis. Solution      i.   We are required to find gradient of […]

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

Question A curve has equation . Find the equation of the tangent to the curve at the point for which . Solution      i.   We are given that curve with equation  and we are required to find the equation of the  tangent to the curve at origin . To find the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Feb-Mar | (P2-9709/22) | Q#4

Question Find the gradient of the curve at the point . Solution We are required to find the gradient of the curve at point . 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 at a particular […]

# 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#7

Question The diagram shows the curve The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on  the curve has x-coordinate q and the gradient of the curve at Q is −m. i.       Find the value of m and hence show that q […]

# 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#3

Question The equation of a curve is . The curve has a stationary point M in the interval . Find the coordinates of M, giving each coordinate correct to 3 significant figures.  Solution We are required to find the x-coordinates of stationary point of the curve. A stationary point on the curve is the point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/21) | Q#7

Question The diagram shows the curve The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on  the curve has x-coordinate q and the gradient of the curve at Q is −m.     i.       Find the value of m and hence show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | May-Jun | (P2-9709/23) | Q#8

Question The diagram shows the curve with parametric equations for . The end-points of the curve are (1, 4) and (3, 3). i.       Show that .    ii.       Find the coordinates of the minimum point, giving each coordinate correct to 3 significant  figures.   iii.       Find the exact gradient of the normal to the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | May-Jun | (P2-9709/22) | Q#8

Question The diagram shows the curve with parametric equations for . The end-points of the curve are (1, 4) and (3, 3). i.       Show that .    ii.       Find the coordinates of the minimum point, giving each coordinate correct to 3 significant  figures.   iii.       Find the exact gradient of the normal to the curve […]

# 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 | (P2-9709/21) | 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 | (P2-9709/22) | 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. […]