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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

Hits: 17   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 […]

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

Hits: 22 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 […]

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#7

Hits: 6 Question The equation of a curve is x2+4xy+2y2=7      i.       Find the equation of the tangent to the curve at the point (-1, 3). Give your answer in the form  ax +by +c = 0, where a, b and c are integers.      ii.       Show that there is no point on the curve […]

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

Hits: 7 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, […]

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

Hits: 6 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 […]

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

Hits: 8 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 […]

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#6

Hits: 8 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 […]

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

Hits: 30 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 […]

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

Hits: 17 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | May-Jun | (P2-9709/21) | Q#8

Hits: 20 Question The diagram shows the curve with equation The curve crosses the x-axis at the point P and has a minimum point M.      i. Find the gradient of the curve at the point P.    ii. Find the exact coordinates of the point M. 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 2017 | May-Jun | (P2-9709/21) | Q#7

Hits: 17 Question The parametric equations of a curve are      i.       Find  and use division to show that  can be written in the form , where a and b are  constants to be found.    ii.       The straight line x − 2y + 9 = 0 is the normal to the curve at the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#11

Hits: 4081 Question The diagram shows the curve  and points A(1,0) and B(5,2) lying on the curve. i.       Find the equation of the line AB, giving your answer in the form y=mx+c.    ii.       Find, showing all necessary working, the equation of the tangent to the curve which is parallel to AB.   iii.       Find the perpendicular distance between the line AB […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#10

Hits: 884 Question A curve has equation y=f(x) and it is given that , where a and b are positive  constants. i.       Find, in terms of a and b, the non-zero value of x for which the curve has a stationary point and  determine, showing all necessary working, the nature of the stationary point.    ii.       It is now given that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 608 Question The function f is such that  for , where k is a constant. Find the  largest value of k for which f is a decreasing function. Solution To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function at that point. If  , the function  is increasing. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 1983   Question The diagram shows part of the curve  and the normal to the curve at the point P(2, 3).  This normal meets the x-axis at Q.      i.       Find the equation of the normal at P.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are required to find the equation of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 653   Question A curve is such that .                    i.       Find the x-coordinate of each of the stationary point on the curve.                  ii.       Obtain an expression for  and hence or otherwise find the nature of each of the  stationary points.   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 782   Question Points A and B lie on the curve . Point A has coordinates (4,7) and B is the  stationary point of the curve. The equation of a line L is , where m is a constant.                             i.       In the case where L passes through the mid-point of AB, find the value of m.                           ii.       Find the set of […]