# 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#7

Question The diagram shows part of the curve The shaded region is bounded by the curve and the two axes.        i. Show that  can be expressed in the form where the values of the constants  and are to be determined.    ii.Find the exact area of the shaded region. Solution      i. […]

# 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#3

Question      i.       Solve the inequality .    ii.       Hence find the largest integer y satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE      i.   Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following […]

# 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#7

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 at which […]

# 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 […]

# 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#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 | 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 | (P2-9709/23) | Q#4

Question Find the equation of the tangent to the curve  at the point on the curve for which x = 0.  Give your answer in the form ax+by +c = 0 where a, b and c are integers. Solution We are given that curve with equation  and we are required 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 | 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 2017 | May-Jun | (P2-9709/22) | Q#4

Question Find the equation of the tangent to the curve  at the point on the curve for which x = 0. Give your answer in the form ax+by +c = 0 where a, b and c are integers. Solution We are given that curve with equation  and we are required to find the equation […]

# 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

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 find the […]

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

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 point P. […]

# 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#2

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following with […]

# 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#3

Question It is given that the variable x is such that  and Find the set of possible values of x, giving your answer in the form a < x < b where the constants a and b are correct to 3 significant figures. Solution First we find the value of x for; Taking natural […]

# 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#2

Question It is given that x satisfies the equation . Find the possible values of Solution SOLVING EQUATION: PIECEWISE Let, . We can write it as; We have to consider two separate cases; When When We have the equation; It  can be written as; We have to consider two separate cases; When ; When […]

# 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#3

Question It is given that the variable x is such that  and Find the set of possible values of x, giving your answer in the form a < x < b where the constants a and b are correct to 3 significant figures. Solution First we find the value of x for; Taking natural […]

# 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

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 and 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#8

Question The diagram shows parts of the graphs of  and  intersecting at points A and  B.      i.       Find by calculation the x-coordinates of A and B.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are required to find the x-coordinates of points A and B which are intersection points of a line  and […]

# 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#2

Question Find the set of values of a for which the curve  and the straight line  meet at two  distinct points. Solution We need to find the equation that satisfies the x-coordinates of the points of intersection of given  curve and line. If two lines (or a line and a curve) intersect each other at a point then that point […]