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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#8

Hits: 45 Question The diagram shows the curve , for and its maximum point M.     i.       Show that    ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#8

Hits: 47 Question The diagram shows the curve , for and its maximum point M. i.       Show that      ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#7

Hits: 33     Question The equation of a curve is 3×2+3xy+y2=3      i.       Find the equation of the tangent to the curve at the point (2, −1), giving your answer in the form  ax +by +c = 0, where a, b and c are integers.    ii.       Show that the curve has no stationary […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#2

Hits: 34 Question Find the gradient of each of the following curves at the point for which x = 0.     i.           ii.         Solution      i.   We are required to find the gradient of the curve at the point for which x = 0. Therefore first we need to find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/23) | Q#3

Hits: 33 Question The equation of a curve is . Find the exact x-coordinate of each of the stationary points of the curve and determine the nature of each stationary point. Solution First we are required to find the exact x-coordinate of each of the stationary points of the curve. A stationary point on the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 32   Question The parametric equations of a curve are      i.       Find the exact value of the gradient of the curve at the point P where y = 6.    ii.       Show that the tangent to the curve at P passes through the point . Solution      i.   We are need  for […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/22) | Q#2

Hits: 8   Question The curve  has one stationary point. Find the coordinates of this stationary point. Solution We are required to find the coordinates of point M which is minimum point of the curve; A stationary point on the curve is the point where gradient of the curve is equal to zero; Since point […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/21) | Q#3

Hits: 26   Question The equation of a curve is . Find the exact x-coordinate of each of the stationary points of the curve and determine the nature of each stationary point. Solution First we are required to find the exact x-coordinate of each of the stationary points of the curve. A stationary point on […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/22) | Q#7

Hits: 41     Question A.  Find the exact area of the region bounded by the curve , the x-axis and the lines   and . The diagram shows the curve , for and its minimum point M. Find the exact x- coordinate of M. Solution A.    We are required to find area under the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/23) | Q#5

Hits: 260     Question The parametric equations of a curve are  ,  ,     i.       Show that .  ii.       Find the equation of the normal to the curve at the point where t = 0. Solution      i.   We are given that; We are required to show that . If a curve is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/21) | Q#5

Hits: 76     Question The parametric equations of a curve are  ,  , i.       Show that .    ii.       Find the equation of the normal to the curve at the point where t = 0. Solution      i.   We are given that; We are required to show that . If a curve is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#7

Hits: 78     Question The equation of a curve is        i.       Show that    ii.       Find the coordinates of each of the points on the curve where the tangent is parallel to the x- axis. Solution      i.   We are given; We are required to find . To find from an […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#2

Hits: 52     Question The curve with equation has one stationary point  in the interval . Find the exact  x-coordinate of this point. Solution We are required to find the x-coordinate of stationary point of the curve with equation; A stationary point on the curve is the point where gradient of the curve is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/23) | Q#8

Hits: 110     Question i.       By differentiating , show that if  θ then .    ii.       Hence show that Giving the values of a and b.   iii.       Find the exact value of Solution      i.   We are given that; We are required to show that; Since   provided that ; Therefore; If […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/23) | Q#4

Hits: 72   Question The parametric equations of a curve are for t < 0.      i.       Show that in terms of t.    ii.       Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution      i.   We are required to find  for the parametric equations […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/21) | Q#8

Hits: 45     Question i.       By differentiating , show that if  θ then .    ii.       Hence show that Giving the values of a and b.   iii.       Find the exact value of Solution      i.   We are given that; We are required to show that; Since   provided that ; Therefore; If […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/21) | Q#4

Hits: 52   Question The parametric equations of a curve are for t < 0.      i.       Show that in terms of t.    ii.       Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution      i.   We are required to find  for the parametric equations […]