Hits: 54

# 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: 54 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: 83   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: 22   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: 60   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: 89     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 | (P2-9709/22) | Q#5

Hits: 81     Question The equation of a curve is x2− 2 x2y+ 3y = 9.      i. Show that    ii. Find the equation of the normal to the curve at the point where x = 2, giving your answer in the  form ax + by + c = 0. Solution      i. […]

# 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: 310     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: 147     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 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#11

Hits: 489 Question The diagram shows the curve . i.       Find the equation of the tangent to the curve at the point . ii.    Show that the x-coordinates of the points of intersection of the line   and the curve are given by the equation . Hence find these x- coordinates. iii.     The region shaded in the diagram is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#9

Hits: 464   Question A curve has equation , where  is a positive constant. Find, in terms of ,  the values of  for which the curve has stationary points and determine the nature of  each stationary point. Solution A stationary point  on the curve  is the point where gradient of the curve is  equal to zero; We are given that; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#9

Hits: 721 Question The diagram shows part of the curve  and three points A, B and C on the  curve with x-coordinates 1, 2 and 5 respectively. i.       A point P moves along the curve in such a way that its x-coordinate increases at  a constant rate of 0.04 units per second. Find the rate at which the y-coordinate […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#6

Hits: 974 Question In the diagram, S is the point  and T is the point . The point Q lies on ST,  between S and T, and has coordinates . The points P and R lie on the x-axis  and y-axis respectively and OPQR is a rectangle.      i.       Show that the area, A, of the rectangle OPQR is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#3

Hits: 459   Question The equation of a curve is  .      i.    Find the gradient of the curve at the point where x = 2.    ii.   Find  and hence evaluate . Solution      i.   We are required to find gradient of the curve at a given point. Gradient (slope) of the curve at the particular […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#10

Hits: 553   Question The diagram shows the curve  and the tangent to the curve at the point . i.         Find the equation of this tangent, giving your answer in the form . ii.       Find the area of the shaded region. Solution i.   We are required to find equation of the tangent to the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/11) | Q#8

Hits: 3990   Question The inside lane of a school running track consists of two straight sections each of  length x metres, and two semicircular sections each of radius r metres, as shown in  the diagram. The straight sections are perpendicular to the diameters of the  semicircular sections. The perimeter of the inside lane is 400 metres. i.       Show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#11

Hits: 494 Question The diagram shows part of the curve  and a point  and  which lie on the curve. The tangent to the curve at B intersects the line  at the point C.      i. Find the coordinates of C.    ii. Find the area of the shaded region. Solution i.   We are required to find the coordinates of point C. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#6

Hits: 472 Question The non-zero variables x, y and u are such that . Given that , find the stationary value of u and determine whether this is a maximum or a  minimum value. Solution We are given that; We are also given; Substituting this value of  in the equation of . A stationary point  on the curve  is the point where gradient of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/12) | Q#11

Hits: 1142   Question The diagram shows the curve  , which intersects the x-axis at A and the  y-axis at B. The normal to the curve at B meets the x-axis at C. Find      i.       the equation of BC,    ii.       the area of the shaded region. Solution      i.   To find the equation of the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/12) | Q#8

Hits: 498 Question The volume of a solid circular cylinder of radius  cm is  cm3.      i.       Show that the total surface area, S cm2, of the cylinder is given by     ii.       Given that  can vary, find the stationary value of .   iii.       Determine the nature of this stationary value. Solution      i.   We are given that volume of solid circular […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#10

Hits: 1145   Question The diagram shows part of the curve  and the point  on the curve. The tangent at A cuts the x-axis at B and the normal at A cuts the y-axis at C.      i.       Find the coordinates of B and C.  ii.   Find the distance AC, giving your answer in the form  , where a and b are integers.   […]