# 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

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 curve is […]

# 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

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 the parametric […]

# 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

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 M is […]

# 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

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 curve […]

# 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

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 curve  , […]

# 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

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.   We […]

# 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

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 given parametrically […]

# 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

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 given parametrically […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#10

Question A curve has equation y=f(x). The point P with coordinates (9,0) lies on the curve. Given that  , a.   Find f(x). b.   Find the x-coordinates of two points on y=f(x) where the gradient of the curve is equal to 10. Solution a.   We are given; We are given coordinates of a point on the […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#1

Question Given , find the value of   when x=3. Solution We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: Therefore; Rule for differentiation is of  is: Rule for differentiation is of  is: Rule for differentiation is […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2013 | June | Q#9

Question  , a.   Show that Where A and B are constants to be found. b.   Find Given that the point (-3,10) lies on the curve with equation y=f(x), c.   Find f(x). Solution a.   We are given; Therefore; b.   We are given; We are required to find . Second derivative is the derivative of the derivative. […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2013 | January | Q#11

Question The curve C has equation , a.   Find , giving each term in its simplest form. The point P on C has x-coordinate equal to . b.   Find the equation of the tangent to C at P, giving your answer in the form y = ax + b, where a and  b are constants. The […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2013 | June | Q#4

Question a.   The polynomial f(x) is given by .                     i.       Use the Factor Theorem to show that x+3 is a factor of f(x).                   ii.       Express f(x) in the form  , where p and q are integers. b.   A curve has equation .                     i.       Find .                   ii.       Show that the x-coordinates of any stationary points of the curve satisfy the equation                  iii.       Use the […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2013 | June | Q#6

Question A curve has equation . The point P with coordinates (-1,6) lies on the curve.  a.   Find the equation of the tangent to the curve at the point P, giving your answer in the form   . b.   The point Q with coordinates (2,k) lies on the curve.                     i.       Find the value of k.                   ii.       Verify that Q also […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2013 | January | Q#6

Question The gradient, , of a curve at the point (x,y) is given by The curve passes through the point P(1,4). a.   Find the equation of the tangent to the curve at the point P, giving your answer in the form . b.   Find the equation of the curve. Solution a.   We are required to find the […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2013 | January | Q#2

Question A bird flies from a tree. At time t seconds, the bird’s height, y metres, above the horizontal ground is given by ,   a.   Find . b.                 i.       Find the rate of change of height of the bird in metres per second when .          ii.       Determine, with a […]

# 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

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 rotated through […]

# 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

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; Therefore, we need […]

# 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

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 of P  […]

# 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

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 given by […]