# 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

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 implicit equation, […]

# 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

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 equal to […]

# 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

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  and  are […]

# 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

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 given below; […]

# 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

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  and  are […]

# 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

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 given below; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#6

Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the value of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#5

Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M can be written in the form , where the value of a is to be  stated.    ii.       Find the exact value of the area of the region enclosed by the curve and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#6

Question The diagram shows the curve , for . The x-coordinate of the maximum point  M is denoted by . i.       Find  and show that  satisfies the equation tan 2x = 2x + 4.    ii.       Show by calculation that  lies between 0.6 and 0.7.   iii.       Use the iterative formula to find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#5

Question The parametric equations of a curve are      i.       Find an expression for in terms of t.      i.       Find the equation of the normal to the curve at the point for which t = 0. Give your answer in  the form ax + by + c = 0, where a, b and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#6

Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the value of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#5

Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M can be written in the form , where the value of a is to be  stated.    ii.       Find the exact value of the area of the region enclosed by the curve and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows the curve with equation . The minimum point on the curve has coordinates  and the x-coordinate of the maximum point is , here  and  are constants.      i.       State the value of .    ii.       Find the value of .   iii.       Find the area of the shaded region.   iv.       The gradient, , of the curve has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#8

Question A curve is such that    i.       Find  ii.    Verify that the curve has a stationary point when  and determine its nature. iii.   It is now given that the stationary point on the curve has coordinates (−1, 5). Find the equation of the curve. Solution i.   Second derivative is the derivative of the derivative. If we have […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#2

Question It is given that , for . Show that  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. If  , the function  is decreasing. If  , the test is inconclusive. We are given that; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#10

Question A curve is defined for x > 0 and is such that  . The point P(4, 8) lies on the curve.     i.       Find the equation of the curve.    ii.     Show that the gradient of the curve has a minimum value when x = 2 and state this minimum value. Solution i.   We are given that; Therefore, for the given case; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows part of the curve  , crossing the y-axis at the point B(0, 3). The point A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C.     i.       Find the equation of the tangent to the curve at A.    ii.    Determine, showing all necessary working, whether C […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#3

Question The diagram shows a plan for a rectangular park ABCD, in which AB = 40m and AD = 60m. Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a triangular playground. The length of DY is  m and the length of XC is  m.     i.       Show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#5

Question A curve has equation  . Verify that the curve has a stationary point at  and determine its nature. Solution i.   A stationary point  on the curve  is the point where gradient of the curve is equal to zero; Therefore first we need gradient of the given curve. Gradient (slope) of the curve is the derivative of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#3

Question An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing […]