# 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#9

Question A curve is such that . The curve has a maximum point at (2, 12). i. Find the equation of the curve. A point P moves along the curve in such a way that the x-coordinate is increasing at 0.05 units per second. ii. Find the rate at which the y-coordinate is changing when x = 3, stating whether […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#7

Question The curve  intersects the x-axis at A. The tangent to the curve at A intersects the y-axis at C.     i.       Show that the equation of AC is .    ii.     Find the distance AC. Solution i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of Equation […]

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

Question The diagram shows part of the curve  which passes through the points A and B. The curve has a maximum point at A and the gradient of the line BA is 2.     i.      Find the coordinates of A and B.    ii.    Find a  and hence evaluate the area of the shaded region. Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#2

Question The equation of a curve is . i.     Obtain an expression for . ii.   A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the y-coordinate when x = 4. Solution i.   We are given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#10

Question It is given that a curve has equation , where .     i.       Find the set of values of  for which the gradient of the curve is less than 5.    ii.       Find the values of  at the two stationary points on the curve and determine the nature of each stationary point. Solution i.   Gradient (slope) of the curve is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#4

Question A watermelon is assumed to be spherical in shape while it is growing. Its mass, Mkg, and radius, r cm, are related by the formula , where  is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and […]