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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

Hits: 285   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 […]

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

Hits: 193   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 […]

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

Hits: 212   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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | May-Jun | (P1-9709/01) | Q#11

Hits: 366   Question The diagram shows the graph of , where  for .      i.       Find an expression, in terms of , for  and explain how your answer shows that  is a decreasing function.    ii.       Find an expression, in terms of , for  and find the domain of .   iii.       Copy the diagram and, on your copy, sketch the graph of , making clear the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#10

Hits: 273   Question Functions  and  are defined by ,        i.       Evaluate .  ii.       Sketch in a single diagram the graphs of  and , making clear the relationship between the graphs.   iii.      Obtain an expression for  and use your answer to explain why  has an inverse.   iv.       Express each of  and , in terms of . Solution i.   We […]

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

Hits: 281   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 […]

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

Hits: 273   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, […]

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

Hits: 526   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. […]

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

Hits: 177   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 […]

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

Hits: 508   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 […]

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

Hits: 281   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, […]

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

Hits: 203   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 […]

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

Hits: 334   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 […]

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

Hits: 354 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 […]

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

Hits: 253 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 […]

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

Hits: 459 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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#8

Hits: 189 Question A curve  has a stationary point at . It is given that, where k is a constant. i.       Show that  and hence find the x-coordinate of the other stationary point, Q.    ii.       Find  and determine the nature of each of the stationary points P and Q.   iii.       Find . Solution i.   A stationary point […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 279 Question The equation of a curve is . Find     i.       an expression for  and the coordinates of the stationary point on the curve,    ii.       the volume obtained when the region bounded by the curve and the x-axis is rotated through  about the x-axis. Solution i.   Gradient (slope) of the curve is the derivative of equation of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#10

Hits: 325 Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.     i.       Find the coordinates of B and C.    ii.       Find the equation of the normal to the curve at C.   iii.       Find the volume obtained when the shaded region is rotated through  about […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#7

Hits: 609 Question The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48m.     i.       Find an expression for  in terms of .    ii.       Given that the area of the garden is A m2, show that .   iii.       Given that  can vary, find the maximum area of the garden, showing that this is […]