# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows parts of the curves  and  and their points of intersection  and . The x-coordinates of  and  are and  respectively.     i.       Show that  and  are roots of the equation . Solve this equation and hence state the value of  and the value of .    ii.       Find the area of the shaded region between the two curves. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#6

Question A curve has equation . It is given that .     i.       Find the set of values of  for which  is an increasing function.    ii.       Given that the curve passes through (1, 3), find . Solution i.   To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#5

Question A curve has equation .     i.       Find  and .    ii.       Find the coordinates of the maximum point A and the minimum point B on the curve. Solution i.   Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation of  is: Second derivative is the derivative of the derivative. If we have derivative of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#10

Question The diagram shows an open rectangular tank of height  meters covered with a lid. The base of the tank has sides of length  meters and  meters and the lid is a rectangle with sides of length  meters  meters. When full the tank holds 4m3 of water. The material from which the tank is made is of negligible thickness. The […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#11

Question The equation of a curve is  .     i.        Find an expression for  and determine, with a reason, whether the curve has any stationary points.    ii.       Find the volume obtained when the region bounded by the curve, the coordinate axes and the line  is rotated through 360o about the x-axis.   iii.       Find the set of values of  for which […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#10

Question The equation of a curve is .     i.       Show that the equation of the normal to the curve at the point (3, 6) is .    ii.       Given that the normal meets the coordinate axes at points A and B, find the coordinates of the mid-point of AB.   iii.       Find the coordinates of the point at which the normal meets […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#8

Question The diagram shows a metal plate consisting of a rectangle with sides  cm and  cm and a quarter-circle of radius  cm. The perimeter of the plate is 60 cm. i.       Express  in terms of .    ii.       Show that the area of the plate,  cm2, is given by . Given that  can vary,   iii.       find the value of  at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#9

Question The diagram shows part of the curve  which has a minimum point at M. The line  intersects the curve at the points A and B.     i.        Find the coordinates of A, B and M.    ii.       Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. Solution i.   It is evident from […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#5

Question The equation of a curve is such that . Given that the curve passes through the point , find     i.       the equation of the normal to the curve at P    ii.       the equation of the curve. Solution     i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#10

Question The equation of a curve is .     i.        Find .    ii.      Find the equation of the tangent to the curve at the point where the curve intersects the y-axis.   iii.       Find the set of values of  for which  is an increasing function of . Solution     i.   Rule for differentiation of  is: Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#8

Question A solid rectangular block has a square base of side  cm. The height of the block is  cm and the total surface area of the block is 96 cm2. i.       Express  in terms of  and show that the volume,  cm3, of the block is given by Given that  can vary,    ii.       find the stationary value of , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#7

Question The diagram shows part of the curve  , which crosses the x-axis at A and the y-axis at B. The normal to the curve at A crosses the y-axis at C.     i.       Show that the equation of the line AC is .    ii.       Find the length of BC. Solution i.   To find the equation of the line either we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#6

Question A curve is such that    and the point (9, 2) lies on the curve.     i.       Find the equation of the curve.    ii.       Find the x-coordinate of the stationary point on the curve and determine the nature of the stationary point. Solution i.   We can find equation of the curve from its derivative through integration; For the given case; […]