# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#8

Question The equation of a curve is x2 + 2xy − y2 + 8 = 0.      i.       Show that the tangent to the curve at the point (-2,2) is parallel to the x-axis.    ii.       Find the equation of the tangent to the curve at the other point on the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve and its maximum point M.      i.       Find the exact coordinates of M.    ii.       Use the trapezium rule with three intervals to estimate the value of giving your answer correct to 2 decimal places. Solution      i.   We are required to find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#8

Question The diagram shows the curve , for . The point  lies on the curve. i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on a line (or […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#8

Question The diagram shows the curve , for . The point  lies on the curve.     i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on a line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/22) | Q#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/21) | Q#8

Question      i.       By differentiating , show that if y = cot x then    ii.       By expressing in terms of and using the result of part (i), show that   iii.       Express cos 2x in terms of sin2 x and hence show that can be expressed as .  Hence using the […]

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