# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#4

Question The sequence of values converges to the value .      i.       Use the iterative formula to find the value of correct to 4 significant figures.  Give the result of each iteration to 6 significant figures.    ii.       State an equation satisfied by and hence determine the exact value of . Solution      i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#6

Question The diagram shows the curve with equation The maximum point is denoted by M.      i.       Find an expression for  and determine the gradient of the curve at the point  where the   curve crosses the x-axis.    ii.       Show that the x-coordinate of the point M satisfies the equation   iii.       Use an […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#5

Question It is given that Where  is a constant.     i.       Show that    ii.       Using the equation in part (i), show by calculation that 0.5 < a < 0.75.   iii.       Use an iterative formula, based on the equation in part (i), to find the value of a  correct to 3 significant figures. Give […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#5

Question It is given that Where  is a constant.     i.       Show that    ii.       Using the equation in part (i), show by calculation that 0.5 < a < 0.75.   iii.       Use an iterative formula, based on the equation in part (i), to find the value of a  correct to 3 significant figures. Give […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#6

Question The diagram shows the curve with parametric equations  ,    for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1.     i.       Show that the value of t at the point P satisfies the equation .    ii.       Use the iterative formula  with t1 =0.7 to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/22) | Q#6

Question The diagram shows the curve with parametric equations  ,    for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1.     i.       Show that the value of t at the point P satisfies the equation .    ii.       Use the iterative formula  with t1 =0.7 to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#5

Question The curve with equation and the point P on the curve has y-coordinate 10. i.               Show that the x-coordinate of P satisfies the equation ii.               Use the iterative formula to find the x-coordinate of P correct to 4 significant figures. Give the  result  of each iteration to 6 significant figures. iii.               Find the gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/22) | Q#5

Question The curve with equation crosses the x-axis at only one point. This point has coordinates .   i.               Show that satisfies the equation ii.               Show by calculation that 0.75 < p < 0.85.  iii.               Use an iterative formula based on the equation in part (i) to find the value of p  correct to 5 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/23) | Q#4

Question The diagram shows the curve with equation . The curve crosses the x-axis at points with coordinates  and .      i.       Use the factor theorem to show that (x +2) is a factor of    ii.       Show that satisfies an equation of the form , and state the  values of p and q.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/21) | Q#4

Question The diagram shows the curve with equation . The curve crosses the x-axis at points with coordinates  and .      i.       Use the factor theorem to show that (x +2) is a factor of    ii.       Show that satisfies an equation of the form , and state the  values of p and q. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/23) | Q#6

Question It is given that where a is a positive constant. i.       Show that    ii.       Use the equation in part (i) to show by calculation that 1.5 < a < 1.6.   iii.       Use an iterative formula based on the equation in part (i) to find the value of a correct to 3  significant […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/22) | Q#6

Question It is given that where a is a positive constant.     i.       Show that    ii.       Use the equation in part (i) to show by calculation that 1.5 < a < 1.6.   iii.       Use an iterative formula based on the equation in part (i) to find the value of a correct to 3  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | May-Jun | (P2-9709/21) | Q#4

Question The diagram shows the curve . The curve crosses the x-axis at the point P and has a maximum point M.      i.       Find the gradient of the curve at the point P.    ii.       Show that the x-coordinate of the point M satisfies the equation   iii.       Use an iterative formula based on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Feb-Mar | (P2-9709/22) | Q#5

Question It is given that a is a positive constant such that i.       Show that    ii.       Use the equation in part (i) to show by calculation that 1.0 < a < 1.5.   iii.       Use an iterative formula based on the equation in part (i) to find the value of a correct […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Feb-Mar | (P2-9709/22) | Q#5

Question It is given that a is a positive constant such that i.       Show that    ii.       Use an iterative formula based on the equation in part (i) to find the value of a correct to 4  significant figures. Give the result of each iteration to 6 significant figures. Solution      i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Feb-Mar | (P2-9709/22) | Q#4

Question The sequence of values given by the iterative formula With initial value , converges to .      i.       Determine the value of correct to 3 decimal places, giving the result of each iteration to 5  decimal places.    ii.       State an equation satisfied by and hence find the exact value of . Solution      […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on  the curve has x-coordinate q and the gradient of the curve at Q is −m. i.       Find the value of m and hence show that q […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/22) | Q#5

Question The diagram shows the curve y=4e−2x and a straight line. The curve crosses the y-axis at the point  P. The straight line crosses the y-axis at the point (0, 9) and its gradient is equal to the gradient of  the curve at P. The straight line meets the curve at two points, one of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/21) | Q#7

Question The diagram shows the curve The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on  the curve has x-coordinate q and the gradient of the curve at Q is −m.     i.       Find the value of m and hence show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | May-Jun | (P2-9709/23) | Q#3

Question      i.       By sketching a suitable pair of graphs, show that the equation has exactly one real root.    ii.       Use the iterative formula to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant  figures. Solution      i.   We are required to show that […]