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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#6

Hits: 14   Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#6

Hits: 6   Question The diagram shows the curve , for . The x-coordinate of the maximum point  M is denoted by . i.       Find  and show that  satisfies the equation tan 2x = 2x + 4.    ii.       Show by calculation that  lies between 0.6 and 0.7.   iii.       Use the iterative formula to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#6

Hits: 19   Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#7

Hits: 18     Question The diagram shows the curve . The curve has a gradient of 3 at the point P.      i.       Show that the x-coordinate of P satisfies the equation    ii.       Verify that the equation in part (i) has a root between x = 3.1 and x = 3.3.   iii. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 33     Question      i.       By sketching a suitable pair of graphs, show that the equation Where x is in radians, has only one root for .      ii.       Verify by calculation that this root lies between x=1.1 and x=1.2.   iii.       Use the iterative formula to determine this root correct to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#6

Hits: 17   Question      i.       Verify by calculation that the cubic equation has a root that lies between x=0.7 and x=0.8.    ii.       Show that this root also satisfies an equation of the form where the values of a and b are to be found.   iii.       With these values of a and b, […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | Q#3

Hits: 17   Question The sequence defined by , Converges to the value .      i.       Find the value of correct to 3 decimal places. Show your working, giving each calculated  value of the sequence to 5 decimal places.    ii.       Find, in the form , an equation of which is a root. Solution      […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/22) | Q#3

Hits: 24   Question The sequence defined by , Converges to the value .      i.       Find the value of correct to 3 decimal places. Show your working, giving each calculated  value of the sequence to 5 decimal places.    ii.       Find, in the form , an equation of which is a root. Solution      […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#7

Hits: 28     Question      i.       By sketching a suitable pair of graphs, show that the equation has exactly two real roots.    ii.       Show by calculation that this root lies between 1.2 and 1.3.   iii.       Show that this root also satisfies the equation   iv.       Use an iteration process based 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#2

Hits: 22   Question The sequence of values given by the iterative formula With initial value , converges to .      i.       Use this iterative formula to find correct to 2 decimal places, giving the result of each iteration  to 4 decimal places.    ii.       State an equation that is satisfied by  , […]

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#6

Hits: 44     Question The curve with equation intersects the line y = x + 1 at the point P.      i.       Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6.    ii.       Show that the x-coordinate of P satisfies the equation   iii.       Use the iterative […]

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#6

Hits: 27     Question The curve with equation intersects the line y = x + 1 at the point P.      i.       Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6.    ii.       Show that the x-coordinate of P satisfies the equation   iii.       Use the iterative […]

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#6

Hits: 27   Question      i.       By sketching a suitable pair of graphs, show that the equation has only one root.    ii.       Verify by calculation that this root lies between 1.3 and 1.4.   iii.       Show that, if a sequence of values given by the iterative formula converges, then it converges […]

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#6

Hits: 32     Question      i.       By sketching a suitable pair of graphs, show that the equation has only one root.    ii.       Verify by calculation that this root lies between 1.3 and 1.4.   iii.       Show that, if a sequence of values given by the iterative formula converges, then it […]

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#7

Hits: 46     Question      i.       By sketching a suitable pair of graphs, show that the equation  has only one root.    ii.       Verify, by calculation that this root lies between x=0 and x=0.5.   iii.       Show that, if a sequence of values given by the iterative formula converges, then it […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/22) | Q#7

Hits: 49   Question The diagram shows the curve y = x2 cos x, for , and its maximum point M.     i.       Show by differentiation that the x-coordinate of M satisfies the equation    ii.       Verify by calculation that this equation has a root (in radians) between 1 and 1.2.   iii. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | May-Jun | (P2-9709/02) | Q#7

Hits: 51     Question The diagram shows the curve and its minimum point M.      i.       Find the exact coordinates of M.    ii.       Show that the curve intersects the line y = 20 at the point whose x-coordinate is the root of  the equation   iii.       Use the iterative formula […]