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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#7

Hits: 44     Question The diagram shows the part of the curve y = sin2 x for  .      i.       Show that    ii.       Hence find the x-coordinates of the points on the curve at which the gradient of the curve is  0.5.   iii.       By expressing sin2 x in terms […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#6

Hits: 24     Question A curve is such that . The point (0, 1) lies on the curve.      i. Find the equation of the curve.    ii. The curve has one stationary point. Find the x-coordinate of this point and determine whether it  is a maximum or a minimum point. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#5

Hits: 23     Question      i.       By sketching a suitable pair of graphs, show that there is only one value of x that is a root of  the equation    ii. Verify by calculation that this root lies between 1 and 2.   iii. Show that this root also satisfies the equation   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#4

Hits: 21 Question The equation of the curve is .     i.       Show that    ii.       Find the equation of the tangent to the curve at the point (2, 4), giving your answer in the form ax+by=c. Solution      i.   We are given that; Therefore; Rule for differentiation of  is: If  and […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#3

Hits: 36   Question i. Express  in the form , where  and , giving exact  value of R and the value of  correct to 2 decimal places.    ii. Hence solve the equation Giving all solutions in the interval . Solution      i.   We are given that; We are required to write it in […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#7

Hits: 6   Question      i. By expanding , and using double-angle formulae, show that    ii. Hence, or otherwise, show that Solution      i.   We have; We apply following addition formula. Therefore; We apply following two formulae. We have the trigonometric identity; Therefore, we can replace; Hence;    ii.   We are required […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#6

Hits: 12     Question The diagram shows the part of the curve  for . The curve cuts the x-axis at A and  its maximum point is M.      i. Write down the coordinates of A.    ii. Show that the x-coordinate of M is e, and write down the y-coordinate of M in terms […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#3

Hits: 6   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, showing the result of each  iteration.    ii.       State an equation satisfied by , and hence find the exact value […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#8

Hits: 43   Question i.       Express  in the form , where  and , giving the exact values of R and .    ii.       Hence show that   iii.       By differentiating , show that if  then .   iv.       Using the results of parts (ii) and (iii), show that Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#7

Hits: 22     Question The diagram shows the curve y = 2ex + 3e-2x. The curve cuts the y-axis at A.      i.       Write down the coordinates of A.    ii.       Find the equation of the tangent to the curve at A, and state the coordinates of the point where  this tangent […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#6

Hits: 16     Question      i.       By sketching a suitable pair ofgraphs, show that there is only one value of x in  the interval  that is a root of the equation    ii.       Verify by calculation that this root lies between 0.8 and 0.9 radians.   iii.       Show that this value […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#5

Hits: 12     Question The curve with equation y = x2 ln x, where x > 0, has one stationary point.      i.       Find the x-coordinate of this point, giving your answer in terms of e.    ii.       Determine whether this point is a maximum or a minimum point. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | Oct-Nov | (P2-9709/02) | Q#3

Hits: 15     Question Find the values of x satisfying the equation 3 sin 2x = cos x, for 0◦ ≤ x ≤ 90◦. Solution We are given; We apply following formula; Using calculator we can find that; Properties of Domain Range Odd/Even Periodicity Translation/ Symmetry It is evident from periodicity and symmetry properties […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#7

Hits: 12   Question      i.By expanding , show that    ii.Hence, or otherwise, show that Solution      i.   We have; We apply following addition formula. Therefore; We apply following two formulae. We have the trigonometric identity; Therefore, we can replace; Hence;    ii.   We are required to show that; As we have […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#6

Hits: 15   Question The parametric equations of a curve are Where t takes all positive values.      i.       Show that    ii.       Find the equation of the tangent to the curve at the point where .   iii.       The curve has one stationary point. Find the y-coordinate of this point, and […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#5

Hits: 12     Question The diagram shows the part of the curve y = xe-x for , and its maximum point M.      i.       Find the x-coordinate of M.    ii.       Use the trapezium rule with two intervals to estimate the value of giving your answer correct to 2 decimal places.   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#4

Hits: 13   Question i.       Express  in the form , where  and , giving the value of  correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval  correct to 1 decimal place.   iii.       Write down the least value of  as  varies. Solution      i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | Q#2

Hits: 15   Question The sequence of values given by the iterative formula With initial value , converges to .      i. Use this iterative formula to find correct to 3 decimal places, showing the result of each  iteration.    ii.  State an equation satisfied by , and hence show that the exact value of […]