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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

Hits: 15 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 […]

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#4

Hits: 10 Question The polynomial p(x) is defined by     i.       Use the factor theorem to show that (x+3) is a factor of p(x).    ii.       Factorise p(x) completely.   iii.       Hence, given that find the value of 2u and, using logarithms, find the value of u correct to 3 significant figures. Solution      i. […]

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#3

Hits: 18 Question      i.       Use the trapezium rule with four intervals to find an approximation to giving your answer correct to 3 significant figures.    ii.       Hence find an approximation to Solution i.   We are required to apply Trapezium Rule to evaluate; The trapezium rule with  intervals states that; We are given that […]

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

Hits: 10 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.   […]

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#1

Hits: 3 Question Solve the equation 2 ln(2x) − ln(x + 3)= ln(3x + 5). Solution We are given that; Power Rule; Multiplication Rule;  and are inverse functions. The composite function is an identity function, with  domain the positive real numbers. Therefore; Taking anti-logarithm of both sides; Now we have two options. If  then will […]

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#6

Hits: 7   Question The diagram shows the part of the curve  for , and the stationary point M.        i.       Find the equation of the tangent to the curve at the origin.    ii.       Find the coordinates of M, giving each coordinate correct to 3 decimal places. 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#3

Hits: 1   Question It is given that k is a positive constant. Solve the equation 2 ln x = ln(3k + x) + ln(2k – x), expressing  x in terms of k. Solution We are given that; Power Rule; Multiplication Rule;  and are inverse functions. The composite function is an identity function, with  domain […]

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#6

Hits: 19 Question The parametric equations of a curve are  ,        i.       Find  in terms of t and hence find the coordinates of the stationary point, giving each  coordinate correct to 2 decimal places.    ii.       Find the gradient of the normal to the curve at the point where the curve crosses the […]

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#4

Hits: 18 Question      i.       Find    ii.       Given that find the value of the positive constant a. Solution      i.   We are required to find; We have the trigonometric identity; It can be rearranged as; Therefore;   provided that It can be rearranged as; Therefore; Rule for integration of  is: Rule for integration […]

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#6

Hits: 5 Question      i.       Find the exact value of    ii.       Given that Find the value of positive constant a. Solution      i.   We are required to find; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:      ii.   We […]

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

Hits: 2 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, […]

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#6

Hits: 3 Question The parametric equations of a curve are  ,        i.       Find  in terms of t and hence find the coordinates of the stationary point, giving each  coordinate correct to 2 decimal places.    ii.       Find the gradient of the normal to the curve at the point where the curve crosses the […]

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#4

Hits: 4 Question      i.       Find    ii.       Given that find the value of the positive constant a. Solution      i.   We are required to find; We have the trigonometric identity; It can be rearranged as; Therefore;   provided that It can be rearranged as; Therefore; Rule for integration of  is: Rule for integration […]

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#5

Hits: 8 Question The variables x and y satisfy the equation , where K and a are constants. The graph of ln y  against x is a straight line passing through the points (0.6, 1.81) and (1.4, 1.39), as shown in the  diagram. Find the values of K and a, correct to 2 significant figures. […]

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

Hits: 10 Question The variables x and y satisfy the equation , where K and a are constants. The graph of ln y against x is a straight line passing through the points (0.6, 1.81) and (1.4, 1.39), as shown in the diagram. Find the values of K and a, correct to 2 significant figures. Solution We […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | May-Jun | (P2-9709/21) | Q#8

Hits: 17 Question The diagram shows the curve with equation The curve crosses the x-axis at the point P and has a minimum point M.      i. Find the gradient of the curve at the point P.    ii. Find the exact coordinates of the point M. Solution      i.   We are required to […]