# 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

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 given that […]

# 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#5

Question Given that find the value of the constant a correct to 3 decimal places. Solution      i.   We are given that; Rule for integration of , or ; Taking logarithm of both sides;  for any

# 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

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 the positive […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#6

Question A curve has parametric equations      i.       Find an expression for in terms of t.    ii.       Find the exact value of at the stationary point.   iii.       Find the gradient of the curve at the point where it crosses the x-axis. Solution      i.   We are required to find  for the parametric […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#5

Question The diagram shows the curve for 0 ≤ x ≤ 6. The region bounded by the curve and the lines x = 0, x = 6 and y = 0 is denoted by R.        i.       Use the trapezium rule with 2 strips to find an estimate of the area of R, giving your […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#4

Question The polynomial is defined by  where  is a constant. It is given that  is a factor of     i.       Use the factor theorem to find the value of .    ii.       Factorise p(x) and hence show that the equation p(x) = 0 has only one real root.   iii.       Use logarithms to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#2

Question The variables x and y satisfy the equation y = Kxp, where K and p are constants. The graph of ln y  against ln x is a straight line passing through the points (1.28, 3.69) and (2.11, 4.81), as shown in  the diagram. Find the values of K and p correct to 2 decimal […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#5

Question The diagram shows the curve and its stationary point M. The x-coordinate of M is m.       i.       Find an expression for and hence show that .    ii.       Use an iterative formula based on the equation in part (i) to find the value of m correct to 4  significant figures. Give the result […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#3

Question The definite integral  is defined by    i.   Show that  .    ii.   Sketch the curve  for .   iii.   State whether an estimate of obtained by using the trapezium rule will be more than or less than 8e −2. Justify your answer. Solution      i.   We are given that; Rule for integration […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#4

Question     i.       It is given that the positive constant a is such that    ii.       Show that   iii.       Use an iterative formula , to find a correct to 3 decimal places. Give the result of each iteration to 5 decimal places. Solution      i.   We are given that; Rule for integration of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#2

Question The variables x and y satisfy the equation y = Aepx, where A and p are constants. The graph of ln y  against x is a straight line passing through the points (5, 3.17) and (10, 4.77), as shown in the diagram. Find the values of A and p correct to 2 decimal places. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#6

Question a)   Find b)  Without using a calculator, find the exact value of giving your answer in the form . c)     The diagram shows the curve for 0 ≤ x ≤ 6. The region bounded by the curve and  the lines x = 0, x = 6 and y = 0 is denoted […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#5

Question The equation of a curve is  . At the point on the curve with x-coordinate p, the gradient of  the curve is .      i.       Show that .      ii.       Show by calculation that 3.3 < p < 3.5.     iii.       Use an iterative formula based on the equation in part (i) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#6

Question a)   Find b)  Without using a calculator, find the exact value of giving your answer in the form . c)     The diagram shows the curve for 0 ≤ x ≤ 6. The region bounded by the curve and  the lines x = 0, x = 6 and y = 0 is denoted by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#5

Question The equation of a curve is  . At the point on the curve with x-coordinate p, the gradient of  the curve is .      i.       Show that .      ii.       Show by calculation that 3.3 < p < 3.5.     iii.       Use an iterative formula based on the equation in part (i) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#7

Question      i.       Find      ii.       Without using a calculator, find the exact value of giving your answer in the form , where a and b are integers.   Solution      i.   We are required to find;   provided that   Rule for integration of  is: We integrate both parts  and  one by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#3

Question Given that 3ex +8e−x = 14, find the possible values of ex and hence solve the equation 3ex +8e−x = 14 correct to 3 significant figures. Solution We are given; Let ; Now we have two options. Since ; Taking logarithm of both sides;  and are inverse functions. The composite function is an identity […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#1

Question Find the gradient of the curve at the point for which x = 0. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is: We are given that; Therefore; Rule for differentiation of  is: Rule for differentiation of natural exponential function is; […]

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

Question      i.       Given that , find the value of 2y.    ii.       Use logarithms to find the value of y correct to 3 significant figures. Solution i.   We are given; We can write it as; Let , then; Now we have two options. Since ; Taking natural logarithm of both […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#1

Question      i.       It is given that x satisfies the equation . Find the value of and, using  logarithms, find the value of x correct to 3 significant figures.    ii.       Hence state the values of x satisfying the equation . Solution i.   We are given; We can write it as; Let […]