# 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#7

Question The diagram shows the curve and its maximum point M.      i.       Find the exact coordinates of M.    ii.       Use the trapezium rule with three intervals to estimate the value of giving your answer correct to 2 decimal places. Solution      i.   We are required to find 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#4

Question a.   Find . b.   Express  in terms of  and hence find . Solution a.     We are required to find; Rule for integration of , or ; b.     We know that , therefore; Hence; Therefore; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:

# 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#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# 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#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# 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

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 to the […]

# 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#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve has […]

# 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

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 to […]

# 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#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve […]

# 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

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 converges to […]

# 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#6

Question The equation of the curve is .     i.       Show that    ii.       Find the equation of the tangent to the curve at the point (1,2), giving your answer in the form  ax+by=c. Solution      i.   We are given that; Therefore; Rule for differentiation of  is: Rule for differentiation of  is: […]

# 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#2

Question Show that Solution We are required to show that; Rule for integration of  is: This integral is valid only when . Division Rule; Power Rule;

# 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#5

Question The variables x and y satisfy the equation y = A(bx), where A and b are constants. The graph of ln y  against x is a straight line passing through the points (1.4, 0.8) and (2.2, 1.2), as shown in the  diagram. Find the values of A and b, correct to 2 decimal […]

# 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#2

Question Use logarithms to solve the equation 5x = 22x+1, giving your answer correct to 3 significant figures. Solution We are given; Taking natural logarithm of both sides; Power Rule;

# 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#2

Question Use logarithms to solve the equation 5x = 22x+1, giving your answer correct to 3 significant figures. Solution We are given; Taking natural logarithm of both sides; Power Rule;

# 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#1

Question Given that 13x = (2.8)y, use logarithms to show that y = kx and find the value of k correct to 3  significant figures. Solution We are given; Taking natural logarithm of both sides; Power Rule;

# 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#1

Question Given that 13x = (2.8)y, use logarithms to show that y = kx and find the value of k correct to 3  significant figures. Solution We are given; Taking natural logarithm of both sides; Power Rule;

# 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#5

Question      i.       Given that y = 2x, show that the equation 2x + 3(2−x) = 4 can be written in the form y2 − 4y + 3 = 0.    ii.       Hence solve the equation 2x + 3(2−x) = 4  giving the values of x correct to 3 significant figures where […]