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

  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.       Use […]

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#4

    Question The parametric equations of a curve are  ,  ,     i.       Show that .    ii.       Hence find the exact value of t at the point on the curve at which the gradient is 2. Solution      i.   We are given that; We are required to show that . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/21) | Q#6

    Question The curve with equation y = x ln x has one stationary point. i.       Find the exact coordinates of this point, giving your answers in terms of e. ii.       Determine whether this point is a maximum or a minimum point. 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 2009 | May-Jun | (P2-9709/02) | Q#8

    Question a)   Find the equation of the tangent to the curve at the point where . b)                  i.       Find the value of the constant A such that           ii.       Hence show that Solution a.     We are given that curve with equation  and […]

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

    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 with initial […]

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#4

    Question The parametric equations of a curve are x = 4 sin θ , y = 3 – 2 cos 2θ , where . Express  in terms of θ, simplifying your answer as far as possible. Solution We are required to express that   in terms of θ for the parametric equations given below; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#10

Question i.       The diagram shows the line  and the curve , which intersect at the points A and B. Find a.   the x-coordinates of A and B, b.   the equation of the tangent to the curve at B, c.   the acute angle, in degrees correct to 1 decimal place, between this tangent and the line .    ii.       Determine […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#7

Question A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is q radians (see diagram). i.       Express  in terms of r and show that the area, Acm2, of the sector is given by    .    […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#7

Question The equation of a curve is . i.       Obtain an expression for .    ii.       Find the equation of the normal to the curve at the point P(1, 3).   iii.       A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#6

Question A curve is such that , where k is a constant.     i.       Given that the tangents to the curve at the points where  and  are perpendicular, find the value of .    ii.       Given also that the curve passes through the point (4, 9), find the equation of the curve. Solution      i.   If two lines are perpendicular […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#4

Question The equation of a curve is .     i.       Find the coordinates of the stationary point on the curve and determine its nature.    ii.       Find the area of the region enclosed by the curve, the x-axis and the lines    and . Solution      i.   Coordinates of stationary point on the curve  can be found from the derivative of equation […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#11

Question The diagram shows the curve  for . The curve has a maximum point at A and a minimum point on the x-axis at B. The normal to the curve at C (2, 2) meets the normal to the curve at B at the point D. i.       Find the coordinates of A and B.    ii.       Find the equation of the normal […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#9

Question The diagram shows part of the curve . i.       Find the gradient of the curve at the point where .    ii.       Find the volume obtained when the shaded region is rotated through 360o about the x-axis, giving your answer in terms of . Solution      i.   Gradient (slope) of the curve at the particular point is the derivative of […]