Hits: 85

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 85     Question The diagram shows part of the curve and its maximum point M. The shaded region is bounded by the curve, the axes and the line  through M parallel to the y-axis.      i.       Find the exact value of the x-coordinate of M.    ii.       Find the exact value of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#3

Hits: 118 Question A curve has equation Find the equation of the normal to the curve at the point (1, 2). Give your answer in the form ax + by  + c = 0, where a, b and c are integers. Solution We are given equation of the curve as; We are required to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#6

Hits: 105 Question The diagram shows part of the curve  and its maximum point M. The x-coordinate of M is denoted by m.      i.       Find  and hence show that m satisfies the equation .    ii.       Show by calculation that m lies between 0.7 and 0.8.   iii.       Use an iterative formula based on the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#4

Hits: 62 Question For each of the following curves, find the exact gradient at the point indicated: i.        at   ii.        at Solution      i.   We are required to find the gradient of the curve at the point . Therefore first we need to find the expression for gradient of the given curve. Gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#6

Hits: 75   Question The diagram shows part of the curve  and its maximum point M. The x-coordinate of M is denoted by m.      i.       Find  and hence show that m satisfies the equation .    ii.       Show by calculation that m lies between 0.7 and 0.8.   iii.       Use an iterative formula based […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#4

Hits: 82 Question For each of the following curves, find the exact gradient at the point indicated: i.        at  ii.        at Solution      i.   We are required to find the gradient of the curve at the point .  Therefore first we need to find the expression for gradient of the given curve. Gradient (slope) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#4

Hits: 86   Question The parametric equations of a curve are Find the equation of the tangent to the curve at the point for which t = 0. Give your answer in the  form ax + by + c = 0, where a, b and c are integers. Solution 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 2014 | May-Jun | (P2-9709/23) | Q#8

Hits: 93 Question The diagram shows the curve , for and its maximum point M.     i.       Show that    ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#8

Hits: 135 Question The diagram shows the curve , for and its maximum point M. i.       Show that      ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#4

Hits: 104   Question The parametric equations of a curve are Find the equation of the tangent to the curve at the point for which t = 0. Give your answer in the  form ax + by + c = 0, where a, b and c are integers. Solution 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 2014 | May-Jun | (P2-9709/21) | Q#7

Hits: 268     Question The equation of a curve is 3×2+3xy+y2=3      i.       Find the equation of the tangent to the curve at the point (2, −1), giving your answer in the form  ax +by +c = 0, where a, b and c are integers.    ii.       Show that the curve has no stationary […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#2

Hits: 168 Question Find the gradient of each of the following curves at the point for which x = 0.     i.           ii.         Solution      i.   We are required to find the gradient of the curve at the point for which x = 0. Therefore first we need to find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#10

Hits: 920 Question a.   The functions  and  are defined for by  , where  and  are positive constants  , Given that  and ,       (i)          calculate the values of  and ,      (ii)         obtain an expression for  and state the domain of .   b.   A point P travels along the curve  in such a way that the x-coordinate of P at time t  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#8

Hits: 604   Question A curve  has a stationary point at  and is such that .       i.       State, with a reason, whether this stationary point is a maximum or a minimum.    ii.       Find  and . Solution i.   Once we have the coordinates of the stationary point  of a curve, we can determine its  nature, whether minimum or maximum, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 743   Question      i.       Express  in the form  .    ii.       Determine whether  is an increasing function, a decreasing function or  neither. Solution i.   We have the expression; We use method of “completing square” to obtain the desired form. We complete the square for the terms which involve . We have the algebraic formula; For the given case we can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 589   Question A curve is such that . The curve has a stationary point at  where .      i.            State, with a reason, the nature of this stationary point.    ii.              Find an expression for .   iii.       Given that the curve passes through the point , find the coordinates […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#6

Hits: 1360   Question The equation of a curve is , where  and  are constants.      i.       In the case where the curve has no stationary point, show that .    ii.       In the case where  and , find the set of values of  for which  is a decreasing               function of . Solution i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#4

Hits: 1818   Question A curve has equation .      i.       Find .   A point moves along this curve. As the point passes through A, the x-coordinate is increasing           at a rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per            second. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#11

Hits: 1189   Question The diagram shows parts of the curves  and  intersecting at points  and . The angle between the tangents to the two curves at  is .      i.       Find , giving your answer in degrees correct to 3 significant figures.    ii.       Find by integration the area of the shaded region. Solution i.   Angle between two curves is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#9

Hits: 645   Question The function  is defined for  and is such that . The curve  passes through the point .      i.       Find the equation of the normal to the curve at P.    ii.       Find the equation of the curve.    iii.     Find the x-coordinate of the stationary point and state with a reason whether this point is a maximum […]