# 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

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  minutes is […]

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

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, by finding […]

# 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

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 compare the […]

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

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 of the […]

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

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 given; Gradient […]

# 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

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.   ii.  Find […]

# 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

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 the angle […]

# 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

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 or a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#9

Question The base of a cuboid has sides of length  cm and  cm. The volume of the cuboid is 288 cm3.      i.       Show that the total surface area of the cuboid, A cm2, is given by     ii.       Given that  can vary, find the stationary value of A and determine its nature. Solution From the given information we can compile following data; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#6

Question A curve is such that  , where a is  constant. The point  lies on the curve and the normal to the curve at  is .      i. Show that .    ii. Find the equation of the curve. Solution i.   If two lines (or one line and a curve) are perpendicular (normal) to each other, then product of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#5

Question A functions  is such that   for   .      i.       Find an expression for  and use your result to explain why  has an inverse.    ii.       Find an expression for , and state the domain and range of . Solution i.   We have; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#9

Question The diagram shows part of the curve   and the tangent to the curve at .      i.       Find expressions for  and .    ii.       Find the equation of the tangent to the curve at P in the form .   iii.       Find, showing all necessary working, the area of the shaded region. Solution i.   First we find the expression […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#8

Question The equation of a curve is such that . Given that the curve has a  minimum point at , find the coordinates of the maximum point. Solution To find the coordinates of a stationary point (in this case a maximum point) we need derivative of equation of the curve. We are given the second derivative of the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/11) | Q#12

Question A curve is such that  . The curve passes through the point .      i.       Find the equation of the curve.    ii.       Find  .   iii.       Find the coordinates of the stationary point and determine its nature. Solution i.   We are given that curve  passes through the point  and we are  required to find the equation of the curve. We […]