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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#10

Hits: 193 Question The curve with equation  passes through the origin.     i.       Show that the curve has no stationary points.    ii.       Denoting the gradient of the curve by m, find the stationary value of m and  determine its nature. Solution i.   We are required to show that curve has no stationary points. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#2

Hits: 45 Question A point is moving along the curve   in such a way that the x-coordinate is  increasing at a constant rate of 0.02 units per second. Find the rate of change of the  y-coordinate when x = 1. Solution i.   We are given that; We are required to find; We know […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#11

Hits: 4262 Question The diagram shows the curve  and points A(1,0) and B(5,2) lying on the curve. i.       Find the equation of the line AB, giving your answer in the form y=mx+c.    ii.       Find, showing all necessary working, the equation of the tangent to the curve which is parallel to AB.   iii.       Find the perpendicular distance between the line AB […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#10

Hits: 1013 Question A curve has equation y=f(x) and it is given that , where a and b are positive  constants. i.       Find, in terms of a and b, the non-zero value of x for which the curve has a stationary point and  determine, showing all necessary working, the nature of the stationary point.    ii.       It is now given that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 623 Question The function f is such that  for , where k is a constant. Find the  largest value of k for which f is a decreasing function. Solution To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function at that point. If  , the function  is increasing. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 2190   Question The diagram shows part of the curve  and the normal to the curve at the point P(2, 3).  This normal meets the x-axis at Q.      i.       Find the equation of the normal at P.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are required to find the equation of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 685   Question A curve is such that .                    i.       Find the x-coordinate of each of the stationary point on the curve.                  ii.       Obtain an expression for  and hence or otherwise find the nature of each of the  stationary points.   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 816   Question Points A and B lie on the curve . Point A has coordinates (4,7) and B is the  stationary point of the curve. The equation of a line L is , where m is a constant.                             i.       In the case where L passes through the mid-point of AB, find the value of m.                           ii.       Find the set of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#4

Hits: 739   Question Machines in a factory make cardboard cones of base radius r cm and vertical height h cm. The volume, V cm3, of such a cone is given by . The machines produce cones for which h+r=18.      i.       Show that .    ii.       Given that r can vary, find the non-zero value of r for which V […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#2

Hits: 710   Question A function f is defined by  for . It is given that f is an increasing  function. Find the largest possible value of the constant a. Solution We are given that function f is increasing function. To test whether a function  is increasing or decreasing at a particular point , we  take derivative of a function […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Oct-Nov | (P1-9709/11) | Q#1

Hits: 679   Question A curve has equation . Find the equation of the tangent to the curve at the  point (4,0). Solution To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#11

Hits: 989   Question The function  is defined for . It is given that  has a minimum value when  and that . (i)          Find . It is now given that ,  and  are the first three terms respectively of an arithmetic progression. (ii)        Find the value of . (iii)       Find , and hence find the minimum value of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#10

Hits: 1175   Question a.     Fig. 1 shows part of the curve  and the line y = h, where h is a constant.        (i)          The shaded region is rotated through 360o about the y-axis. Show that the volume of                         revolution, V, is given by […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#6

Hits: 843   Question The line 3y+x=25 is a normal to the curve y= x2 – 5x + k. Find the value of the constant k. Solution If a line  is normal to the curve , then product of their slopes  and  at that point (where line is normal to the curve) is; Therefore, by finding slopes of both the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/12) | Q#9

Hits: 780   Question The equation of a curve is  .       i.       Find the coordinates of the stationary point of the curve.    ii.       Find an expression for  and hence, or otherwise, determine the nature of the                                 stationary point.   iii.       Find the values of x […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/12) | Q#5

Hits: 733   Question A curve has equation .      i.       Find the equation of the tangent to the curve at the point where the curve crosses the x-axis.     ii.       A point moves along the curve in such a way that the x-coordinate is increasing at a constant  rate of 0.04 units per second. Find the rate of change […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#10

Hits: 595   Question The diagram shows the curve  defined for x>0. The curve has a minimum point at A and  crosses the x-axis at B and C. It is given that  and that the curve passes through the  point . i.       Find the x-coordinate of A. ii.       Find . iii.       Find the x-coordinates of B and C. iv.       Find, showing all […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#9

Hits: 704   Question The point A(2,2) lies on the curve .                     i.       Find the equation of the tangent to the curve at A. The normal to the curve at A intersects the curve again at B.                   ii.       Find the coordinates of B. The tangents at A and B intersect each other at C.                  iii.        Find the coordinates of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#7

Hits: 567   Question The function f is defined for  by .                     i.       Find  and . The first, second and third terms of a geometric progression are respectively ,   and .                   ii.       Find the value of the constant k. Solution      i.   We are given the function; Therefore; Rule for differentiation of  is: Therefore; Second derivative is the derivative […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#3

Hits: 1588   Question The diagram shows a water container in the form of an inverted pyramid, which is such that when  the height of the water level is h cm the surface of the water is a square of side  cm.      i.       Express the volume of water in the container in terms of h. [The volume of a […]