Hits: 575

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#10

Hits: 575 Question The function  is defined by  for .      i.       Find  and  and hence verify that the function  has a minimum value at . The points  and  lie on the curve , as shown in the diagram.      i.       Find the distance AB.    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   We are given; Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#9

Hits: 1376 Question A curve passes through the point A(4,6) and is such that  . A point P is moving along  the curve in such a way that the x-coordinate of P is increasing at a constant rate of 3 units per  minute.      i.       Find the rate at which the y-coordinate of P is increasing when P is at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 560 Question i.       Express  in the form , where a, b and c are constants.  ii.   The function, where , is defined for . Find  and state, with       a reason, whether  is an increasing function, a decreasing function or neither. Solution i.   We have the expression; We use method of “completing square” to obtain […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 1520 Question The diagram shows part of the curve . The point P(2,1) lies on the curve and the  normal to the curve at P intersects the x-axis at A and the y-axis at B.      i.       Show that B is the mid-point of AP. The shaded region is bounded by the curve, the y-axis and the line y […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#9

Hits: 792 Question The curve y=f(x) has a stationary point at (2,10) and it is given that      i.       Find f(x).    ii.       Find the coordinates of the other stationary point.   iii.       Find the nature of each of the stationary points. Solution      i.   We are given; We are required to find . We can find equation of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#3

Hits: 2200 Question Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are  identical isosceles triangles. Angle ABE = angle BAE = 30o. The length of AD is 40 cm. The tank is  fixed in position with the open top ABCD horizontal. Water is poured into the tank […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#11

Hits: 1627 Question The diagram shows part of the curve  and a point P(6,5) lying on the curve. The line  PQ intersects the x-axis at Q(8,0).      i.       Show that PQ is a normal to the curve.   ii.       Find, showing all necessary working, the exact volume of revolution obtained when the shaded  region is rotated through 360o about the x-axis. [In part […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#5

Hits: 668 Question A curve has equation  .      i.         Find  and     ii.        Find the coordinates of the stationary points and state, with a reason, the nature of each                   stationary point. Solution i.   We are given that; Gradient (slope) of the curve is the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/13) | Q#10

Hits: 1508 Question Points A(2,9) and B(3,0) lie on the curve y=9+6x−3×2, as shown in the diagram. The tangent at A  intersects the x-axis at C. Showing all necessary working,     i.       find the equation of the tangent AC and hence find the x-coordinate of C,    ii.       find the area of the shaded region ABC. Solution      i.   To find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/13) | Q#8

Hits: 1368 Question The function  is defined by   for x > −1.      i.       Find .    ii.       State, with a reason, whether f is an increasing function, a decreasing function or neither. The function  is defined by  for x < −1   iii.       Find the coordinates of the stationary point on the curve . Solution      i.   We are given that; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#10

Hits: 1182 Question The equation of a curve is  .      i.       Find, showing all necessary working, the volume obtained when the region bounded by the               curve, the x-axis and the lines x=1 and x=2 is rotated through 360O about the x-axis.    ii.     Given that the line  is a normal to the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/12) | Q#4

Hits: 783   Question Variables u, x and y are such that  and . Express u in terms of x and  hence find the stationary value of u. Solution We are given that; We can fin expression for y from this equation and substitute in expression of u to write u in terms of  x. Therefore; Substituting this in expression of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#10

Hits: 2622   Question The diagram shows part of the curve  . The curve intersects the y-axis at A . The  normal to the curve at A intersects the line  at the point B.      i.       Find the coordinates of B.    ii.       Show, with all necessary working, that the areas of the regions marked P and Q are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#9

Hits: 4944   Question The equation of a curve is , where  is a positive constant.      i.       Show that the origin is a stationary point on the curve and find the coordinates of the other  stationary point in terms of .    ii.          Find the nature of each of the stationary points. Another curve has equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/11) | Q#2

Hits: 2730 Question   The diagram shows the curve  and the points  and . The point Q lies on the  curve and PQ is parallel to the y-axis.   i. Express the area, A, of triangle XPQ in terms of .   The point P moves along the x-axis at a constant rate of 0.02 units per second and Q […]