Hits: 125

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#7

Hits: 125 Question The equation of a curve is      i.       Show that;    ii.       Find the x-coordinate of each stationary point of the curve in the interval . Give each answer correct to 3 significant figures. Solution      i.   We are required to show that; We are given; We utilize Quotient Rule to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 48 Question Find the x-coordinates of the stationary points of the following curves:      i.      ii.   Solution      i.   First we are required to find the x-coordinate of the stationary points of the following curve. A stationary point on the curve is the point where gradient of the curve is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/21) | Q#7

Hits: 126   Question The parametric equations of a curve are for .  The curve crosses the x-axis at points B and D and the stationary points are A and C, as shown in the diagram. i.       Show that .    ii.       Find the values of t at A and C, giving each answer correct […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/21) | Q#2

Hits: 37     Question The equation of a curve is Find the coordinates of the points on the curve at which the gradient is −4. Solution We are given equation of the curve as; We need expression for gradient of the curve. Gradient (slope) of the curve is the derivative of equation of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#7

Hits: 91 Question A.  Find the gradient of the curve At the point (1,1). B.  The parametric equations of a curve are Find the gradient of the curve at the point (−3, 3). Solution A.    We are given equation of the curve as; We are required to find expression for gradient of the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#4

Hits: 156 Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M is .    ii.       The region shaded in the diagram is enclosed by the curve and the lines ,   and  . Use integration to show that the area of the shaded region is . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#7

Hits: 129 Question A.  Find the gradient of the curve At the point (1,1). B.  The parametric equations of a curve are Find the gradient of the curve at the point (−3, 3). Solution A.    We are given equation of the curve as; We are required to find expression for gradient of the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#4

Hits: 126     Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M is .    ii.       The region shaded in the diagram is enclosed by the curve and the lines ,   and  . Use integration to show that the area of the shaded region […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#7

Hits: 457 Question The equation of a curve is y3+4xy=16      i.       Show that .    ii.       Show that the curve has no stationary points.   iii.       Find the coordinates of the point on the curve where the tangent is parallel to the y-axis. Solution      i.   We are required to find . Hence; […]

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: 576 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: 1378 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: 1629 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: 669 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 […]