Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#7

Question i.   A straight line passes through the point (2, 0) and has gradient . Write down the equation of the line. ii.  Find the two values of  for which the line is a tangent to the curve . For each value of , find the coordinates of the point where the line touches the curve.   iii. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#3

Question The diagram shows the curve  and the line  intersecting at points A, O and B.     i.       Show that the x-coordinates of A and B satisfy the equation .    ii.       Solve the equation  and hence find the coordinates of A and B, giving your answers in an exact form. Solution i.   We are given that points A & B are […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows a quadrilateral ABCD in which the point A is , the point B is  and the point C is . The diagonals AC and BD intersect at M. Angle   and . Calculate     i.        the coordinates of M and D,    ii.       the ratio . Solution     i.   First we find the coordinates […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#7

Question A curve is such that   . The line  is the normal to the curve at the point  on the curve. Given that the x-coordinate of  is positive, find     i.       the coordinates of P,    ii.       the equation of the curve. Solution i.   We are given that equation of the line is; We can rearrange the equation of the line as; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#4

Question The equation of a curve is  and the equation of a line is , where  is a constant.     i.       In the case where , find the coordinates of the points of intersection of the line and the curve.    ii.       Find the value of  for which the line is a tangent to the curve. Solution     i.   For the case […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#10

Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.     i.       Find the coordinates of B and C.    ii.       Find the equation of the normal to the curve at C.   iii.       Find the volume obtained when the shaded region is rotated through  about the y-axis. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#9

Question A line has equation  and a curve has equation , where  is a constant.     i.       For the case where , the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB.    ii.       Find the two values of  for which the line is a tangent to the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#3

Question      i.       Sketch, on a single diagram, the graphs of  and  for .    ii.       Write down the number of roots of the equation  in the interval .   iii.       Deduce the number of roots of the equation  in the interval . Solution     i.        ii.   If two lines (or a line and a curve) intersect each other at a […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#9

Question A curve is such that  and P(9, 5) is a point on the curve.     i.       Find the equation of the curve.    ii.       Find the coordinates of the stationary point on the curve.   iii.       Find an expression for  and determine the nature of the stationary point.   iv.       The normal to the curve at P makes an angle of  with the positive x-axis. Find […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#3

Question The line  , where a and b are positive constants, meets the x-axis at P and the y-axis at Q. Given that   and that the gradient of the line PQ is  , find the values of a and b. Solution We are given that points P and Q are x and y intercepts, respectively. First we find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#2

Question Find the set of values of  for which the line  intersects the curve  at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#7

Question The line passes through the points  and . The line  is parallel to  and passes through the origin. The point C lies on  such that AC is perpendicular to . Find     i.       the coordinates of C,    ii.       the distance AC. Solution     i.   Consider the diagram below. It is evident that point  is the intersection of lines  and AC. If […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/11) | Q#10

Question i.       Express  in the form  and hence state the coordinates of the minimum point, A, on the curve . The line  intersects the curve  at points P and Q. It is given that the coordinates of P are (3,7).    ii.       Find the coordinates of Q.   iii.       Find the equation of the line joining Q to the mid-point of […]