# 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#9

Question The diagram shows parts of the graphs of  and  intersecting at points A and B.      i.       Write down an equation satisfied by the x-coordinates of A and B. Solve this equation and  hence find the coordinates of A and B.    ii.             Find by integration the area of the shaded region. Solution i. […]

# 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#6

Question  is the point and is the point , where  is a constant.     i.       Find, in terms of a, the gradient of a line perpendicular to .    ii.       Given that the distance  is , find the possible values of . Solution      i.   If two lines are perpendicular (normal) to each other, then product of their slopes  and  is; Since we are […]

# 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#9

Question The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is . The coordinates of A, B and C are ,  and  respectively.      i.       Find the equation of AD.    ii.       Find, by calculation, the coordinates of D. The point E is such that ABCE is a parallelogram.   iii.       Find the length […]

# 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 | Oct-Nov | (P1-9709/11) | Q#5

Question Find the set of values of  for which the line  meets 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 curve).  Therefore, […]

# 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#4

Question The line  passes through the points  and , where  and  are constants.      i.       Find the values of  and .    ii.       Find the coordinates of the mid-point of . Solution i.   Since the line  through the points  and , coordinates of both points must satisfy equation of the line. For point For point Substituting  in equation ; ii.   We are given the coordinates […]

# 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#11

Question The diagram shows a parallelogram ABCD, in which the equation of AB is  and the equation  of AD is . The diagonals AC and BD meet at the point E . Find, by calculation, the coordinates of A, B, C  and D. Solution It is evident from the diagram that point A is the intersection point of sides AD […]

# 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#10

Question The diagram shows the curve  and the line . Find, showing all necessary working, the area of the shaded region. Solution It is evident from the diagram that; First we find area under the curve. We are given equation of the curve as; We are also given equation of the line as; To find the area of region under the […]

# 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 | (P1-9709/12) | 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#1

Question Find the coordinates of the point at which the perpendicular bisector of the line  joining  to  meets the x-axis. Solution We are required to find the coordinates of the x-intercept of the perpendicular  bisector of the line joining  to . The point  at which curve (or line) intercepts x-axis, the value of . So we  can find the value […]

# 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#11

Question A line has equation  and a curve has equation .      i.       For the case where the line is a tangent to the curve, find the value of the  constant .    ii.              For the case where , find the x-coordinates of the points of intersection  of the line and the curve. Find […]

# 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#10

Question The diagram shows the function  defined for , where      i.       State the range of .    ii.       Copy the diagram and on your copy sketch the graph of .   iii.       Obtain expressions to define the function , giving also the set of values for which each expression is valid. Solution i.   To find the range of  we can substitute […]

# 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#7

Question The coordinates of points A and B are  and  respectively, where  and  are constants. The distance AB is  units and the gradient of the line AB is 2. Find the possible  values of  and of . Solution Expression for slope of a line joining points  and ; Therefore, slope of line AB with points  and ; Expression to find […]

# 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#4

Question A curve has equation  . Find the equation of the tangent to the curve at the point where the line  intersects the curve. Solution We are required to find the equation of the tangent to the curve at the point where curve  and line  intersect. To find the equation of the line either we need coordinates of […]